Blurred quantum Darwinism across quantum reference frames
BBlurred quantum Darwinism across quantum reference frames
Thao P. Le, ∗ Piotr Mironowicz,
2, 3, 4 and Paweł Horodecki
3, 4 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT Department of Algorithms and System Modeling, Faculty of Electronics,Telecommunications and Informatics, Gdańsk University of Technology National Quantum Information Centre, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk, Poland International Centre for Theory of Quantum Technologies,University of Gdansk, Wita Stwosza 63, 80-308 Gdansk, Poland (Dated: August 11, 2020)Quantum Darwinism describes objectivity of quantum systems via their correlations with theirenvironment—information that hypothetical observers can recover by measuring the environments.However, observations are done with respect to a frame of reference. Here, we take the formalism of[Giacomini, Castro-Ruiz, & Brukner.
Nat Commun , 494 (2019)], and consider the repercussionson objectivity when changing quantum reference frames. We find that objectivity depends on non-degenerative relative separations, conditional state localisation, and environment macro-fractions.There is different objective information in different reference frames due to the interchangeability ofentanglement and coherence, and of statistical mixing and classical correlations. As such, objectivityis subjective across quantum reference frames. I. INTRODUCTION
The emergence of the classical world from theunderlying quantum mechanics remains a fundamentalriddle. Quantum Darwinism is one particular approachthat describes the emergence of objectivity throughthe spread of information [1]. A system state S is objective (or inter-subjective [2]) when manyindependent observers can determine the state of S independently, without perturbing it, and arrive at thesame result [3, 4].Quantum Darwinism can be seen as an extensionof the decoherence theory. As systems interact withtheir surrounding environments, decoherence theorydescribes how quantum systems lose their coherenceand decohere into a preferred pointer basis [5–7]. Theenvironment is not unchanged through this process—thesystem becomes correlated with the environment.Quantum Darwinism occurs if the information aboutthe system has been proliferated into multiple fractionsof the environment, such that many observers canaccess independent environments and gain equivalentinformation about the system. Objective states can bedescribed either with Żurek’s quantum Darwinism [8],strong quantum Darwinism [9], or spectrum broadcaststructure [4]. The emergence of these states have beenstudied extensively (for example, recent works includeRefs. [10–24]).A key component of quantum Darwinism is themeasurement performed by observers—which in physics,is done relative to some reference frame. However,in works thus far, one implicitly assumes that allobservers share the same classical frame. However, isthe objectivity still consistent if observers do not sharethe same frame? ∗ [email protected] While classical reference frames are well established,there are numerous different proposals for describingquantum reference frames that focus on different aspects(for example, [25–31]). In this paper, we apply theframework of Giacomini et al. [30], in which quantumreference frames are associated with a physical quantumstate and vice-versa.We examine objective states in different quantumreference frames. The method of Giacomini et al. [30] allows us to move to the reference frameassociated with any particular environment state whichin turn is associated with the hypothetical observerframe. Entanglement and coherence have becomeinterchangeable frame-dependent properties; as arestatistical mixing and classical correlations. Suchcorrelations are an intrinsic part of quantum Darwinism,hence, in general, objectivity does not remain the same indifferent quantum reference frames. However, there are certain conditions in which objectivity is consistent, andconditions in which some kind of objectivity exists. Toclearly show this, we consider static particles, such thatchanging quantum reference frames requires only changesin relative position, and we use the clear state structureafforded by spectrum broadcasting [4].We show that, if all system and environment positionsare exactly localised and randomly distributed (say, duerandom noise) then objectivity is consistent in all frames.We demonstrate that non-matching relative positionsbetween all states is key to this consistency.However, by allowing the system and environments tohave a non-zero, continuous spread, objectivity distortsand blurs when changing quantum reference frames.The internal statistical mixedness and coherences of theenvironment states now play a crucial role in distributingnew correlations. We find that the distinguishability ofthe other environment states depends on an interplayof relative distance separations and relative spreads;and how large macro-fractions of environments may be a r X i v : . [ qu a n t - ph ] A ug required to recover objectivity.Finally, we analyse scenarios with a system interactingwith environments to show how objectivity canarise dynamically, and to show how these factors—coherences, spectrum broadcasting, mixedness, and stateseparation—affect the level of objectivity in differentframes.This paper is structured as follows. In Sec. II, wedescribe the frameworks of spectrum broadcast structureand quantum reference frames. In Sec. III, wedepict some states that have consistent objectivity inall relevant quantum reference frames. In Sec. IV weexamine objective states with a Gaussian-like spread.We describe the distortion of objectivity, and investigatethe requirements for environment-state distinguishabilitythat is a necessary component of quantum Darwinism.In Sec. V, we prove the precise conditions for perfectobjectivity in all quantum reference frames. In Sec.VI, we numerical investigate a fully coherent modelinvolving a dynamic interaction between a system andtwo environments. We conclude in Sec. VII. II. PRELIMINARIESA. Spectrum broadcast structure
In quantum Darwinism, we consider a centralsystem S that interacts and becomes correlated withits surrounding environment E . Typically, only afragment, F ⊂ E of the environment is measuredand evaluated against the conditions for objectivity—asthe full pure system-environment will retain coherencesand entanglement under a global unitary evolution[11]. There are number of different frameworks thatdescribe the properties of an objective state [4, 8,9], each corresponding to slightly different strengthsof objectivity. In this paper, we are focused onspectrum broadcast structure [4], because it has a clearstate structure that allows us to explicitly calculatehow the state changes under quantum reference frametransformations. Note that from here, when we speak of“environment”, we refer to the observed environment. Definition 1. Spectrum broadcast structure (SBS) [4]. A system-environment has spectrumbroadcast structure when the joint state can be writtenas ρ SE = (cid:88) i p i | i (cid:105)(cid:104) i | S ⊗ ρ E | i ⊗ · · · ⊗ ρ E N | i , (1)where {| i (cid:105) S } is the pointer basis, p i are probabilities,and all states ρ E k | i are perfectly distinguishable, i.e. Tr (cid:0) ρ E k | i ρ E k | j (cid:1) = 0 ∀ i (cid:54) = j, for each observed environment E k .These states have zero discord [32] between thesystem and environments, feature maximal classicalcorrelations between the system and environments, andimplicitly satisfy strong independence (in which theenvironments do not interact amongst themselves, seeDefinition 2 further in the text) [9]. All states withspectrum broadcast structure are objective, though notall objective states have spectrum broadcast structure[4, 9]. B. Quantum reference frames
As we noted, there is a number of different prescrip-tions for reference frames and quantum information (e.g.Refs. [25–28]). In this paper, we apply the framework ofGiacomini et al. [30], which is inherently relational.We consider the system and environments to be static( i.e. without momentum) and distributed across space.Thus, a general reference frame transformation, ˆ S ( C → A ) position ,is defined here as position only, as follows [30]: ˆ S ( C → A ) position (cid:90) dx A dx B Ψ( x A , x B ) | x A (cid:105) A | x B (cid:105) B = (cid:90) dq B dq C Ψ( − q C , q B − q C ) | q B (cid:105) B | q C (cid:105) C , (2) i.e. there is a coordinate transformation, x A → − q C , x B → q B − q C . We will always start in an implicitlaboratory reference frame ( C ) , and move to the quantumreference frames centered on a particular quantum state.For our purposes, SBS is inherently mixed. Hence,if the initial state ρ ( C ) SE ··· E N in the ( C ) reference frame(laboratory frame) is ρ ( C ) SE ··· E N = (cid:90) dx S dx (cid:48) S (cid:32) N (cid:89) i =1 (cid:90) dx E i dx (cid:48) E i (cid:33) ρ (cid:0) x S , x E , . . . , x E N , x (cid:48) S , x (cid:48) E , . . . , x (cid:48) E N (cid:1) | x S (cid:105)(cid:104) x (cid:48) S | S ⊗ N (cid:79) j =1 | x E j (cid:105)(cid:104) x (cid:48) E j | E j , (3)then the transformation to the environment E reference frame (without loss of generality) is: ρ ( E ) SCE ··· E N = (cid:90) dq S dq (cid:48) S (cid:90) dq C dq (cid:48) C (cid:32) N (cid:89) i =2 (cid:90) dq E i dq (cid:48) E i (cid:33) | q S (cid:105)(cid:104) q (cid:48) S | S ⊗ | q C (cid:105)(cid:104) q (cid:48) C | C ⊗ N (cid:79) j =2 | q E j (cid:105)(cid:104) q (cid:48) E j | E j × ρ (cid:0) q S − q C , − q C , q E − q C , . . . , q E N − q C , q (cid:48) S − q (cid:48) C , − q (cid:48) C , q (cid:48) E − q (cid:48) C , . . . , q (cid:48) E N − q (cid:48) C (cid:1) . (4)Entanglement and coherences in the position basisare quantum reference frame dependent [30]. Fur-thermore, statistical (incoherent) mixtures and classicalcorrelations are also frame dependent. Given thatobjectivity is built up from correlations between systemand environment, and given that the environment statescan contain coherences and statistical mixture, changingreference frames can have a serious effect on theobjectivity of the system. III. PERFECT LOCALISATION ANDOBJECTIVITY IN ALL QUANTUM REFERENCEFRAMES
We consider a system S and collection of environments { E i } i =1 ,...,N , such that they are objective in thelaboratory frame C , and in particular have spectrumbroadcast structure. The system and environments arelocated in a one-dimensional, continuous space, withpositions x X , X = S, E i . In the idealised situation, thesepositions are perfectly localised, i.e. existing at isolatedpoints in space, and this allows us to gain insight intoone of factors that contribute to consistent objectivity inall quantum reference frames— non-degenerative relativepositions. We begin with section III A, where we first examine thesimplest, illustrative situation where the objective stateshave GHZ-like structure. In section III B, we considergeneral perfectly localised objective SBS states.
A. GHZ-like objective states
One of the simplest objective states possible is thereduced Greenberger–Horne–Zeilinger state (GHZ state);it is simpler yet again if its elements are incoherent in theposition basis as follows: ρ ( C ) SE ··· E N = (cid:88) i p i | x Si (cid:105)(cid:104) x Si | S ⊗ N (cid:79) j =1 | x E j i (cid:105)(cid:104) x E j i | E j , (5)which is objective provided that all (cid:8) x Si (cid:9) i , { x E i } i aredistinct. The implicit laboratory reference frame ( C ) is perfectly localised and product with the system andenvironments. The objective information is characterisedby the probability distribution { p i } i .In the frame of any of the environments—we take E without loss of generality—the joint state now involvesthe laboratory C as one of its subsystems, and now E is implicit, perfectly localised and product with all othersubsystems: ρ ( E ) SCE ··· E N = (cid:88) i p i | x Si − x E i (cid:105)(cid:104) x Si − x E i | S ⊗ |− x E i (cid:105)(cid:104)− x E i | C ⊗ N (cid:79) j =2 | x E j i − x E i (cid:105)(cid:104) x E j i − x E i | E j . (6) In order for this to still be objective, and with thesame information as in the laboratory frame C , { p i } i ,we require that all { x Si − x E i } i are distinct, and all { x E j i − x E i } i are distinct—that is, these terms are non-matching or non-degenerate .The majority of states of the form Eq. (5) remainconsistently objective in all quantum reference frames,in the following sense: If all the various positions { x Si } i , { x E i } i , etc. are randomly chosen from a continuousinterval, for example with probability mass function f uni ( x ) = 1 , x ∈ [0 , , then the probability that anytwo are equal is zero: P ( x i = x j ) = 0 , due to thenature of discrete sampled numbers from uncountablyinfinite interval. Hence, any randomly drawn (cid:8) x Xi (cid:9) i,X , X = S, E , . . . , E N will produce an objective state forEq. (5). By the same argument, the probability thatany relative separations { x Si − x E i } i are equal is zero: P ( x Si − x E i = x Sj − x E j ) = 0 , and hence all the terms inthe system-environment state in any quantum referenceframe, Eq. (6), are distinct and hence remains objectivewith the same spectrum probabilities { p i } i .Randomly sampled positions of the system andenvironment describe disorganised and noisy scenariosand models. However, solid state materials andlattices can have a rigid structure and hence potentiallydegenerate distances between state positions. In thesesituations, SBS and objectivity may become trivial incertain quantum reference frames. Example 1.
Consider the typical reduced GHZ state,where x i = i for i = 0 , : ρ ( C ) SE ··· E N = p | (cid:105)(cid:104) | ⊗ N +1 + p | (cid:105)(cid:104) | ⊗ N +1 . (7)In the quantum reference frame of environment E , thestate has the form ρ ( E ) SCE ··· E N = | · · · (cid:105)(cid:104) · · · | SE ··· E N ⊗ ( p | (cid:105)(cid:104) | C + p |− (cid:105)(cid:104)− | C ) . (8)The system and the remaining environments are trivially“objective” and uncorrelated. Meanwhile, the oldinformation about the system has been shifted into thequantum system of the laboratory reference frame C . Observation 1.
If all positions are perfectly localised,the non-degeneracy of the relative positions of the systemand environments is crucial in ensuring the consistentobjectivity in all quantum reference frames. If some ofthe relative positions x i , . . . are not distinct, then theybecome non-distinguishable and thus degrade the originalobjectivity.In Appendix A, we consider GHZ-like states withcontinuous objective probabilities, leading to ananalogous requirement of non-degeneracy (in particular,continued injectivity of the functions mapping thecontinuous positions of the system and environment).If instead the various positions { x Xi } i,X are pickeduniformly from a finite set of N positions, then theprobability of two being the same is P ( x i = x j ) = 1 /N .This goes to zero as N → ∞ . This situation cancorrespond to the case when there is a finite precisionof a measurement device, and where any spread in thepositions is much smaller than the device precision. InSec. IV, we will consider when there is an inherentspread in the position, and in Sec. VI, the positionsof the system and environment are limit to a finiteset. But firstly, in the following subsection, we considergeneral coherent—albeit still localised—objective stateswith spectrum broadcast structure. B. Perfectly localised spectrum broadcast statesand new objectivity
States with the SBS form typically contain coherencesand mixtures in the conditional environment states.Under transformations of quantum reference frames,these can turn into global correlations. Combined withperfect localisation, we show how this produces a new,more complex objective information in different frames.In general, a perfectly localised objective state withthe SBS can be written as ρ ( C ) SE ··· E N = (cid:88) i p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 ρ E j | i , (9)where we have general coherent states: | ψ Si (cid:105) = (cid:88) k q k,i | x Sk | i (cid:105) S , (10) ρ E j | i = (cid:88) k j t k j ,i | ϕ E j i,k j (cid:105)(cid:104) ϕ E j i,k j | E j , (11) | ϕ E j i,k j (cid:105) = (cid:88) a ij r a ij ,i,j,k j | x E j a ij ,k j | i (cid:105) E j . (12)Objectivity requires that these states are orthogonal: (cid:104) ψ Si | ψ Si (cid:48) (cid:105) = 0 ∀ i (cid:54) = i (cid:48) and (cid:104) ϕ E j i,k j | ϕ E j i (cid:48) ,k (cid:48) j (cid:105) = 0 ∀ ( i, k j ) (cid:54) = (cid:0) i (cid:48) , k (cid:48) j (cid:1) . It is sufficient (though not necessary) if we let allthe values (cid:8) x k | i (cid:9) i,k , { x E j a ij ,k j | i } i,k j ,a ij be randomly chosennumbers from a continuous interval, in which case theprobability that any are equal is zero, hence all termsare orthogonal.In the frame of environment E , the joint state has thefollowing form: ρ ( E ) SCE ··· E N = (cid:88) i,k p i t k ,i (cid:88) a i ,a (cid:48) i r a i ,i, ,k r ∗ a (cid:48) i ,i, ,k | ˜ ψ Si,k ,a i (cid:105)(cid:104) ˜ ψ Si,k ,a (cid:48) i | S ⊗ |− x E a i ,k | i (cid:105)(cid:104)− x E a (cid:48) i ,k | i | C ⊗ N (cid:79) j =2 (cid:88) k j t k j ,i | ˜ ϕ E j i,k j ,a i (cid:105)(cid:104) ˜ ϕ E j i,k j ,a (cid:48) i | E j , (13) where | ˜ ψ Si,k ,a i (cid:105) = (cid:88) k q k,i | x Sk | i − x E a i ,k | i (cid:105) S (14) | ˜ ϕ E j i,k j ,a i (cid:105) = (cid:88) a ij r a ij ,i,j,k j | x E j a ij ,k j | i − x E a i ,k | i (cid:105) . (15)Due to the coherences and statistical mixedness ofthe original environment E state, there is nowentanglement and correlations between the system andthe environment in the ( E ) frame. In particular,much of the entanglement is tied with the laboratorysubsystem C —and to the indices a i and a (cid:48) i that camefrom the original E state. Hence if the positions (cid:110) x E a i ,k | i (cid:111) i,k ,a i are distinct, then we can trace out C and remove the system-environment entanglement: ρ ( E ) SE ··· E N = (cid:88) i,k ,a i p i t k ,i | r a i ,i, ,k | × | ˜ ψ Si,k ,a i (cid:105)(cid:104) ˜ ψ Si,k ,a i | S ⊗ N (cid:79) j =2 ˜ ρ E j | i,k ,a i , (16) ˜ ρ E j | i,k ,a i := (cid:88) k j t k j ,i | ˜ ϕ E j i,k j ,a i (cid:105)(cid:104) ˜ ϕ E j i,k j ,a i | E j . (17)From the assumption that all the { x ··· } ··· arerandomly sampled from a continuous distribution,all the relative differences { x Sk | i − x E a i ,k | i } k,i,k , (cid:110) x E j a ij ,k j | i − x E a i ,k | i (cid:111) i,j,k j ,k are unique, hence theconditional states of the system and the environmentsare perfectly distinguishable, and the reduced state ρ ( E ) SE ··· E N has the SBS. However, the objectiveinformation is now encoded by the probabilities (cid:110) p i t k ,i | r a i ,i, ,k | (cid:111) i,k ,a i . Although the original systeminformation can still be recovered by taking the relevantmarginal distribution, we see that in each differentreference frame corresponding to environment E j , we willhave a different set of objective information. Observation 2.
Coherences in the environment cancreate entanglement between the system, lab, andremainder environments. This can typically be“removed” by tracing out the laboratory subsystem.
Observation 3.
Incoherent mixedness in the envi-ronment creates new classical correlations between thesystem, lab, and remainder environments. This can leadto new objective information, which includes the originalinformation which can be recovered from the marginalsby summing over terms associated with the environment.Hence, while entanglement and coherence are frame-dependent properties, it is equally relevant thatincoherent mixedness and classical correlations are alsoframe-dependent. Only a very small class of objectivestates retain the same objectivity in different quantumreference frames: and more generally, the systemobjectivity transforms to a more complicated objectivity,of which the original system information is embeddedwithin.
IV. CONTINUOUS SPREAD AND BLURREDOBJECTIVITY
Thus far, we have shown how non-degeneracy ofrelative positions plays a crucial role in objectivity,when positions are perfectly localised. However, ingeneral, systems and environments have a non-zerospread. In this section, we examine systems andenvironments with a continuous spread described byGaussian distributions across space, characterised bymean µ and standard deviation σ . Objectivity becomesblurred and distinguishability reduces as states become“smeared” across space in different reference frames.In Section IV A, we describe the error probabilityof distinguishing conditional states, and how that isbounded by the fidelity. This fidelity becomes ourmeasure for a perceived objectivity. In Section IV B,we consider incoherent objective states, in which theconditional environment states are single Gaussians forsimplicity, and in Section IV C, we consider generalcoherent objective states. A. Effective perceived objectivity and the fidelityof measurement
One method to quantify compliance with the SBS iswith a distance measure to the set of the SBS states. Forexample, some of us [2] have developed a computabletight bound η [ ρ SF ] on the trace distance (where F denotes a subset of environment states): T SBS ( ρ SF ) = 12 min ρ SBS SF (cid:13)(cid:13) ρ SF − ρ SBS SF (cid:13)(cid:13) ≤ η [ ρ SF ] (18) η [ ρ SF ] ≡ Γ( t ) + (cid:88) i (cid:54) = j √ p i p j F (cid:88) k =1 B (cid:0) ρ E k | i , ρ E k | j (cid:1) , (19)where Γ( t ) is the decoherence factor, and B ( ρ i , ρ j ) = (cid:13)(cid:13) √ ρ i √ ρ j (cid:13)(cid:13) (20)is the fidelity describing the distinguishability of theconditional environment states and p ii are probabilitiesof a decohered system ( i.e. the system can be writtenas ρ S = (cid:80) i p i | i (cid:105)(cid:104) i | + (cid:80) i (cid:54) = j p ij | i (cid:105)(cid:104) j | , and the { p ij } termsare encoded in the decoherence factor Γ( t ) ). The abovebound, however, implicitly assumes strong independenceof the environments: Definition 2. Strong independence [4]. Sub-environments { E k } k have strong independence relative to the system S if their conditional mutual informationis vanishing: I ( E j : E k | S ) = 0 , ∀ j (cid:54) = k. (21)Unlike the work in [2], strong independence is not maintained in general when changing quantum referenceframes. However, strong independence is not required fora more general objectivity [9].Here, we focus on the distinguishability of theconditional states. For an ensemble { p i , ρ i } i , and a setof measurement operators { Π i } i , (cid:80) i Π i = , to pick out i , the average probability of successful measurement is: P ( success ) = (cid:88) i p i Tr[ ρ i Π i ] , (22)and the average probability of failure is then P ( error ) =1 − P ( success ) . The minimum error of distinguishing thestates is bounded by the fidelity of the conditional states[33, 34]: (cid:88) i Consider the following incoherent objective state withthe SBS, ρ ( C ) SE ··· E N = (cid:88) i p i | x Si (cid:105)(cid:104) x Si | S ⊗ N (cid:79) j =1 ρ E j | i , (25)where the environment states are unmixed (in the senseof consisting of a single Gaussian-distributed state ratherthan a discrete sum of Gaussians): ρ E j | i = (cid:90) dx E j f (cid:0) x E j | µ E j | i , σ E j | i (cid:1) | x E j (cid:105)(cid:104) x E j | . (26)We have defined the Gaussian (normal) probabilitydensity f ( x | µ, σ ) = 1 √ π σ exp (cid:34) − (cid:18) x − µσ (cid:19) (cid:35) . (27) - - - - - 20 0 20 400.00.10.20.30.4 Figure 1. (Color on-line) Top: if the peaks for the statesare separated much further than their standard deviations(here, ∆ µ = 10 σ ), then there is very little overlap andthese states are distinguishable. Bottom: Alternatively, ifthe central peaks are the same or very close, varying greatlystandard deviations (here, five times or more) allows for gooddistinguishability, as measurement at further locations will,with high probability, correspond to the wider distributions. This allows us to focus on the effects of the Gaussianspread on the objectivity. From the very beginning,there is no perfect objectivity: the fidelity between twoconditional environment states for i, i (cid:48) is: (cid:13)(cid:13) √ ρ E j | i √ ρ E j | i (cid:48) (cid:13)(cid:13) = exp − (cid:0) µ E j | i − µ E j | i (cid:48) (cid:1) (cid:16) σ E j | i + σ E j | i (cid:48) (cid:17) (cid:113) σ E j | i + σ E j | i (cid:48) (cid:112) σ E j | i σ E j | i (cid:48) , (28)which is always non-zero. As we impose that our originalstate in the laboratory frame is objective, this fidelitymust be sufficiently small for all i (cid:54) = i (cid:48) . Hence, for anypair of i (cid:54) = i (cid:48) , we must either have µ E j | i − µ E j | i (cid:48) (cid:29) (cid:113) σ E j | i + σ E j | i (cid:48) , i.e. the peak separations are larger thanthe standard deviation, or σ E j | i (cid:29) σ E j | i (cid:48) (or vice-versa), i.e. one conditional state must have a larger spreadthan the others—this allows for the detection of thewide-spread distribution outside the bulk to the sharperdistribution. These two cases are depicted in Fig. 1. Observation 4. Objectivity requires the distinguisha-bility of conditional states. If the conditional statesare described with a Gaussian distribution, then thedistinguishability requires a combination of sufficientlyfar separated peaks { µ } , or otherwise sufficiently differentspreads { σ } .Without loss of generality, we change to the quantum Figure 2. (Color on-line) In the frame of ( E ) , the originalpeaks of environment E j , at µ E j | i are shifted by q C , whichranges over the entire space, but with a Gaussian envelopecentered at − µ E | i . Every curve corresponds to a differentenvironment state conditioned on a different q C (for a fixed i ). Different q C give curves that overlap a lot, and hence arenot distinguishable. reference frame of the first environment E : ρ ( E ) SCE ··· E N = (cid:88) i p i (cid:90) dq C f (cid:0) − q C | µ E | i , σ E | i (cid:1) × | x Si + q C (cid:105)(cid:104) x Si + q C | S ⊗ | q C (cid:105)(cid:104) q C |⊗ N (cid:79) j =2 (cid:90) dq E j f (cid:0) q E j − q C | µ E j | i , σ E j | i (cid:1) | q E j (cid:105)(cid:104) q E j | . (29)In the new frame, the system is centered around x Si − µ E | i , the old laboratory C is centered around − µ E | i , and the other environments have a complexdistribution with a continuum of multiple peaks, at q C + µ E j | i , where q C is centered around − µ E | i . Whilethe original system-objective information still exists,there are now extra classical correlations given acrossby (cid:82) dq C . This continuum across q C means that we do not have objectivity for the continuous distribution (cid:8) p i f (cid:0) − q C | µ E | i , σ E | i (cid:1)(cid:9) i,q C , as the states given by q C versus q C + δ are not well distinguished. This is depictedin Fig. 2Thus, the most immediate, and preferred, candidatefor objectivity is the original information indexed by i . Firstly, the new conditional system states must bedistinguishable. The local system state is ρ ( E ) S = (cid:80) i p i ρ ( E ) S | i , where the conditional states are: ρ ( E ) S | i := (cid:90) dq C f (cid:0) − q C | µ E | i , σ E | i (cid:1) | x Si + q C (cid:105)(cid:104) x Si + q C | S . (30)The fidelity of conditional system states is (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) S | i (cid:113) ρ ( E ) S | i (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) = exp − (cid:0) x Si − µ E | i − x Si (cid:48) + µ E | i (cid:48) (cid:1) (cid:16) σ E | i + σ E | i (cid:48) (cid:17) (cid:113) σ E | i + σ E | i (cid:48) (cid:46)(cid:112) σ E | i σ E | i (cid:48) . (31)Distinguishability requires a low fidelity, which occurseither if the shifted distances are non-degenerate with asufficiently large separation, or if one of σ E | i (cid:29) σ E | i (cid:48) . The reduced state on environment E j is ρ ( E ) E j = (cid:80) i p i ρ ( E ) E j | i , with conditional states ρ ( E ) E j | i := (cid:90) dq C (cid:90) dq E j f (cid:0) − q C | µ E | i , σ E | i (cid:1) × f (cid:0) q E j − q C | µ E j | i , σ E j | i (cid:1) | q E j (cid:105)(cid:104) q E j | . (32)The fidelity of the conditional states is: (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E j | i (cid:113) ρ ( E ) E j | i (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) = √ (cid:104)(cid:16) σ E | i + σ E | i (cid:48) (cid:17)(cid:16) σ E j | i + σ E j | i (cid:48) (cid:17)(cid:105) / (cid:113) σ E | i + σ E | i (cid:48) + σ E j | i + σ E j | i (cid:48) exp − (cid:0) µ E | i − µ E | i (cid:48) − µ E j | i + µ E j | i (cid:48) (cid:1) (cid:16) σ E | i + σ E | i (cid:48) + σ E j | i + σ E j | i (cid:48) (cid:17) . (33)Once again, distinguishability requires low fidelity,which occurs if the shifted differences are verynon-degenerate: µ E | i − µ E | i (cid:48) − µ E j | i + µ E j | i (cid:48) (cid:29) (cid:113) σ E | i + σ E | i (cid:48) + σ E j | i + σ E j | i (cid:48) , or if at least one ofthe standard deviations σ ∈ (cid:8) σ E | i , σ E | i (cid:48) , σ E j | i σ E j | i (cid:48) (cid:9) isseparated from the others by orders of magnitude, so that (cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E j | i (cid:113) ρ ( E ) E j | i (cid:48) (cid:13)(cid:13)(cid:13) ∼ / √ σ → for σ → ∞ . Observation 5. For objectivity of the originalinformation, a necessary condition is a good localdistinguishability (local perceived objectivity). Thisrequires a combination of very non-degenerate relativeseparations and very localised conditional states; orconditional spreads that vary by orders of magnitude.Suppose the conditional fidelity (cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E j | i (cid:113) ρ ( E ) E j | i (cid:48) (cid:13)(cid:13)(cid:13) is not close to zero. In this case, we can take macro-fractions in order to increase distinguishability. Supposewe have a fraction F = { E j } j ∈ F . Then, the conditionalfidelity is (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) F | i (cid:113) ρ ( E ) F | i (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:89) j ∈ F (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E j | i (cid:113) ρ ( E ) E j | i (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) . (34)Provided (cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E j | i (cid:113) ρ ( E ) E j | i (cid:48) (cid:13)(cid:13)(cid:13) < , which is true providedthat there is non-degeneracy in the relative positions, µ E | i − µ E | i (cid:48) − µ E j | i + µ E j | i (cid:48) (cid:54) = 0 , then the product ofincreasingly many of them takes (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) F | i (cid:113) ρ ( E ) F | i (cid:48) (cid:13)(cid:13)(cid:13)(cid:13) → . Observation 6. Information becomes less distin-guishable in different frames. Provided that thereis non-degeneracy in the relative peak-positions,distinguishability can be achieved by taking a suitablylarge collection of sub-environments (macrofractions).In Fig. 3, we demonstrate the interplay betweenlocalisation and macrofraction size and their contributionto the distinguishability of two conditional states.In general, the conditional environment states can bemixed, e.g. ρ E | i = (cid:80) k q k ρ E | i,k from Eq. (25) can be a Figure 3. (Color on-line) Plot of the conditional state fidelity (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) F | (cid:113) ρ ( E ) F | (cid:13)(cid:13)(cid:13)(cid:13) , versus the amount of localisation ( σ thesame for all Gaussian states) and macrofraction size | F | , forthe case of the state in Eq. (25). Here, the peak positions (cid:8) µ E i | , µ E i | (cid:9) i are picked randomly from the interval [ − , and the graph is averaged over collections of randomsamples. Sharp localisation σ → , and large macrofractions | F | lead to low conditional state fidelity and hence greaterdistinguishability. mixture of distinguishable Gaussian states. As previouslydetermined in Section III, this leads to a new objectiveinformation given by the distribution { p i q k } i,k , where theoriginal information is recovered through the marginalobtained by summing over all values of q k . C. Coherent objectivity states and the rise of newclassical and quantum correlations In general, objective states have coherence. Whenmoving to the reference frame of one of thoseenvironments, this coherence turns into entanglementbetween the other subsystems. Consider the followingstate, in which the system and environments are coherentrelative to the position basis: ρ ( C ) SE ··· E N = (cid:88) i p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 ρ E j | i , (35)where all the pure states are Gaussian wave-packets: | ψ Si (cid:105) = (cid:90) dx S f (cid:0) x S | µ S | i , σ S | i (cid:1) | x S (cid:105) S (36) ρ E j | i = (cid:88) k j t k j ,i | ϕ E j i,k j (cid:105)(cid:104) ϕ E j i,k j | E j (37) | ϕ E j i,k j (cid:105) = (cid:90) dx E j f (cid:0) x E j | µ E j | i,k j , σ E j | i,k j (cid:1) | x E j (cid:105) E j . (38)Note that f ( · ) = (cid:112) f ( · ) is the square-root of a Gaussian(which may include a potential phase). In the referenceframe of environment E , ρ ( E ) SCE ··· E N = (cid:88) i,k p i t k ,i (cid:90) dq S dq (cid:48) S dq C dq (cid:48) C f (cid:0) q S − q C | µ S | i , σ S | i (cid:1) × f ∗ (cid:0) q (cid:48) S − q (cid:48) C | µ S | i , σ S | i (cid:1) f (cid:0) − q C | µ E | i,k , σ E | i,k (cid:1) × f ∗ (cid:0) − q (cid:48) C | µ E | i,k , σ E | i,k (cid:1) | q S (cid:105)(cid:104) q (cid:48) S | S ⊗ | q C (cid:105)(cid:104) q (cid:48) C | C ⊗ N (cid:79) j =2 (cid:88) k j t k j ,i (cid:90) dq E j dq (cid:48) E j f (cid:0) q E j − q C | µ E j | i,k j , σ E j | i,k j (cid:1) f ∗ (cid:16) q (cid:48) E j − q (cid:48) C | µ E j | i,k j , σ E j | i,k j (cid:17) | q E j (cid:105)(cid:104) q (cid:48) E j | E j . (39)Coherence in the environment states (relative to theposition basis in which we change reference frames) leadsto an entanglement between the laboratory frame stateand the system-environments. This entanglement can beremoved by tracing out the laboratory state. The smallchanges in q C will not be distinguishable. Instead, thebest candidate for the perceived objective informationis { p i t k ,i } i,k — i.e. , the original objectivity informationmixed with the E incoherent statistical mixedness thathas now turned into classical correlations in the newframe as we have seen with previous examples.The local system state is ρ ( E ) S = (cid:80) i,k p i t k ,i ρ ( E ) S | i,k ,where the conditional states are: ρ ( E ) S | i,k := (cid:90) dq C f (cid:0) − q C | µ E | i,k , σ E | i,k (cid:1)(cid:20)(cid:90) dq S f (cid:0) q S − q C | µ S | i , σ S | i (cid:1) | q S (cid:105) (cid:21)(cid:20)(cid:90) dq (cid:48) S f ∗ (cid:0) q (cid:48) S − q C | µ S | i , σ S | i (cid:1) (cid:104) q (cid:48) S | S (cid:21) . (40)The system is conditionally centered around µ S | i − µ E | i,k , with a spread of approximately (cid:113) σ S | i + σ E | i,k . Observation 7. In other reference frames, theconditional system states is typically no longer pure, butthey can still be distinguishable. We can consider this a generalised objectivity , in which the conditional systemstates are mixed (instead of conditionally pure) andperfectly distinguishable in the manner the environmentstates are.Heuristically, provided that these new peaks aresufficiently separated, or that different standarddeviations separated by orders of magnitude, then theconditional states will be distinguishable. Since wecannot calculate the eigendecomposition for ρ ( E ) S | i,k ingeneral, we will instead calculate the overlap/linearfidelity, which is a lower bound to the fidelity: Tr (cid:104) ρ ( E ) S | i,k ρ ( E ) S | i (cid:48) ,k (cid:48) (cid:105) = 2 σ S | i σ S | i (cid:48) exp − (cid:0) µ E | i,k − µ S | i − µ E | i (cid:48) ,k (cid:48) + µ S | i (cid:48) (cid:1) (cid:16) σ E | i,k + σ E | i (cid:48) ,k (cid:48) + σ S | i + σ S | i (cid:48) (cid:17) (cid:114)(cid:16) σ S | i + σ S | i (cid:48) (cid:17)(cid:16) σ E | i,k + σ E | i (cid:48) ,k (cid:48) + σ S | i + σ S | i (cid:48) (cid:17) . (41)The linear fidelity is small when the relative differencesare greater than the standard deviations, or if σ E | i,k are large compared to σ S | i .Similarly, the environment states, ρ ( E ) E j = (cid:80) i,k p i t k ,i ρ ( E ) E j | i,k , have conditional states ρ ( E ) E j | i,k := (cid:88) k j t k j ,i (cid:90) dq C f (cid:0) − q C | µ E | i,k , σ E | i,k (cid:1)(cid:20)(cid:90) dq E j f (cid:0) q E j − q C | µ E j | i,k j , σ E j | i,k j (cid:1) | q E j (cid:105) (cid:21)(cid:20)(cid:90) dq (cid:48) E j f ∗ (cid:16) q (cid:48) E j − q (cid:48) C | µ E j | i,k j , σ E j | i,k j (cid:17) (cid:104) q (cid:48) E j | E j (cid:21) . (42)We could calculate their linear fidelity (not shownhere): provided they are separated in position, or iftheir standard deviations are very different, then theconditional environment states will be distinguishable.When the environment states have coherence, the fullsystem-environment state gains entanglement in otherquantum reference frames. However, this entanglementcan be decohered into classical correlations by tracing outthe (transformed) laboratory system. Distinguishabilityrequires the locations of the new peaks in the newreference frame to be sufficiently separated, or thatthe size of the spreads in the new reference framebe sufficiently different, and can be enhanced withmacrofractions. Once distinguishable, the information { p i t k ,i } i,k can be recovered from the environments, andin turn the original system information. However, the t k ,i component is unique to the ( E ) frame. Observation 8. The system information { p i } i from thelaboratory frame is unique, in that it is recoverable in allframes.Note though this is not the same as saying that { p i } i is objective in all frames, as all the previous and followingexamples have shown. Observation 9. All the information in the system-environment remains when changing reference frames.However, this information can become scrambled andprevent the system information { p i } i from being theonly objective information in the new frames. Instead,the internal information of the new environment frame( i.e. mixedness and coherence in the conditional states)produces new correlations that augment the originalobjective system information. Thus, to keep the exactsame objective information, there should be as littleinternal conditional information in the environment aspossible. V. PRECISE CONDITIONS FOR PERFECTOBJECTIVITY IN ALL QUANTUM REFERENCEFRAMES In general, a discrete SBS state (i.e. containingcountably many terms) can be written as follows: ρ ( C ) SE ··· E N = (cid:88) i p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 ρ E j | i , (43) (cid:104) ψ Si | ψ Si (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) (44) ρ E j | i ρ E j | i (cid:48) = 0 , ∀ i (cid:54) = i (cid:48) , ∀ j, (45)where we have general coherent states on the systemand mixed states on the environment that are perfectlydistinguishable under different index i . We can write | ψ Si (cid:105) and ρ E j | i in general as: | ψ Si (cid:105) = (cid:88) x S ψ (cid:16) x S (cid:12)(cid:12)(cid:12) i (cid:17) | x S (cid:105) S , (46) ρ E j | i = (cid:88) x Ej ,x (cid:48) Ej t (cid:16) x E j , x (cid:48) E j (cid:12)(cid:12)(cid:12) i, j (cid:17) | x E j (cid:105)(cid:104) x (cid:48) E j | E j . (47)The objective information here is { p i } i . However, asthe cases above show, SBS states do not always remainSBS in different frames, and if they do, they will oftenhave a different objective information. In the followingtheorem, we give the particular SBS structure requiredfor the same objective information in all relevant frames: Theorem 1. A discrete SBS state ρ ( C ) SE E ··· E N (Eq. (43))is perfectly objective, with the same objective information { p i } i , in all lab and environment reference frames if andonly if it can be written in the following reduced form: ρ ( C ) SE ··· E N = (cid:88) i p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 | x E j | i (cid:105)(cid:104) x E j | i | E j , (48) - - - - - - Figure 4. Top: Curves representing the system conditionalstates in the original lab frame: they are separated andhence distinguishable. Bottom: Both curves are shifted bya different amount as we move to an environment frame, yetthe curves overlap and are no longer distinguishable. and satisfying the perfect distinguishability conditions inthe original lab frame: (cid:104) ψ Si | ψ Si (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , (49) (cid:104) x E j | i | x E j | i (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , ∀ j, (50) and all environment frames: (cid:104) ˜ ψ Si,j | ˜ ψ Si (cid:48) ,j (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , ∀ j (51) (cid:104) x E j | i − x E k | i | x E j | i (cid:48) − x E k | i (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , ∀ j (cid:54) = k, (52) where | ˜ ψ Si,j (cid:105) = (cid:16)(cid:80) q S ψ (cid:0) q S + x E j | i | i (cid:1) | q S (cid:105) S (cid:17) . The proof is given in Appendix B 1: it proceeds byconsidering the general transformed state of the system-environment in frame E (without loss of generality) andimposes that the system spectrum remains { p i } i (whichenforces the environment states ρ E j | i conditioned on i be pure) and that SBS is preserved (which gives thedistinguishability conditions).Thus, not only are the conditional environmentstates pure, they must also non-degenerate separations[Eq. (52)]. This can be easily achieved by introducingrandomness to the precise (cid:8) x E j | i (cid:9) terms. An example ofperfect objective states is given in Sec. III A.However, the orthogonality conditions Eq. (51) forthe system states (cid:110) | ˜ ψ Si,j (cid:105) (cid:111) i in frame E j are muchmore nontrivial. It is possible that the shifts in thewavefunction from ψ ( x S | i ) → ψ (cid:0) x S + x E j | i | i (cid:1) can causeoverlaps in the conditional system states in the newframe. We depict this in Fig. 4.Suppose we start of with a general continuous SBS0 Table I. Summary of the minimal specialised SBSstate structure required for perfect objectivity in otherquantum reference frames (QRFs), aside from detaileddistinguishability conditions.Objectivity type State structure requirementObjective in all QRFs, withthe same classicalinformation { p i } All environment conditionalstates are pure in x basisand localised (Thm. 1, Sec.III A)Objective in all QRFs, butwith different objectiveinformation All environment conditionalstates are incoherent andmixed in x basis (Prop. 1,Sec. III B, Sec. IV B)A reduced state isobjective in all QRFs, withdifferent objectiveinformation Environment conditionalstates can be coherent (Cor.2, Sec. IV C) state instead of a discrete one: ρ ( C ) SE ··· E N = (cid:90) di · p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 ρ E j | i , (53) | ψ Si (cid:105) = (cid:90) dx S · ψ (cid:16) x S (cid:12)(cid:12)(cid:12) i (cid:17) | x S (cid:105) S , (54) ρ E j | i = (cid:90) dx E j dx (cid:48) E j · t (cid:16) x E j , x (cid:48) E j (cid:12)(cid:12)(cid:12) i, j (cid:17) | x E j (cid:105)(cid:104) x (cid:48) E j | E j , (55)satisfying distinguishability conditions: (cid:104) ψ Si | ψ Si (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , (56) ρ E j | i ρ E j | i (cid:48) = 0 , ∀ i (cid:54) = i (cid:48) , ∀ j. (57) Corollary 1. A continuous SBS state is perfectlyobjective, with the same (possibly continuous) objectiveinformation { p i } i , in all lab and environment referenceframes if and only if it satisfies the same state structureas given in Theorem 1 (up to a continuous i ), that is,with form ρ ( C ) SE ··· E N = (cid:90) di · p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 | x E j | i (cid:105)(cid:104) x E j | i | E j , (58) and all other conditions given in Theorem 1.Proof. Take the continuous limit on the sums on thesystem and environment states from the discrete SBSstate, (cid:80) x S → (cid:82) dx S , (cid:80) x Ej → (cid:82) dx E j and follow thesame proof: perfect objectivity collapses those sums todiscrete states and all other conditions follow.If we relax the requirement that same objectiveinformation appears, then we can relax the conditionallypure environment states to incoherent environment statesin the x basis: Proposition 1. A discrete SBS state ρ ( C ) SE E ··· E N of thefollowing form can be perfectly objective in all frames ( C , E j ), albeit with different objective information: ρ ( C ) SE ··· E N = (cid:88) i p i | ψ S | i (cid:105)(cid:104) ψ S | i | S ⊗ N (cid:79) j =1 (cid:88) x Ej t (cid:16) x E j (cid:12)(cid:12)(cid:12) i, j (cid:17) | x E j (cid:105)(cid:104) x E j | E j , (59) provided it satisfies the perfect distinguishability condi-tions in the original lab frame: (cid:104) ψ S | i | ψ S | i (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) (60) ρ E j | i ρ E j | i (cid:48) = 0 , ∀ i (cid:54) = i (cid:48) (61) and all environment frames: (cid:104) ˜ ψ ( E j ) ( i,q C | i ) | ˜ ψ ( E j ) (cid:16) i (cid:48) ,q (cid:48) C | i (cid:48) (cid:17) (cid:105) = 0 , ∀ (cid:0) i, q C | i (cid:1) (cid:54) = (cid:16) i (cid:48) , q (cid:48) C | i (cid:48) (cid:17) , (62) ρ ( E j ) E k | ( i,q C | i ) ρ ( E j ) E k | (cid:16) i (cid:48) ,q (cid:48) C | i (cid:48) (cid:17) = 0 ∀ (cid:0) i, q C | i (cid:1) (cid:54) = (cid:16) i (cid:48) , q (cid:48) C | i (cid:48) (cid:17) , ∀ k (cid:54) = j, (63) where | ˜ ψ ( E j ) ( i,q C | i ) (cid:105) = (cid:88) q S ψ (cid:16) q S − q C | i (cid:12)(cid:12)(cid:12) i (cid:17) | q S (cid:105) S (64) ρ ( E j ) E k | ( i,q C | i ) = (cid:88) q Ej t (cid:16) q E j − q C | i (cid:12)(cid:12)(cid:12) i, j (cid:17) | q E j (cid:105)(cid:104) q E j | E j . (65) Note that the values q C | i = q ( E ) C | i can take depends on theindex i and the original states on E . The proof is given in Appendix B 2. Note thatthis proposition is not an if-and-only-if : we havechosen that the objective information in frame E , forexample, is (cid:110) p i t ( − q C | i , − q C | i (cid:12)(cid:12)(cid:12) i, j = 1) (cid:111) ( i,q C | i ) , leadingto the conditions in the proposition. An example ofProposition 1 is depicted in Fig. 5.In the continuous case, this proposition will hold onlyup to some error, e.g. (cid:104) ˜ ψ ( E j ) ( i,q C | i ) | ˜ ψ ( E j ) (cid:16) i (cid:48) ,q (cid:48) C | i (cid:48) (cid:17) (cid:105) = δ > .With continuous environments states—even if they areincoherent—will result in a reduced distinguishability asgiven in Fig. 2. Corollary 2. Suppose we have a general discreteSBS state ρ ( C ) SE E ··· E N from Eq. (43) can be objectivein all other environment frames without particularstate-structure restrictions (unlike in Theorem 1 andProposition 1), provided we allow for partial trace andthe reduced state satisfies the nontrivial distinguishabilityconditions. If we move from frame C to frame E k , then traceout the subsystem C Tr C (cid:104) ρ ( E k ) SCE ...E N (cid:105) , then the reduced1 Figure 5. (Color on-line) Example of an objective SBS state described by Proposition 1. Top: the original objective informationis { p , p } . The environment E has mixed, incoherent states conditioned on i = 0 ( p , top left), and i = 1 ( p , top right).Bottom: When moving into the quantum reference frame of E , the system states have been shifted such that they still remaindistinguishable. The new objective information is { p t , p t , p t , p t } . state could be objective (provided it satisfies thenontrivial distinguishability conditions). We did notneed to restrict the conditional environment states to belocalised or incoherent—sometimes, simply tracing out asubsystem can give an SBS state. For example, the GHZstate ( | (cid:105) + | (cid:105) ) / √ is entangled and not SBS, buttracing out a single subsystem and we are left with anSBS state, ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | ) / .This corollary implies that the conditional environ-ment states can have coherences, and also shows howimportant, intricate, and nontrivial the distinguishabilityconditions are to the objectivity of a state, andemphasises our focus on the indistinguishability ofconditional states in other parts of this paper. Proof. From the general discrete SBS state in the frameof E , Eq. (B1), we trace out the C -subsystem: ρ ( E ) SE ··· E N = (cid:88) i,q C | i p i t (cid:16) − q C | i , − q C | i (cid:12)(cid:12)(cid:12) i, j = 1 (cid:17) × | ˜ ψ S ( i,q C | i ) (cid:105)(cid:104) ˜ ψ S ( i,q C | i ) | S ⊗ N (cid:79) j =2 ρ ( E ) E j | i,q C | i (66) | ˜ ψ S ( i,q C | i ) (cid:105) = (cid:88) q S ψ (cid:16) q S − q C | i (cid:12)(cid:12)(cid:12) i (cid:17) | q S (cid:105) S (67) ρ ( E ) E j | i,q C | i = (cid:88) q Ej ,q (cid:48) Ej t (cid:16) q E j − q C | i , q (cid:48) E j − q C | i (cid:12)(cid:12)(cid:12) i, j (cid:17) × | q E j (cid:105)(cid:104) q (cid:48) E j | E j . (68)By choosing (cid:110) p i t (cid:16) − q C | i , − q C | i (cid:12)(cid:12)(cid:12) i, j = 1 (cid:17)(cid:111) ( i,q C | i ) as thenew objective information, this reduced state will haveSBS provided it satisfies distinguishability conditions: (cid:104) ˜ ψ S ( i,q C | i ) | ˜ ψ S (cid:16) i (cid:48) ,q (cid:48) C | i (cid:17) (cid:105) = 0 , ∀ (cid:0) i, q C | i (cid:1) (cid:54) = (cid:16) i (cid:48) , q (cid:48) C | i (cid:17) , (69) ρ ( E k ) E j | i,q C | i ρ ( E k ) E j | i (cid:48) ,q (cid:48) C | i = 0 , ∀ (cid:0) i, q C | i (cid:1) (cid:54) = (cid:16) i (cid:48) , q (cid:48) C | i (cid:17) , ∀ j (cid:54) = k. (70)Table I summaries the results in this section. VI. OBJECTIVITY IN A DYNAMIC SYSTEMAND TWO ENVIRONMENTS In the prior sections, we focused primarily oncalculating the distinguishability of conditional systemand environment states. This distinguishability forms alower bound to an ideal objective state; however, it ismissing a quantification of the non-objective correlationsbetween the system and environments. In this section,we consider a numerical model that allows us to fullyexplore the divergence from an ideal objective state withthe SBS.2We analyse the broadcast probabilities which showthat the information is different in different referenceframes. The investigation of mutual information betweenenvironments illustrates that strong independence ofenvironments is also not conserved between referenceframes. We consider a couple of cases as illustration ofphenomena occurring when changing between differentreference frames in an information broadcasting scenario.We performed a series of numerical experiments. Notethat a computer’s memory cannot store an infinitenumber of data needed to fully describe quantum systemsin a continuum of space coordinates.To provide an conceptual image of the dynamicalscenario, we consider a toy-model where the coordinatesystem is discrete and organized as a ring of size D with all coordinates from the finite set { , · · · , D − } with the metric of the finite field Z D . This coordinatesimplification is similar in spirit to the lattice Ising model.The process of information propagation is governedby relevant Hamiltonians describing the time evolutionof interacting subsystems. Here we consider a simplescenario with a central system S , interacting with twoenvironments E and E observed from the point of viewof a non-interacting laboratory frame C . The referenceframe transformation shifts from the point of view of C to the point of view of E .The general interaction H between subsystems S and { E i } Ni =1 can be decomposed into several terms: H = N (cid:88) i =1 D − (cid:88) s =0 | s (cid:105)(cid:104) s | S ⊗ H ( s ) E i (cid:124) (cid:123)(cid:122) (cid:125) central interaction + N (cid:88) i =1 H E i (cid:124) (cid:123)(cid:122) (cid:125) self-evolution + N (cid:88) i (cid:54) = j =1 H E i ,E j (cid:124) (cid:123)(cid:122) (cid:125) environment interaction + H S,E , ··· E N (cid:124) (cid:123)(cid:122) (cid:125) global interaction , (71)where sub-indices enumerates subsystems on which thegiven part of the total Hamiltonian acts. We note thatthe form of the central interaction part ensures that theevolution of each of the environments depends on thestate of the central system and thus is responsible forimprinting information about it.In a typical measurement scenario one usuallyassumes that the evolution is dominated by the centralinteraction, and then the so-called generalized vonNeumann measurement is performed [35, 36]. It isreasonable to assume that this part is acting only fora limited period of time, as one expect the measurementto occur after a finite number of time units.We define the time unit t = 1 as the time over whichthe central interaction is active. We also define theenergy scale as relative to the strength of the centralinteraction. We assume that the self-evolution and theinteraction of environments is of two orders weaker andthe global interaction (that is in most cases a sort ofenvironmental noise) to be weaker of three orders thanthe central interaction. Since in this paper the Hilbert space is assumed to form a coordinate basis it is naturalto pay a particular attention to environment interactionswith strengths depending on the distance of subsystems.To be more specific, the central interaction H ( s ) S,E i isdefined in a way that after a unit of time the state | k (cid:105) E i is transformed to | k ⊕ D s (cid:105) E i , where ⊕ D is the additionmodulo D . The environment interaction Hamiltonian H E i ,E j is defined in a way that propagates jumps ofstates of a pair of interacting subsystems towards eachother with rate of the jumps given by . r , where r is thedistance between subsystems, and . is the couplingconstant (two order of magnitudes less than the self-evolution and measurement interaction). A self-evolutionof environments allows for jumps towards neighbouringstates, leading to a slow spread of the localization.Since global interaction is conceptualized as beingcaused by unintended jumps beyond control, the rateof each possible jump is regarded as a uniform randomnumber between and the coupling constant equal to . to model the assumption that this kind of force isof three orders weaker than the measurement interaction.It has been observed [37] that the capacity of anenvironment to receive information about the centralsystem depends on its purity: the higher is the entropyof the subsystem, the less additional information it cangain. In particular one expects that the completely mixedstate is not able to perceive the observed entity.In our investigation we consider various joint statesof the central system with two environments. The jointstate that maximizes the information flow, and thus ismost interesting, is the state: ρ mpp := ρ mix S ⊗ | (cid:105)(cid:104) | E ⊗ | (cid:105)(cid:104) | E , (72)where ρ mix := D (cid:80) D − i =0 | i (cid:105)(cid:104) i | is the maximally mixed stateon the D -dimensional ring. To see how mixedness ofenvironments influences information flow we consider asystem with slightly blurred environments: ρ mbb := ρ mix S ⊗ ρ blur E ⊗ ρ blur E , (73)where ρ blur := 0 . · | (cid:105)(cid:104) | + 0 . · | (cid:105)(cid:104) | + 0 . · | D − (cid:105)(cid:104) D − | . (74)We consider also the cases when only one of theenvironments is mixed: ρ mmp := ρ mix S ⊗ ρ mix E ⊗ | (cid:105)(cid:104) | E , (75a) ρ mpm := ρ mix S ⊗ | (cid:105)(cid:104) | E ⊗ ρ mix E . (75b)Another case that we find interesting to investigateis the situation when the environments are maximallyentangled, as this case revealed new phenomena whenchanging frames in Ref. [30]. We consider the state: ρ mEE := ρ mix S ⊗ | Φ (cid:105)(cid:104) Φ | E E , (76)3 Table II. Considered dynamical scenarios for a systeminteracting with two environments E , E . Interaction detailsfor H E , H E , and H E ,E are in main text (followingEq. (71)). The various initial states are given in the maintext from Eqs. (72) to (77), where the labelling ρ SE E denotes where that subsystem is mixed (m), pure (p),blurred/partially mixed (b), entangled (E) or pure and shiftedrelatively to each other (s).Caselabel self-evolution environmentinteraction globalinteraction initialstate1.1 - - - ρ mpp ρ mbb ρ mEE ρ mmp ρ mpm H E + H E - - ρ mpp ρ mpp H E ,E - ρ mpp ρ mpp H E ,E - ρ mps ρ mps H E + H E H E ,E - ρ mps ρ mps H E + H E H E ,E random ρ mpp ρ mpp where | Φ (cid:105) E E := √ D (cid:80) Di =0 | i (cid:105) E | i (cid:105) E . Since we areconcerned with the relative properties of states, we alsoconsider the case when both environments are pure, butshifted relatively to each other by half of the size of thering: ρ mps := ρ mix S ⊗ | (cid:105)(cid:104) | E ⊗ |(cid:100) D/ − (cid:101)(cid:105)(cid:104)(cid:100) D/ − (cid:101)| E . (77)This state allows us to observe the dependence ofstrength of environmental interactions on the distancebetween system and environment, and thus provide thespacial meaning to the D dimensional Hilbert space.We summarize all cases we investigate in the dynamicalscenario in the Tab. II.From the perspective of external observer C , the time-dependent tripartite state consists of the central object S , and two environments E and E . From frame ofthe first environment, E , the relevant state consists ofthe central object, S , the external observer, C , and thesecond environment, E . We refer to these states as ρ ( C ) SE E and ρ ( E ) SCE , respectively.The core part of the SBS is the spectrum of theprobability distribution that is broadcast from systemto environments. This spectrum is given by p ( C ) i := (cid:104) i | S Tr E E (cid:16) ρ ( C ) SE E (cid:17) | i (cid:105) S , (78a) p ( E ) i := (cid:104) i | S Tr CE (cid:16) ρ ( E ) SCE (cid:17) | i (cid:105) S , (78b)where Tr E E and Tr CE denotes partial trace oversubsystems E and E , and C and E , respectively. The conditional states, c.f. Eq. (30), are ρ ( C ) E | i := (cid:16) /p ( C ) i (cid:17) · (cid:104) i | S Tr E (cid:16) ρ ( C ) SE E (cid:17) | i (cid:105) S , (79a) ρ ( C ) E | i := (cid:16) /p ( C ) i (cid:17) · (cid:104) i | S Tr E (cid:16) ρ ( C ) SE E (cid:17) | i (cid:105) S , (79b) ρ ( E ) C | i := (cid:16) /p ( E ) i (cid:17) · (cid:104) i | S Tr E (cid:16) ρ ( E ) SCE (cid:17) | i (cid:105) S , (79c) ρ ( E ) E | i := (cid:16) /p ( E ) i (cid:17) · (cid:104) i | S Tr C (cid:16) ρ ( E ) SCE (cid:17) | i (cid:105) S , (79d)where Tr E , Tr E and Tr C denotes partial trace overrelevant subsystems. For the conditional states { ρ ( · ) ·| i } D − i =0 we calculate their two averages, weighted, c.f. Eq. (23): D − (cid:88) i (cid:54) = j =0 (cid:113) p ( · ) i p ( · ) j (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( · ) ·| i (cid:113) ρ ( · ) ·| j (cid:13)(cid:13)(cid:13)(cid:13) , (80)and unweighted: D ( D − D − (cid:88) i (cid:54) = j =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( · ) ·| i (cid:113) ρ ( · ) ·| j (cid:13)(cid:13)(cid:13)(cid:13) . (81)For the sake of clarity, cf. Eq. (20), we denote the fidelityterms as: B ( C ) E ( i, j ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( C ) E | i (cid:113) ρ ( C ) E | j (cid:13)(cid:13)(cid:13)(cid:13) , (82a) B ( C ) E ( i, j ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( C ) E | i (cid:113) ρ ( C ) E | j (cid:13)(cid:13)(cid:13)(cid:13) , (82b) B ( E ) C ( i, j ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) C | i (cid:113) ρ ( E ) C | j (cid:13)(cid:13)(cid:13)(cid:13) , (82c) B ( E ) E ( i, j ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:113) ρ ( E ) E | i (cid:113) ρ ( E ) E | j (cid:13)(cid:13)(cid:13)(cid:13) . (82d)In order to contrast strong versus weak independencebetween observing subsystems we also calculate the meanconditional quantum mutual information (Definition 2): I ( C ) mean := D − (cid:88) i =0 p ( C ) i (cid:104) H (cid:16) ρ ( C ) E | i (cid:17) + H (cid:16) ρ ( C ) E | i (cid:17) − H (cid:16) ρ ( C ) E E | i (cid:17)(cid:105) , (83a) I ( E ) mean := D − (cid:88) i =0 p ( E ) i (cid:104) H (cid:16) ρ ( E ) C | i (cid:17) + H (cid:16) ρ ( E ) E | i (cid:17) − H (cid:16) ρ ( E ) CE | i (cid:17)(cid:105) , (83b)4where H ( · ) is von Neumann entropy and the conditionalstates are: ρ ( C ) E E | i := (cid:16) /p ( C ) i (cid:17) · (cid:104) i | S ρ ( C ) SE E | i (cid:105) S , (84a) ρ ( E ) CE | i := (cid:16) /p ( E ) i (cid:17) · (cid:104) i | S ρ ( E ) SCE | i (cid:105) S . (84b)There are dynamical situations when the mean mutualinformation reaches some value and does not deviatesignificantly (up to some fluctuations) from it further intime. We refer to this value as the value of saturation : I ( C ) sat := lim T →∞ T (cid:90) T I ( C ) mean ( t ) dt, (85a) I ( E ) sat := lim T →∞ T (cid:90) T I ( E ) mean ( t ) dt, (85b)from the point of view of C and E , respectively. The fluctuations of the mean mutual information are definedas: σ ( C ) I := (cid:34) lim T →∞ T (cid:90) T (cid:16) I ( C ) mean ( t ) − I ( C ) sat (cid:17) dt (cid:35) , (86a) σ ( E ) I := (cid:34) lim T →∞ T (cid:90) T (cid:16) I ( E ) mean ( t ) − I ( E ) sat (cid:17) dt (cid:35) , (86b)for reference frames of C and E , respectively. We alsodefine the time of saturation t ( C ) sat from the perspective C ( t ( E ) sat from the perspective E ), as a time when themean mutual information reaches the value I ( C ) sat − σ ( C ) I ( I ( E ) sat − σ ( E ) I ) for the first time.We performed a series of numerical simulations in orderto investigate how well the SBS form is preserved inboth the frames of the external observer, C , and thefirst environment, E . For the cases discussed belowwe take D = 12 . This dimension has been chosenas a compromise between computational effort andmodelling the dependence of behaviours of subsystemson their spatial separations. The long time averages arecalculated over time points between and with time step .The reference frame transformation ˆ S ( C → E ) position satisfyingEq. (4) is, in this case, a permutation of indicesof rows and columns of a tripartite D dimensionaldensity matrix. As such the spectrum of the initialdensity matrix remains invariant. For the case ofdimension D = 12 we have found that the unitarytransformation over D = 1728 dimensional joint tri-coordinate space has character (trace) , contains irreducible subspaces of dimension , and irreduciblesubspaces of dimension . We have directly checked that p r o b a b ili t y i n E time1.1 p Figure 6. (Color on-line) Selected probabilities in thereference frame E for Cases 1.1 and 1.2. Note that for Cases1.3 and 1.4 the probability p ( E )0 = D ≈ . and is constantin time. For the Case 1.5 p ( E )0 is identical as in the Case 1.1.(Cases given in Tab. II) χ (cid:104) ˆ S ( C → E ) position (cid:105) = D , where χ [ · ] is the character (trace)of a transformation, for D ≤ and conclude that thisis a general property of reference frame transformationsas D → ∞ .Below we summarize how the properties relevantfor perceived objectivity behave in our dynamicalmodel. The change in the broadcast spectrum ofprobabilities (Sec. VI A), the dynamics and overallvolume of mutual information (Sec. VI B), the varyingdistinguishability of subsystems (Sec. VI C), and theHolevo and quantum mutual information between centralsystem and subenvironments (Sec. VI D) in differentframes provides a concrete illustration of the mainpremise of this paper: what is objective from one pointof view may not be objective from another point of view.All the cases referred to in the following section havebeen labelled in Tab. II. The full details are described inAppendix C. A. Probabilities The probabilities of the central system are key forthe spectrum in an objective state. From our numericalcalculations, we find that all probabilities (of the centralsystem spectrum) { p ( C ) i } D − i =0 from laboratory C ’s pointof view are constant and uniform over time, except inCases 5.1 and 5.2 (cases given in Tab. II) where theadded global interaction influences the central subsystemand thus modifies its spectrum.The system-environment generalised measurementinteraction has been designed such that the effectivemeasurement occurs by time t = 1 . This occurs withoutdisturbance in Cases 1.1 and 1.5 (Tab. II), where thereare no other interactions, and where the initial state of E in C ’s frame of reference, ρ ( C ) E ( t ) := Tr SE (cid:16) ρ ( C ) SE E ( t ) (cid:17) ,is pure (see Fig. 6). In these two cases, p ( E )0 ≈ at5 p r o b a b ili t y i n E time2.12.23.13.23.33.44.14.25.15.2 0 0.5 1 0 1 Figure 7. (Color on-line) Probabilities p ( E )0 for cases fromthe groups 2, 3, 4 and 5 in the reference frame of E , whichhave various interactions in addition to central measurementHamiltonian. Inset: probabilities during time [0 , where themeasurement-like Hamiltonian dominates. (Cases given inTab. II) time t = 1 , i.e. implying that the central system stateis close to pure at the end of the measurement, whichcorresponds to a trivial kind of objectivity in the frameof E (much like Example 1). Alternatively, we can saythat E ’s capacity was not used .If the initial environment state ρ ( C ) E is slightly mixed inCase 1.2, the value of the probability p ( E )0 in the frameof E diverges from , proportional to the mixedness inthe original E state. In Cases 1.3 and 1.4, when p ( E )0 was either maximally mixed or a part of a maximallyentangled state (which means that the local state of E is identical) the values of probabilities { p ( E ) i } D − i =0 remain uniform at the time t = 1 , meaning there wasno information transfer at all.The probabilities in the frame of E for all other casesbehave very similarly to the Case 1.1 up to time t = 1 ,despite their interactions, since their evolution on theshort time scale is still dominated by the measurementinteraction (see Fig. 7). B. Strong independence and the conditionalmutual information The mean mutual information between the conditionalenvironment informs us on whether the subsystems havestrong or weak independence, the former of which isa condition of spectrum broadcast structure: a small(ideally zero) mean mutual information denotes strongindependence.In Figs 8 and 9, we give the plots for how the meanmutual information I ( E ) mean behaves over time, in the frameof E . We find that the mean mutual informationtypically starts from the value I mean = 0 , reaches localmaximum close to . exactly at the time t = 0 . andreturns close to for the time t = 1 . The only exception m u t u a l i n f o r m a t i o n [ b i t s ] time1.11.21.31.4 Figure 8. (Color on-line) Plot of I ( E ) mean (the mean mutualinformation seen from the reference frame of the firstenvironment, E ) for Cases 1.1, 1.2, 1.3 and 1.4. In the Case1.5 the function is constant and equal to . This shows thecases when I ( E ) mean ( t = 1) = 0 . The plot for the Case 1.1 isidentical for Cases 2.1, 3.1, 3.3, 4.1. (Cases given in Tab. II).When the mutual information is low, the environments havestrong independence. m u t u a l i n f o r m a t i o n i n E [ b i t s ] time2.23.24.25.15.2 0 0.5 1 1.5 0 0.5 1 Figure 9. (Color on-line) Plot of mean mutual informationin the reference frame of E , I ( E ) mean for Cases 2.2, 3.2, 4.2,5.1 and 5.2, cf. Tab. III. The function in the Case 3.4 isalmost identical as in the Case 3.2. Inset: mean mutualinformation during times t = [0 , where the measurement-like Hamiltonian dominates. (Cases given in Tab. II).As the mutual information is large as time increases, theenvironments do not have strong independence. from this behaviour is the Case 1.4, i.e. when we observethe system ρ ( E ) SCE from a point of view of a completelyrandom observer ( i.e. , E was mixed in the lab frame C ). In that case the mutual information starts with themaximal value equal . and gradually drops. For allcases, I ( E ) mean ( t = 1) ≈ (with very small non-zero valuein the cases with random interactions).The Cases 1.1, 2.1, 3.1, 3.3 and 4.1 have very similarvalues of I ( E ) mean in each moment of time, which can beseen in Fig. 8. Furthermore, the mutual information is after time t = 1 in reference frame E . Thus, while theobserving subsystems C , E initially develop conditionalcorrelations during times t = [0 , , they satisfy strong6 Table III. Dynamics of mean mutual information I ( E ) sat in thereference of frame of the first environment, E ; σ ( E ) I describesthe fluctuation and t ( E ) sat is the time of saturation (cf. Fig. 9and Eqs. (85), (86)).2.2 3.2 3.4 4.2 5.1 5.2 I ( E ) sat . 513 2 . 867 2 . 872 2 . 862 3 . 508 3 . σ ( E ) I . 168 0 . 022 0 . 014 0 . 026 0 . 004 0 . t ( E ) sat 150 50 50 50 500 25 Table IV. Dynamics of mean mutual information in the frameof reference of the external laboratory C . Values at thetimes t = 0 . , shows the gradual increase (cf. Fig. 12 inthe Appendix C).3.1 3.2 3.3 3.4 4.1 4.2 5.1 5.2 I ( C ) mean (0 . 5) 0 . 002 0 . 023 0 . 000 0 . 023 0 . 000 0 . 023 0 . 004 0 . I ( C ) mean (1) 0 . 006 0 . 070 0 . 001 0 . 071 0 . 001 0 . 071 0 . 012 0 . I ( C ) sat . 769 5 . 740 2 . 942 5 . 730 5 . 845 5 . 739 3 . 521 3 . σ ( C ) I . 949 0 . 032 1 . 025 0 . 038 0 . 127 0 . 033 0 . 004 0 . t ( C ) sat 30 50 200 40 250 50 60 25 independence thereafter, a necessary condition for idealobjectivity (Definition 2).The second common pattern is shown in Fig. 9, wheremean mutual information in the reference frame of E ,after approaching the value at the time t = 1 ,will gradually increase to some saturation level, withslight fluctuations. This happens in Cases 2.2, 3.2,3.4, 4.2, 5.1 and 5.2, where some random Hamiltonianis present (cf. Tab. III). In all other cases the meanmutual information in this reference frame is after thetime t = 1 .On the other hand, from the point of view of theexternal laboratory C , the mean mutual informationis constant when there is no interaction betweenenvironments, i.e. in Case groups 1 and 2. Inthis reference frame, cases with inter-environmentalinteraction (groups 3, 4 and 5) lead to the mean mutualinformation gradually increasing (without local maxima)till the point of saturation, t ( C ) sat . The parameters of thisbehaviour are given in Tab. IV.We see that the tripartite states typically have strongindependence while in the lab frame C , but this weakenswhen moving to the environment frame E .We observe that in the short time range, before theinteraction of environments with central system has fullyoccurred, environments are more independent in thelaboratory’s frame, whereas after the interaction theindependence is stronger for environmental frame.For long time scale we distinguish two situations: Ifthe environmental interaction decreases with distance(Cases 3.1, 3.3 and 4.1) then the strong independencebetween C and E occurs in the environment frame E ,but not between E and E in frame C . Meanwhile, if theenvironment interaction is random and does not decrease Table V. Upper bound to distinguishability error, Eq. (23), atthe time t = 1 for cases without random inter-environmentalinteractions, in the reference frame of C for subsystems E and E and in the reference frame of E for subsystems C and E .frame subsystem 2.1 2.2 3.1 3.3 4.1C E ( C )1 . 000 0 . 28 0 . 050 0 . 018 0 . C E ( C )2 . 000 0 . 025 0 . 050 0 . 018 0 . E C ( E ) . 040 0 . 109 0 . 051 0 . 018 0 . E E ( E )2 . 000 0 . 000 0 . 011 0 . 005 0 . with distance (Cases 3.2, 3.4, 4.2), there is no strongindependence in either frames. In this second situation,the mutual information happens to be almost exactlytwice as large (differing by at most) in C ’s framecompared to E ’s frame, suggesting that there is greaterindependence in frame E . When global interaction ispresent, then the mutual information is the same in bothframes (differing by . at most) for all times.Overall, in the cases we consider, strong independencecan be maintained in original frame C , but not theenvironment frame E . Furthermore, environment-environment interactions destroy strong independence. C. Distinguishability Distinguishability is crucial for observers to determinethe central system’s spectrum. In the reference frameof C the upper bound that describes the error indistinguishing conditional states, Eq. (23), is equal atthe time t = 1 for Cases 1.1, 2.1, and close to for Cases2.2, 3.1, 3.3 and 4.1, i.e. for cases without random inter-environmental interactions (see Tab. V). It is also forthe subsystem E in Case 1.4 (when E is pure and E is maximally mixed) and for subsystem E in Case 1.5(when E is pure and E is maximally mixed).For the other cases which do have random interactions,error the bound is significantly higher, viz. for the Case3.2, 3.4, 4.2 the bound is between . and . , equal to . for the Case 5.1, and . for the Case 5.2.Similarly, from the perspective of E , the distinguisha-bility error upper bound Eq. (23) is low (below . for atleast one subsystem C or E ) when there are no randominteractions and the initial state of E is pure: thesecorrespond to Cases 1.1, 2.1, 2.2, 3.1, 3.3 and 4.1, andalso 1.5 (that was for E and above for the initiallymaximally mixed E ). This is summarised in Tab. V.Overall, the effect of moving frame is thus: while it iseasier to distinguish E in lab C frame, while E has thelower distinguishability error when in the E frame.7 m u t u a l i n f o r m a t i o n [ b i t s ] time2.1, 2.2, 3.3 and 4.1 in C3.1 in C3.2, 3.4 and 4.2 in C2.1, 3.1, 3.3 and 4.1 in E Figure 10. The quantum mutual information between thecentral system S and subsystem E ( C ) in the reference frameof C ( E ). A high quantum mutual information often suggestsobjectivity. (Cases given in Tab. II) D. Holevo information and mutual informationbetween central system and environments The analysis in the previous subsections have shownthat the SBS structure is generally not preserved betweenreference frames. Another aspect of quantum Darwinismand SBS is the information propagation measured bythe quantum mutual information, and by the (classical)Holevo information transfer from the central system toenvironments. To this end, we calculate the quantummutual information, as well as the Holevo informationby considering a hypothetical measurement of the centralsystem in its eigenbasis, and the ensemble of steeredstates of each of the environments.Fig. 10 shows the quantum mutual informationbetween the central system S and subsystem E ( C )in the reference frame of C ( E ). Both Holevoinformation (figure given in Appendix C as it is extremelysimilar) and quantum mutual information of both sub-environments are equal for Cases 1, 2, 3 and 4 inframe C and equal for the environment E in frame E , thus satisfying one requirement for SBS. The lowquantum mutual information in frame E comparedto high quantum mutual information in frame C alsosuggests that the central system and C are triviallyobjective, i.e. that only one probability in the spectrumis substantial (cf. Sec. VI A).On the other hand, the Holevo and quantum mutualinformation differ for the subsystem C in frame E inmost Cases. For the Case 1.3 the quantum mutualinformation is constant and is equal to the half of themutual information of a maximally entangled state, andHolevo information is . In Cases 2, 3 and 4 thequantum mutual information is also significantly higherthat Holevo information, with exception for the Case 2.1when they are equal. In general, the state C in frame E develops quantum correlations with the central system,reflecting the results from the prior sections. VII. CONCLUSION We examined the transformation of objectivity indifferent quantum reference frames. We used thequantum reference frame formalism of Giacomini et al. [30], in which local and global properties are frame-dependent and can interchange, and analysed thestructure of objective states in the frame of theirenvironments.Under perfect localisation, we showed how non-degenerate relative positions is the key factor in ensuringthat objectivity is consistent across different quantumreference frames. This can be done by randomly choosingall required positions from a continuous interval—asthe probability of two random real numbers beingequal is zero. Environment-state coherences introduceentanglement between system and environments, whichthen require discarding of environments to remove.Meanwhile, environment-state statistical mixedness, i.e. internal classical noise, introduces new classicalcorrelations in addition to the original classical objectivecorrelations. Thus, in general, each reference frame hasa different set of objective information that involvesthe original statistics of the environment. Nevertheless,the original system information is recoverable by takingthe appropriate marginal of the information distribution.Hence, the objective information of the system inthe original laboratory frame is unique and existsconsistently in all frames—although typically the system-environment state no longer has an objective state formin other frames.We then considered systems and environments with anon-zero spread over position. Distinguishability requireseither sufficiently separated (non-degenerate) relativepositions, or large differences in spread of differentconditional states. Objectivity becomes “blurred” indifferent frames due to the continuous spread in theenvironment states, and the greater the blurring, thelarge the macrofractions are required in the new framein order to recover a form of objectivity.We also found that the best candidate of objectiveinformation finds the system in conditional mixed statesthat are distinguishable, rather than the conditional pure system states strictly required of quantumDarwinism. This suggests a generalised objectivity, i.e. where the system is conditionally mixed and perfectlydistinguishable like the environments. Furthermore, onlya strict subset of all objective states are perfect robustacross quantum reference frames—when its observedenvironments are also objective , that is having aninvariant SBS [13]. This suggests that objective statesis consistent under different frames only if it system andenvironments have similar structure, i.e. all conditionallypure, or all conditionally mixed.We also proved that perfect objectivity holds in all laband environment frames if and only if the environmentstates are pure and perfectly localised in the positionbasis conditioned on i of the system information. An8open question is how states close to this specialisedform of SBS behave under quantum reference frametransformations.Finally, we examined the dynamical emergenceof objectivity in different quantum reference frames,explicitly demonstrating how changing frames affectsthe objective probabilities and degrades spectrumbroadcast structure by weakening strong independenceand reducing distinguishability.Quantum Darwinism is an approach towards under-standing the quantum-to-classical transition, examininghow hypothetical observers may acquire objectiveinformation about a common system. Observers mayhave different reference frames, which subsequentlyaffects the information they can obtain. We have shownthat the system’s objective information is recoverablein all frames, despite the interchangeability of the localand global statistics. At the same time, our workdemonstrates the rise of extra frame-dependent objectiveinformation, and the robustness (or otherwise) of variousobjective state structures. In the strictest sense,objectivity is subjective across quantum reference frames.This work opens up the pathway to understandingquantum Darwinism and its intersection with relativity.On a more philosophical level the considerations aretightly related to the problem of Wigner’s friend [38]gedanken experiment.After the completion of this manuscript, we were madeaware of independent work on a similar topic: Ref. [39]examines dynamical aspects of quantum reference frametransformations and objectivity, and thus their resultsare complementary to our results presented here. ACKNOWLEDGEMENTS The support by the Foundation for Polish Sciencethrough IRAP project co-financed by the EU withinthe Smart Growth Operational Programme (contractno. 2018/MAB/5) is acknowledged. This work wasalso supported by the Engineering and Physical SciencesResearch Council [grant number EP/L015242/1]. Appendix A: Objectivity with continuousprobabilities In the main paper, the branch structure of theobjective state is typically discrete, indexed by { i } andsummed. However, we can consider the more generalsituation in which this { i } becomes continuous, even inthe laboratory frame. A GHZ-like continuous objectivestate is the following: ρ ( C ) SE E ··· E N = (cid:90) dx S ψ ( x S ) | x S (cid:105)(cid:104) x S | S ⊗ | φ ( x S ) (cid:105)(cid:104) φ ( x S ) | E ⊗ · · · ⊗ | φ N ( x S ) (cid:105)(cid:104) φ N ( x S ) | E N , (A1) where ψ ( x S ) are the continuous probabilities, and { φ i } i are bijective (one-to-one and onto) functions. One-to-oneness is necessary so that the environments E i positions are unique for any different system position x S ,and hence are always perfectly correlated with the systemposition x S . For simplicity, we will also take { φ i } i beingonto, which ensures that the inverse is also one-to-one.In the quantum reference frame of the environment E ,the joint state has form: ρ ( E ) SCE ··· E N = (cid:90) dq C ψ (cid:0) φ − ( − q C ) (cid:1) × | φ − ( − q C ) + q C (cid:105)(cid:104) φ − ( − q C ) + q C | S ⊗ | q C (cid:105)(cid:104) q C | C ⊗ N (cid:79) j =2 | φ j (cid:0) φ − ( − q C ) (cid:1) + q C (cid:105)(cid:104) φ j (cid:0) φ − ( − q C ) (cid:1) + q C | E j . (A2)The original probabilities ψ ( · ) still exist—as the originalconditional environment states are pure in the positionbasis (c.f. Thm. 1). In the frame E , there is an integralover q C rather than x S . In this particular scenario,objectivity requires distinguishability for different q C .Hence φ − ( − q C ) + q C and (cid:8) φ j (cid:0) φ − ( − q C ) (cid:1) + q C (cid:9) j mustalso be one-to-one functions—so that none of the newpositions become degenerate (and subsequently reduceconditional distinguishability).Note that the composition of one-to-one functionsis one-to-one; but sum of one-to-one functions isnot necessarily one-to-one: a sufficient but notnecessary condition is that ddx φ i (cid:0) φ − ( x ) (cid:1) > or that ddx φ i (cid:0) φ − ( x ) (cid:1) < for all x . Hence the state given inEq. (A1) is not always objective for any set of one-to-onefunctions ψ ( · ) , φ j ( · ) . Appendix B: Proof of conditions for perfectobjectivity1. Proof of Theorem 1 Here, we prove the state structure and conditions foran SBS state to be objective, with the same objectiveinformation, in all laboratory and environment quantumreference frames. Proof. In the new environment frame E , the system-environment state has general form: ρ ( E ) SCE ··· E N = (cid:88) i p i (cid:88) q C ,q (cid:48) C (cid:88) q S ,q (cid:48) S ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q (cid:48) C | i ) | q S (cid:105)(cid:104) q (cid:48) S | S ⊗ t ( − q C , − q (cid:48) C | i, j = 1) | q C (cid:105)(cid:104) q (cid:48) C | C ⊗ N (cid:79) j =2 (cid:88) q Ej ,q (cid:48) Ej t (cid:16) q E j − q C , q (cid:48) E j − q (cid:48) C (cid:12)(cid:12)(cid:12) i, j (cid:17) | q E j (cid:105)(cid:104) q (cid:48) E j | E j , (B1)where note that in t ( − q C , − q (cid:48) C | i, j = 1) , only j = 1 isfixed. The general reduced system state is: ρ ( E ) S = (cid:88) i p i (cid:88) q C t ( − q C , − q C | i, j = 1) × (cid:88) q S ,q (cid:48) S ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q C | i ) | q S (cid:105)(cid:104) q (cid:48) S | S . (B2)If the same initial objective information { p i } i ismaintained, the system must have { p i } i as its spectrum,even in the frame E . Hence, we must be able todecompose it as ρ ( E ) S ! = (cid:80) i p i | ˜ ψ Si (cid:105)(cid:104) ˜ ψ Si | S , where | ˜ ψ Si (cid:105) = (cid:80) q S T ( q S ) | q S (cid:105) S are the new eigenstates of S in the frame E with some coefficients T ( q S ) : ρ ( E ) S | i = (cid:88) q C t ( − q C , − q C | i, j ) × (cid:88) q S ,q (cid:48) S ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q C | i ) | q S (cid:105)(cid:104) q (cid:48) S | S (B3) = (cid:32)(cid:88) q S | q S (cid:105) S (cid:33)(cid:88) q (cid:48) S (cid:104) q (cid:48) S | S × (cid:88) q C (cid:18) ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q C | i ) × t ( − q C , − q C | i, j = 1) (cid:19) (B4) ! = (cid:88) i p i | ˜ ψ Si (cid:105)(cid:104) ˜ ψ Si | S (B5) = (cid:32)(cid:88) q S T ( q S ) | q S (cid:105) S (cid:33)(cid:88) q (cid:48) S T ∗ ( q (cid:48) S ) (cid:104) q (cid:48) S | S . (B6)The coefficient (cid:80) q C ( · · · ) term in Eq. (B4) must befactorisable into independent terms involving q S and q (cid:48) S : T ( q S ) T ∗ ( q (cid:48) S ) ! = (cid:88) q C ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q C | i ) × t ( − q C , − q C | i, j = 1) . (B7)The only way to factorise this is if there is only one nonzero term in the sum, i.e. if and only if t ( − q C , − q C | i, j = 1) ! = δ (cid:0) − q C − x E j =1 | i (cid:1) . (B8)Thus, the original conditional environment states ρ E | i ofenvironment E are pure and incoherent in the x basis: ρ ( C ) E | i = (cid:88) x E ,x (cid:48) E t (cid:0) x E , x (cid:48) E | i, j = 1 (cid:1) | x E (cid:105)(cid:104) x (cid:48) E | E ! = | x E | i (cid:105)(cid:104) x E | i | . (B9) This holds analogously for objectivity in any otherenvironment quantum reference frame. Hence, werequire that all conditional environment states are pure,leaving us with system-environment state structure inthe original laboratory frame and in the environment E frame respectively: ρ ( C ) SE ··· E N = (cid:88) i p i | ψ Si (cid:105)(cid:104) ψ Si | S ⊗ N (cid:79) j =1 | x E j | i (cid:105)(cid:104) x E j | i | E j . (B10) ρ ( E ) SCE ··· E N = (cid:88) i p i | ˜ ψ Si (cid:105)(cid:104) ˜ ψ Si | ⊗ |− x E | i (cid:105)(cid:104)− x E | i | C ⊗ N (cid:79) j =2 | x E j | i − x E | i (cid:105)(cid:104) x E j | i − x E | i | E j , (B11) | ˜ ψ Si, ( j =1) (cid:105) := (cid:88) q S ψ (cid:16) q S + x E j =1 | i (cid:12)(cid:12)(cid:12) i (cid:17) | q S (cid:105) S . (B12)The conditional states already have strong independence,and so the final requirement of SBS objectivity is perfectdistinguishability: (cid:104) x E j | i − x E k | i | x E j | i (cid:48) − x E k | i (cid:48) (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) , j (cid:54) = k, (B13) (cid:104) ˜ ψ Si,j | ˜ ψ Si (cid:48) ,j (cid:105) = 0 , ∀ i (cid:54) = i (cid:48) . (B14)We should already have that (cid:104) ψ Si | ψ Si (cid:48) (cid:105) = 0 , (cid:104) x E j | i | x E j | i (cid:48) (cid:105) = 0 ∀ i (cid:54) = i (cid:48) as this is necessary for thestate in the lab frame to be objective. 2. Proof of Proposition 1 Here, we prove that a particular SBS state structure—with relevant distinguishability conditions—will beobjective in all laboratory and environment frames, albeitwith different objective information. Proof. Recall the general reduced state of the system inthe frame E , Eq. (B2). If we now have a new objectiveinformation, then there will be some eigendecompositionof the system with new probabilities ˜ p ˜ i : ρ ( E ) S = (cid:88) ˜ i ˜ p ˜ i | ˜ ψ ˜ i (cid:105)(cid:104) ˜ ψ ˜ i | . (B15)The reduced system state is ρ ( E ) S = (cid:88) i p i (cid:88) q C t ( − q C , − q C | i, j = 1) × (cid:88) q S ,q (cid:48) S ψ ( q S − q C | i ) ψ ∗ ( q (cid:48) S − q C | i ) | q S (cid:105)(cid:104) q (cid:48) S | S . (B16)Note that j = 1 is fixed, while i is not fixed. Recallhow in the situation when we had the same objective0information, the coefficient t ( − q C , − q C | i, j = 1) was adirac delta. However, as we are not requiring the same objective information here, we can relax this condition.Instead, we can identify (cid:80) q S ψ ( q S − q C | i ) | q S (cid:105) S =: | ˜ ψ ˜ i (cid:105) as the conditionally pure states, and impose that they areorthogonal, such that the system decomposition is ρ ( E ) S = (cid:88) i,q C | i ∈ C E | i ˜ p (cid:0) i, q C | i (cid:1) | ˜ ψ ( i,q C | i ) (cid:105)(cid:104) ˜ ψ ( i,q C | i ) | (B17) ˜ p (cid:0) i, q C | i (cid:1) = p i t (cid:0) − q C | i , − q C | i | i, j = 1 (cid:1) (B18) | ˜ ψ ( i,q C | i ) (cid:105) = (cid:88) q S ψ (cid:0) q S − q C | i | i (cid:1) | q S (cid:105) S , (B19)with new objective information (cid:8) p i t ( − q C | i , − q C | i | i, j = 1) (cid:9) ( i,q C | i ) , and where q C = q ( E ) C | i has a dependence on i and on the environment frame(here, E ). Hence, the sets C E | i describe the possiblevalues q C can take given index i and environmentframe E . The conditional system states need to bedistinguishable for this to be a valid decomposition: (cid:104) ˜ ψ ( i,q C | i ) | ˜ ψ ( i (cid:48) ,q (cid:48) C | i ) (cid:105) = 0 , ∀ (cid:0) i, q C | i (cid:1) (cid:54) = ( i (cid:48) , q (cid:48) C | i ) . (B20)Returning to the full system-environment-lab state inframe E , we now have that ρ ( E ) SCE ··· E N = (cid:88) i p i (cid:88) q C | i ,q (cid:48) C | i t ( − q C | i , − q (cid:48) C | i | i, j = 1) × | ˜ ψ ( i,q C | i ) (cid:105)(cid:104) ˜ ψ (cid:16) i,q (cid:48) C | i (cid:17) | ⊗ | q C | i (cid:105)(cid:104) q (cid:48) C | i | C ⊗ N (cid:79) j =2 (cid:88) q Ej ,q (cid:48) Ej t ( q E j − q C | i , q (cid:48) E j − q (cid:48) C | i | i, j ) | q E j (cid:105)(cid:104) q (cid:48) E j | E j . (B21)For SBS structure, the system cannot have extracoherences beyond the (cid:110) | ˜ ψ ( i,q C | i ) (cid:105) (cid:111) i,q C basis we haveestablished, and the conditional environment states mustbe distinguishable. Hence, t ( − q C | i , − q (cid:48) C | i | i, j ) ! = 0 if q C | i (cid:54) = q (cid:48) C | i , i.e the environment E is incoherent in the x basis. This means that all the environments must beincoherent in the x basis in order for this to hold in allenvironment frames, ρ ( E ) SCE ··· E N = (cid:88) i (cid:88) q C | i ∈ C E | i p i t (cid:0) − q C | i | i, j = 1 (cid:1) × | ˜ ψ ( i,q C | i ) (cid:105)(cid:104) ˜ ψ ( i,q C | i ) | ⊗ | q C | i (cid:105)(cid:104) q C | i | C ⊗ N (cid:79) j =2 (cid:88) q Ej t (cid:0) q E j − q C | i | i, j (cid:1) | q E j (cid:105)(cid:104) q E j | E j , (B22)where we have written t ( q, q | i, j ) = t ( q | i, j ) . Note that q C = q ( E ) C | i is dependent on i and dependent on the H o l e v o i n f o r m a t i o n [ b i t s ] time2.1, 2.2, 3.3 and 4.1 in C3.1 in C3.2, 3.4 and 4.2 in C2.1, 3.1, 3.3 and 4.1 in E Figure 11. (Color on-line) The Holevo information betweencentral system S and subsystem E ( C ) in the reference frameof C ( E ). Compared with Fig. 10, the Holevo information isvery similar to the quantum mutual information, aside fromthe last 3 cases in frame E . original environment E states: this is encoded in thecoefficients t ( − q C | i | i, j = 1) , as well as the sets C E | i forextra clarity.Lastly, we impose the distinguishability conditions onthe environment states: ρ ( E ) E j | ( i,q C ) ρ ( E ) E j | ( i (cid:48) ,q (cid:48) C ) = 0 , ∀ ( i, q C ) (cid:54) = ( i (cid:48) , q (cid:48) C ) . (B23)Note that this implies any specific q C value is assignedonly to one unique index i , since we must have | q C | i (cid:105) being distinguishable for each nonzero combination of (cid:0) i, q C | i (cid:1) . In other words, knowing q C automatically givesknowledge of i , like a many-to-one function. (Equally,the sets { C E | i } i are disjoint across i .) Appendix C: Detailed discussion of numericalresults of dynamic system and two environments In this Appendix we provide a detailed discussion ofnumerical experiments of Sec. VI. Subsec. C 1 considerswhen the measurement-limit Hamiltonian is the onlyinteraction present. Subsec. C 2 considers additionalself-evolution in the environment. Subsec. C 3 considersenvironment-environment interactions (without self-evolution), while Subsec. C 4 considers the combined self-evolutions and environment-environrment interactions.Finally, Subsec. C 5 considers the effect of adding a moregeneral global interaction. 1. Measurement limit Hamiltonian We first consider the case when only the centralinteraction is present, as described in the main text. Weinvestigate how the system-to-environment informationtransfer varies given different initial states (Cases 1.1to 1.5), and we see how the SBS form is preservedwhen we transform from the reference frame of the1external observer, C , to the reference frame of the firstenvironment, E .Note that this interaction Hamiltonian acts only tilltime t = 1 , and there is no further evolution afterwards.When there is only the central interaction, we find thatin frame C , all probabilities remain unchanged, and thereis strong independence at all times, except for Case 1.3(where the initial environment states are entangled). Case 1.1 - information broadcasting with pure environments We start with a scenario when the initial state isgiven by Eq. (72), i.e. when both environments arepure in state | (cid:105) , and thus has the largest capacityto obtain information about the central system S . Inboth reference frames C and E , the states are fullySBS—though note that in the frame of E , this SBS is degenerated and thus trivial, since it has only one non-zero probability, p ( E )0 = 1 at time t = 0 . External laboratory perspective. The numerical calcu-lations showed that in frame C , all conditional environ-ments states, E and E , are initially indistinguishable.As the system interacts with the environments, thefidelity of all conditional states of environments decreasesmonotonically to at the time t = 1 , meaning that theconditional states are now distinguishable. Since bothenvironments are symmetric, the fidelity is identical forboth of them. First environment’s perspective. Fig. 6 shows how allprobabilities different from p converge monotonically to until time t = 1 , corresponding to a trivial objectivity.The plot of conditional environment-environmentmutual information I ( E ) mean is show in Fig. 8 in themain text, where a high value of I ( E ) mean denotes lack ofstrong independence. In frame E , the subsystems C ( E ) and E ( E )2 initially gain correlations amongst themselvesup to time t = 0 . , before decaying back to strong-independent state by time t = 1 .The fidelity of individual conditional states of C remains large, but less than . Despite this, the upperbound Eq. (23) to the distinguishability error is small,and is zero at the time moment t = 1 . The fidelity ofindividual conditional states of E remains small, butgreater than . Comparing the upper bounds Eq. (23)over both subsystems C and E we see that the bound isslightly smaller for the latter. Case 1.2 - information broadcasting with slightly blurredenvironments The second case we consider deals with environmentalstates that are not pure, but has a small admixture ofpositions other than the dominant one, given in Eq. (73). External laboratory perspective. In this case the finalfidelity B ( C ) E is non-zero between the states close to each other, and near to zero for the others. For example atthe time t = 0 . when the information propagationprocess is almost done B ( C ) E (0 , ≈ . whereas for B ( C ) E (0 , ≈ . and B ( C ) E (0 , ≈ . . First environment’s perspective. The probabilitydistribution tends towards distribution corresponding tothe initial distribution of the environment, i.e. p = 0 . , p = p D − = 0 . , c.f. Eq. (73). The exact shape of thefunction is shown in Fig. 6.Again, mutual information is slightly higher for thecentral system’s state . The value of I mean is again thelargest at the time t = 0 . and converges to at the timemoment t = 1 for all Cases 1. The plot of I ( E ) mean is shownin Fig. 8.The fidelity of conditional states of both C and E remain non-zero all the time, but fidelity in the case of E is smaller. At the time moment t = 1 , fidelity ofstates , and D − is equal to . The average fidelityfor C is at the time . equal to . , and for E isequal to . . Note that for the time moment t = 1 the probabilities different than , and D − are equalto , and thus ρ ( E ) C | i and ρ ( E ) E | i for other values of i areundefined. Case 1.3 - information broadcasting with maximallyentangled environments The Case 1.3 investigates the initial joint state withmaximally entangled environments, given in Eq. (76). External laboratory perspective. Here, the mutualinformation is constant in time, and equal to . , suchas it results from the maximally entangled state in thedimension under consideration, D = 12 . The fidelityof conditional states is also constant in time and equalto between all states. This shows that there is noinformation transfer in this scenario. First environment’s perspective. All probabilities p ( E ) i are identical, equal to D ≈ . and constant overtime. The fidelity all the time equal to between allconditional states.The mutual information is identical for all centralsystem’s states. At the time moments t = 0 , , themutual information is equal to , while the maximumof . is attained at the time moment t = 0 . . We notethat I ( E ) mean behaves similarly like in the Case 1.1, but hasdifferent values. The plot of I ( E ) mean is shown in Fig. 8. Case 1.4 - information broadcasting with first environmentmaximally mixed In this case the first environment E is initiallymaximally mixed, and the second one pure. One mayexpect that there is no information transfer to E .2 External laboratory perspective. As expected, condi-tional states on E , { ρ ( C ) E | i } are completely indistinguish-able. On the other hand, the conditional states on E are fully distinguishable at the time moment t = 1 . First environment’s perspective. From the point ofview of E , all probabilities p ( E ) i are identical andconstant over time. This supports the conclusion thatfrom this perspective no knowledge regarding the centralsystem S is obtained. The mutual information is identicalfor all states of the central system; it is initially quitelarge ( . at the time t = 0 ), but monotonicallydecreasing to at the time moment t = 1 . The plotof I ( E ) mean is shown in Fig. 8.The conditional states of C are completely indistin-guishable. The conditional states of E are initiallyindistinguishable, but monotonically tends to the fulldistinguishability. Case 1.5 - information broadcasting with secondenvironment maximally mixed This case is similar to the previous one, where one ofthe environments is initially maximally mixed, but thistime, the second environment, E is initially maximalymixed, as given in Eq. (75b). External laboratory perspective. As implicated by thesymmetry of the scenario, fidelity is identical in values asin the Case 1.4 from C ’s reference frame, with swappedenvironments. First environment’s perspective. In this case allprobabilities { p ( E ) i } are initially identical, and tend to for p and for the others at the time moment t = 1 .Furthermore, the values of probabilities are the same asin the Case 1.1 (from E ’s point of view). This meansthat the different state of the second environment E didnot influence the perceptions of E .The mutual information is all the time for all centralsystem’s states. The fidelity of all conditional statesremain quite high for both C and E ( . and ,respectively), nonetheless the error probability upperbound decreases to for both C and E . This is causednot by the distinguishability of conditional states but bythe degeneracy of the SBS when only one probability isnon-zero. 2. Self-evolution of environments In the previous subsection, we dealt with the situationwhen the only time evolution was caused by theinteraction of the environments with the central system.In the second group of cases we add a self-evolutionHamiltonian to each environment. As described in themain text, the self-evolution Hamiltonians allows forjumps between neighbouring states in the Case 2.1, and for random jumps with random rates in the Case 2.2. Forboth cases the initial joint state ρ mpp is given in Eq. (72). Case 2.1 - self-evolution of environments External laboratory perspective. From symmetry itfollows both environments E and E are identical. Thevalue of mean mutual information, I ( C ) mean = 0 for allmoments of time. The fidelities decreases monotonicallyfrom the value at the time moment t = 0 to the value at the time t = 1 . Afterwards the fidelities remainconstantly . The last observation is a direct consequenceof the fact that the fidelity is invariant under unitaryrotations, such like the self-rotations of the environments. First environment’s perspective. Initially in the E reference frame, all probabilities { p ( E ) i } are identicaland equal to D , and tend to for p ( E )0 at the timemoment t = 1 . However, for the times of order of , theprobabilities adjacent to p ( E )0 ( i.e. { p ( E )1 } and { p ( E ) D − } )begin to be non-negligible. At the times of order ,each probability becomes significant, and p ( E )0 drops tothe value . (and will revive). We can see this in Fig. 7.With long-term averaging all probabilities are the same.The conditional mutual information, H (cid:16) ρ ( C ) E | i (cid:17) + H (cid:16) ρ ( C ) E | i (cid:17) − H (cid:16) ρ ( C ) E E | i (cid:17) , (C1)is the largest for the state i = 0 of the central system.The mean mutual information I ( E ) mean is zero at thebeginning and at the time t = 1 , while its maximum of . occurs at time t = 0 . . After the time moment t ≥ , the error upper bound is constantly equal to . Thedifference between I ( E ) mean in this case and in the Case 1.1do not exceed . (cf. Fig. 8).The fidelities of C are initially equal to . The upperbound of the guessing probability in Eq. (23) drops toa small value of . at the time moments close to ,but for large times it is high and becomes trivial at thetime t ≈ . On the other hand, for E fidelities are alsoinitially equal to , but afterwards they fall quite quicklyto the value of at the time moments close to andremain equal to for all time moments after . We notethat before time moment most of the probabilities { p ( E ) i } are close to and their fidelity does not influencethe upper bound, but after this moment their fidelitiesare well defined and equal to . Case 2.2 - random self-evolution of environments Since in this case the jumps are not distance-dependent, it is expected that locality-related phenom-ena does not occur in this case, in contrast to the previousCase 2.1.3 External laboratory perspective. In the C ’s perspec-tive, the mutual information is always equal to . Thefidelities of both E and E are very similar (up torandom fluctuations), and are at a similar level as Case2.1 (from perspective C ), but does not converge to at the time t = 1 , but instead to low values between . − . (depending on the choice of the pair ofconditional states). For the same reason as in theprevious case, they are constant after the time t ≥ . First environment’s perspective. From the point ofview of E all the probabilities initially identical. p ( E )0 tend to for the time t = 1 . Comparing to the Case2.1 (from E perspective), other probabilities becomesnon-negligible faster, already at the time of order ,see Fig. 7. Another difference is that, as expected, allof these probabilities behave similarly, unlike the localspread to neighbours in the Case 2.1.The mean mutual information starts with the value , with a local maximum . at the time t = 0 . ,and drops back down to at the time moment t = 1 .Afterwards it increases and reaches saturation level ofabout . at the time t ≈ —see Fig. 9 in the maintext.Subsystem C ’s fidelities remains very high all the time,although for the time moments close to the upperbound on the probability of error in Eq. (23) is low( . ) but it increases quickly (and becomes trivial forthe time t = 8 ). For subsystem E , the fidelities anderror probability upper bound drops to for the timeclose to and begin to increase from time t = 10 , tobecome trivial at the time of order . 3. Mutual information transfer betweenenvironments This group of cases investigates interactions betweenenvironments (without self-evolution). Cases 3.1 and 3.3have distance dependent interactions (as described in themain text), and Cases 3.2 and 3.4 have random rates ofjumps. We contrast two different initial states: one withboth E and E located at the same position, given inEq. (72) (Cases 3.1 and 3.2), and one with the positionof E shifted by half of the ring, given in Eq. (77) (Cases3.3 and 3.4). Case 3.1 - simple mutual information transfer External laboratory perspective. In Fig. 12, we seethat the mutual information is initially and graduallyincreases to significant values (above ) for the time oforder , then fluctuates, sometimes reaching even thevalue . .As expected from the symmetry, the fidelities areidentical between the two environments, and are thegreater the closer the states are. The upper bound inEq. (23) decreases to a value close to ( . at minimum) m u t u a l i n f o r m a t i o n i n C [ b i t s ] time3.13.23.33.44.14.25.15.2 0 0.025 0.05 0.075 0 1 Figure 12. (Color on-line) The values of mean mutualinformation between environments I ( C ) mean as a function of timefor cases from groups 3, 4 and 5 (cf. Tab. IV.) p r o b a b ili t y i n E f o r . timep p p p p Figure 13. (Color on-line) Probabilities { p ( E ) i } i =0 , , , , forthe Case 3.1. p ( E )0 is dominating at the time t = 1 , afterwardsthe probabilities neighbouring at the -dimensional ring, i.e. p ( E )1 and p ( E )11 starts to be non-negligible. After some timealso more distant probabilities, p ( E )2 and p ( E )10 , are increasing. at the time moments close to , afterwards it increasesquickly and becomes trivial at the time close to andabove. First environment’s perspective. Here all probabili-ties initially identical, for the moment p ( E )0 = 1 .Afterwards at the time of order the share of p ( E )1 and p ( E ) D − (the locally neighbouring states) graduallyincreases. At the time of the order some share of p ( E )2 and p ( E ) D − (more distant, but still close states) becomesimportant. The remaining probabilities even for longtimes (of order ) stay close to . Detailed plotsof these probabilities is shown in Fig. 7 and Fig. 13.Mean mutual information starts with the value ,reaches its maximum of . at the time t = 0 . , dropsagain to for the time t ≈ and afterwards remains .The I ( E ) mean in this case and in Case 1.1 are very similar,and never differ more than . (cf. Fig. 8).The upper bound on the probability of error Eq. (23)drops to a value close to for the time of order ( . . for C and E , respectively). For the subsystem C it increases quickly, and becomes trivial at the time oforder . For the subsystem E remains low even forlarge times (average . for a time interval of up to ). Case 3.2 - mutual information transfer with randomHamiltonian External laboratory perspective. Mean mutual infor-mation is initially and gradually increases to significantvalues for the time of order , then reaches saturationand maintains a high constant level of about . startingfrom time about (and does not drop even for the timeof order ) (see Fig. 12).The error probability upper bound is very high, andthus upper bound is only useful for times close to (whenit is equal to . ), and the bound becomes trivial aftertime t = 3 . First environment’s perspective. At time t = 0 , allprobabilities p ( E ) i are equal. The probability p ( E )0 dominates for times close to , then all other probabilitiesincrease to a similar degree. After time of order , allprobabilities become very similar (see Fig. 7).The mean mutual information is initially , themaximum of . is attained at the time t = 0 . , returnsclose to for at times close to , gradually increases tosaturation level of . from time of order and does notdrop (see Fig. 9).For C ’s conditional states the upper bound for errorprobability Eq. (23) is very high all the time (giving anon-trivial bound only for the time between . and ),and minimal (equal to . ) at the time t = 1 . For E ’sconditional states the bound is also low ( . ) only nearthe moment , and becomes trivial after time t = 9 . Thefidelity states of both subsystems is high all the time. Case 3.3 - mutual information transfer with shifted state In this case the state of the second environment E is shifted by (cid:100) D/ (cid:101) − , equal to for D = 12 ,given in Eq. (77). This particular case illustrates thespatial dependence in Hamiltonian. We also cnsider anadditional case of the state shifted in opposite directionon the D -dimensional ring: ρ mix S ⊗ | (cid:105)(cid:104) | E ⊗ | (cid:105)(cid:104) | E . (C2) External laboratory perspective. Similarly to Case 3.1( C ), the mean mutual information is initially andgradually increases. At the time about its valuereaches values above and then fluctuates, but this timeit reaches large values (between and ) later, for a timeof order , and for the time t < is negligible. Thelong time fluctuations are significant, the mean value overtime period till is . with variation . .The function of I ( C ) mean is shown at Fig. 12. for the time moments starting about to , fidelitiesand the upper bound in Eq. (23) are small. From timeabout the bound becomes trivial. Fidelities havesimilar values for both subsystems E and E .Now, when the state Eq. (C2) is considered, the meanmutual information exceeds faster, at the time momentabout , and exceeds for at time t = 200 . Its long timeaverage is . with variation . . For both statesEqs. (77) and (C2), the conditional mutual informationof Eq. (C1) is almost the same for all states of the centralsystem S in all moments of time, with variance at anyfixed time moment less than . . First environment’s perspective. For the stateEq. (77) the probability p ( E )0 dominates betweentime moments and (it is above . for thetime in range . and ), as it is shown at Fig. 7;afterwards the probabilities { p ( E ) i } i =7 begin to increase.They exceed the value . at the times t = 9 , , , , for p ( E )11 , p ( E )10 , p ( E )9 , p ( E )8 and p ( E )7 ,respectively. The probabilities { p ( E ) i } i =1 remainnegligible during the whole considered period at least tilltime moment . The average for large times (upto ) is the highest for p ( E )0 and p ( E )7 ( . ),slightly smaller for p ( E )8 and p ( E )11 ( . ), and evensmaller for p ( E )9 and p ( E )10 ( . ).In comparison, for the state Eq. (C2), p ( E )0 is abovethe value . for the time range . − and dominatestill time t ≈ . The probabilities that start to be non-negligible are { p ( E ) i } i =1 , (above . at the time t =10 ), then gradually p ( E )2 (at the time t = 40 ), p ( E )3 (at the time t = 90 ) and p ( E )4 (at the time t = 200 ). { p ( E ) i } i =1 , . Again, the long time averages group intothree categories: of averages . , . and . forstates in sets { p ( E ) i } i =0 , , { p ( E ) i } i =4 , and { p ( E ) i } i =1 , ,respectively.We note that the distance between initial states ofthe environments in Eq. (77) was (towards increasingnumber ), whereas for the state of Eq. (C2) it was (towards decreasing number − mod ). From thepoint of view of E we see that the distance betweenthe peak probability of and the other peak was thesame, viz. in the former case it was (towards decreasingnumber − mod ), and in the latter (towardsincreasing number )The mean mutual information I ( E ) mean for both initialstates is initially , with maximum . at time t = 0 . ,then dropping to for the time t ≈ and remaining thereafter. The difference between I ( E ) mean in this case andin Case 1.1 do not exceed . (cf. Fig. 8).The upper bound on the error Eq. (23) is small bothfor subsystems C and E for both states. For the stateEq. (77) and subsystem C , the bound is . at thetime t = 1 , and remains non-trivial till time . Forsubsystem E it is equal to . and remains small, withlong time average . . For the state Eq. (C2) the values5at the time t = 1 are . and . , for C and E ,respectively, with the former non-trivial till time t = 50 ,and long time average of the latter equal to . . Case 3.4 - mutual information transfer with shifted state andrandom Hamiltonian Here, we consider when the second environment E state shifted relatively to E , but with environmental-interaction Hamiltonian that does not depend ondistances between environments, and thus we do notexpect this scenario to differ significantly from the Case3.2, where E and E were at the same position. External laboratory perspective. Indeed, the directcalculation shows that the behaviour is very much like3.2, both from C ’s point of view. Again, the meanmutual information initially is equal to and graduallyincreases to significant values (above ) starting for thetime t = 10 , then reaches saturation of about . andmaintains a high constant level starting from time t = 30 (see Fig. 12).As in 3.2 the error probability upper bound is veryhigh, only makes sense for the time close to (when thevalue of the bound is . ), and once again it becomestrivial after time t = 3 . First environment’s perspective. Again, at the time all probabilities p ( E ) i are equal, and p ( E )0 dominatesfor the time close to , finally all probabilities fluctuateat the same level with variance below . after timemoment t = 50 (see Fig. 7).Similarly to Case 3.2, here the mean mutualinformation starts at , increases till time moment t = 0 . (reaching the value . , the same as in 3.2), drops to alow level ( . ) for times close to , reaches high valuesfrom time moment t = 20 , and attains saturation ofabout . at the time t ≈ . The values of I ( E ) mean inthis case do not differ from those in the Case 3.2 by morethan . till time moment t = 1 , and than . overtill time t = 1000000 .Exactly as in Case 3.2, for conditional states of C havea very high upper bound for error probability for all thetimes. For E , the upper bound it is low ( . ) onlynear the time t = 1 , and fidelities of conditional states ofboth subsystems high all the time. 4. Self-evolution with interactions Now, we analyse cases when both self-evolution andenvironmental interactions are present. In Case 4.1 theHamiltonians described in the main text (paragraphsfollowing Eq. (71)), and in Case 4.2 we have randomHamiltonians. Case 4.1 - simple self-evolution with mutual informationtransfer External laboratory perspective. In Fig. 12, we seethat the mean mutual information is initially and,at first, increases very slowly (it stays at a level . until time t = 3 ) before reaching saturation at thetime t = 500 and staying at a constantly high valueof . with slight fluctuations. Thus, in the long timeregime, there is no strong independence between theenvironments.The upper bounds of Eq. (23) for both E and E arelow only for the time near t ≈ , but after time t = 80 becomes trivial. The fidelities are relatively low untiltime moment t = 10 . First environment’s perspective. The dynamics of p ( E )0 is shown in Fig. 7. Initially all probabilities areidentical. For the time t = 1 we have p ( E )0 = 1 , then theprobabilities of neighbouring locations begin to increase, viz. p ( E )1 and p ( E )11 exceed . at the time t = 3 , andbegin to be significant (above . ) for the time t = 10 .For times of order , all probabilities are significant,and the long time average is the same for all states.The mean mutual information is initially equal to ,with maximum of . for the time t = 0 . , afterwardsit falls to the value for the time moment t = 1 andremains below . at least till time t = 1000000 . Thedifference between I ( E ) mean in this case and in Case 1.1 donot exceed . (cf. Fig. 8). Thus, in this frame, there is strong independence.The upper bound in Eq. (23) for C is small only forthe time close to , for E it remains low until time t =10 ; they become trivial at the time close to and ,respectively. The fidelities are in both cases large, butfor C much larger than for E . Case 4.2 - random self-evolution with mutual informationtransfer External laboratory perspective. In Fig. 12, we seethat the mean mutual information increases from veryquickly reaching . for the time moment t = 1 , andsaturates at ≈ . at the time t ≈ and remainsvirtually constant, without significant fluctuations, thusthere is no strong independence.The upper bound in Eq. (23) for both C and E are ata very similar high level, and become trivial at the time t ≈ . The fidelities are small only for a time close to .Both environments behave in almost the same manner,up to random fluctuations. First environment’s perspective. The values of { p ( E ) i } D − i =0 are initially identical. For the time t = 1 we have p ( E )0 ≈ . , and quite quickly the remainingprobabilities increase evenly. Starting from thetime t = 50 , all probabilities fluctuate at the same level.Fig. 7 contains the evolution of p ( E )0 .6In Fig. 9, the mean mutual information is initially , with maximum of . for the time t = 0 . , anddecreases to the value . for the time moment .Afterwards it increases and reaches the saturation valueof . by time t = 40 and remains at this level. Hence,there is no strong independence.The upper bound on the probability of error is non-trivial only near time t = 1 (equal to . for C and . for E ), very high fidelities for both C and E . 5. Slightly disturbed global evolution In this last group of analysed cases, we modify thecases from the group 4 by adding a distortion with arandom Hamiltonian over the tripartite state of C , E and E , as discussed above. This is the only group ofcases where the probabilities { p ( C ) i } D − i =0 evolve in timeand the decoherence factors, Γ ( C ) := D − (cid:88) i (cid:54) = j =0 (cid:12)(cid:12)(cid:12) (cid:104) i | S Tr E E (cid:16) ρ ( C ) SE E (cid:17) | j (cid:105) S (cid:12)(cid:12)(cid:12) , (C3a) Γ ( E ) := D − (cid:88) i (cid:54) = j =0 (cid:12)(cid:12)(cid:12) (cid:104) i | S Tr CE (cid:16) ρ ( E ) SCE (cid:17) | j (cid:105) S (cid:12)(cid:12)(cid:12) , (C3b)have to be taken into account. Case 5.1 - global evolution with simple self-evolution andmutual information transfer External laboratory perspective. The fluctuationsfrom the uniform probability distribution of { p ( C ) i } D − i =0 are only visible from time t = 50 , but nevertheless thestandard deviation of the probabilities at any given time(at least till time t = 1000000 ) does not exceed . .The mean mutual information between the conditionalenvironments increases from , by the value . fortime t = 1 , to the value of saturation equal to about . , and after time t = 50 it fluctuates around this value,see Fig. 12. Hence, the environments do not have strongindependence.The upper bound in Eq. (23) is large for both E and E , the bound is reasonable only close to time t = 1 , andbecomes trivial before time t = 20 . Fidelities remainslow till time around t = 10 . The long time average of Γ ( E ) is . , its value at the time t = 1 is . . First environment’s perspective. In Fig. 7, thedistribution of { p ( E ) i } D − i =0 is initially uniform, and bytime t = 1 it becomes dominated by p ( E )0 ≈ . . Theadjacent probabilities p ( E )1 and p ( E ) D − begin to increaseand become significant for the time t ≈ . From time t ≈ the distribution is again uniform with almost nofluctuation (the standard deviation is . ).The mean mutual information starts from the value , the local maximum of . is attained for the time t = 0 . , then it decreases to . at the time t = 1 , andafterwards it reaches saturation about . for the time t ≈ and remains at this level. The plot of I ( E ) mean isshown in Fig. 9. Hence, in the long time regime, there isno strong independence.The error probability upper bound is low only aroundtime t = 1 ( . for C and . for E ), and thefidelities are high for both environments, meaning thatthe conditional states are not very distinguishable. Thelong time average of Γ ( E ) is . , its value at the timemoment is . . Case 5.2 - global evolution with random self-evolution andmutual information transfer External laboratory perspective. The fluctuationsfrom the uniform distribution of { p ( C ) i } D − i =0 are noticeableonly from time t ≈ onwards, nonetheless the standarddeviation of the probabilities at any fixed time momentdoes not exceed . .The mean mutual information gradually increases from to a value of . at the time t = 1 , and to . at time t = 40 , after which it falls to levels close to . by time t ≈ and remains at this level withessentially no fluctuation (at least till time t = 1000000 )(see Fig. 12). Hence, in the long time regime, there is nostrong independence here.The upper bound on the probability of error is smallonly around time t ≈ ( . for C and . for E )and trivial for times t > , although the fidelities issmall ( . at the time t = 1 , . at the time t = 3 ,below . till time t = 20 ) and almost identical for bothsubsystems. The long time average for Γ ( C ) is . ,while its value at time t = 1 is . . First environment’s perspective. 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