From clocks to cloners: Catalytic transformations under covariant operations and recoverability
aa r X i v : . [ qu a n t - ph ] A ug From clocks to cloners: Catalytic transformations under covariant operations andrecoverability
Iman Marvian and Seth Lloyd
1, 2 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (Dated: November 6, 2018)There are various physical scenarios in which one can only implement operations with a certainsymmetry. Under such restriction, a system in a symmetry-breaking state can be used as a catalyst,e.g. to prepare another system in a desired symmetry-breaking state. This sort of (approximate)catalytic state transformations are relevant in the context of (i) state preparation using a bounded-size quantum clock or reference frame, where the clock or reference frame acts as a catalyst, (ii)quantum thermodynamics, where again a clock can be used as a catalyst to prepare states whichcontain coherence with respect to the system Hamiltonian, and (iii) cloning of unknown quantumstates, where the given copies of state can be interpreted as a catalyst for preparing the new copies.Using a recent result of Fawzi and Renner on approximate recoverability, we show that the achievableaccuracy in this kind of catalytic transformations can be determined by a single function, namelythe relative entropy of asymmetry, which is equal to the difference between the entropy of stateand its symmetrized version: if the desired state transition does not require a large increase of thisquantity, then it can be implemented with high fidelity using only symmetric operations. Our lowerbound on the achievable fidelity is tight in the case of cloners, and can be achieved using the Petzrecovery map, which interestingly turns out to be the optimal cloning map found by Werner.
PACS numbers:
I. INTRODUCTION
A wide range of seemingly different physical and in-formation theoretic problems can be understood and for-malized as special cases of the following abstract problem:we want to transform a known input state to a desiredoutput state, or a state close to it, under the restrictionthat we can only implement operations which respect acertain symmetry. As a result of this restriction on theallowed operations, many state transformations are notpossible. The simplest example is the impossibility oftransforming a symmetric state to a symmetry-breakingone, which is reminiscent of the Curie’s principle. Therelevant property of states that determines whether suchtransformations are possible or not, can be interpreted asthe symmetry-breaking, or the asymmetry of state rela-tive to the symmetry under consideration. The resourcetheory of asymmetry is a framework for classifying andquantifying this property, and answering this sort of ques-tions about state transitions under symmetric operations[1–4].In this paper we consider a special case of this generalproblem, namely catalytic transformations under sym-metric operations. More specifically, we consider the casewhere a system in a symmetry-breaking state is used asa catalyst to transform another system in an initial sym-metric state to a desired asymmetric state, such that thecatalyst remains (approximately) unchanged. We arguethat this scenario naturally arises in the context of (i)state preparation using a bounded size quantum clockor quantum reference frame [5], (ii) quantum thermody-namics [6], and (iii) cloning of unknown quantum states[7]. Then, using a recent result of Fawzi and Renner [8] on approximate recoverability, we show that the achiev-able accuracy in this kind of catalytic transformationscan be determined by a single measure of asymmetry,namely the relative entropy of asymmetry , which turnsout to be equal to the the entropy of the symmetrizedversion of state minus the entropy of the original state.If this quantity does not increase considerably in the de-sired state transformation then the transformation canbe implemented with high fidelity using only symmetricoperations. As we explain later, this is somewhat sur-prising, because for general state transformations undersymmetric operations, just by looking to a single measureof asymmetry one cannot guarantee the possibility of astate transition, or an approximate version of it. We dis-cuss some applications of this result in various contexts.In particular, we show that in the case of cloning of quan-tum states our lower bound on the achievable fidelity canbe achieved using the Petz recovery map [9, 10], whichinterestingly turns out to be the optimal cloning maporiginally found by Werner [7].The article is organized as follows. In Sec. II we firstprovide a formal presentation of the problem of catalytictransformations under symmetric operations, and thenwe show examples of different physical scenarios whichall can be formalized as special examples of this problem.Then, in Sec. III we review the concept of measures ofasymmetry, and in particular, we discuss about the rela-tive entropy of asymmetry, which is the measure of asym-metry we are concerned with in this paper. In Sec. IV Aand Sec. IV B we present upper and lower bounds onachievable accuracy of catalytic transformations in termsof the increase of the relative entropy of asymmetry inthe desired transformation. In Sec. V we present appli-cations of these bounds in the context of state prepara-tion using bounded size clocks, and cloning of unknownquantum states. Finally, in Sec. VI we present a gener-alization of our lower bound in Sec IV B, which shows ifduring a symmetric process the relative entropy of asym-metry does not drop considerably, then the process canbe approximately inverted using a symmetric operation.
II. CATALYTIC TRANSFORMATIONS
In this paper we study (inexact) catalytic state trans-formations in the following form τ R ⊗ ρ S sym Covariant −−−−−−→ τ R ⊗ σ S . (1)Here τ R and ρ S sym are initial states of systems R and S ,which we sometimes refer to them as the reference frame and system respectively, and τ R ⊗ σ S is the desired jointfinal state of these systems. We assume ρ S sym is invariantunder the action of a certain symmetry. Then, by apply-ing a time evolution which also has this symmetry, or is covariant with respect to this symmetry, we try to mapthe initial state τ R ⊗ ρ S sym to a state close to the desiredstate τ R ⊗ σ S .The most general physical time evolution that canbe implemented on a quantum system (with probabil-ity one) can be described by a trace-preserving com-pletely positive linear map, also known as quantum chan-nel . This includes unitary transformations, measure-ments, and adding systems or discarding subsystems. Weconsider quantum channels which are covariant with re-spect to an arbitrary symmetry. Let G be the groupcorresponding to the symmetry under consideration, U Xg denote the unitary representation of the group element g ∈ G on system X , and U Xg ( · ) = U Xg ( · ) U Xg † be the cor-responding super-operator. Then, under the action of thegroup element g ∈ G state ρ X of system X transformsto U Xg ( ρ X ). The representation of symmetry on the jointsystem X and X is given by U X X g = U X g ⊗ U X g . Aswe see in the following examples, the form of represen-tation is often dictated by the physical interpretation ofthe symmetry.Then, state ρ X is symmetric with respect to the sym-metry under consideration, or G-invariant , if for all g ∈ G , U Xg ( ρ X ) = ρ X . Similarly, a quantum channel E X → Y from system X to system Y is called G-covariant ,or symmetric, if it commutes with the action of group,i.e. E X → Y ◦ U Xg = U Yg ◦ E X → Y , ∀ g ∈ G . (2)Note that to specify the set of G-covariant channels fromsystem X to system Y we need to know the representa-tion of symmetry G on both spaces. We often suppressthe superscript corresponding to the label of the system,when there is no chance of confusion. Also, since group G is often clear from the context, we sometimes use theterm covariant or symmetric instead of G -covariant. We are interested to determine whether using only co-variant operations, one can implement the desired statetransition in Eq.(1), or an approximate version of it. Butsince, by assumption state ρ S sym in the input is invariantunder the action of the group, i.e. U Sg ( ρ S sym ) = ρ S sym forall g ∈ G , it turns out that the answer to this ques-tion is independent of this state. This is because for anystate τ R and symmetric state ρ S sym , there exists covariantchannels which transform state τ R to τ R ⊗ ρ S sym , and viceversa. Therefore, the transformation in Eq.(1) can be im-plemented with covariant channels with certain error ifand only if the state transition τ R Covariant −−−−−−→ τ R ⊗ σ S (3)can be implemented with the same error. Hence, in thefollowing we sometimes consider this more concise formof the problem.Unless state σ S in the output is also a symmetric state,in general, using only covariant operations the state tran-sition τ R → τ R ⊗ σ S cannot be implemented exactly.Roughly speaking, this is because covariant channels can-not generate asymmetry (relative to the symmetry underconsideration), and therefore their output state cannotcontain more asymmetry than their input. Later we seea more precise and quantitative version of this statementin terms of measures of asymmetry .To explain the physical relevance of the catalytic trans-formations described in Eqs.(1, 3), in the following weconsider some illustrative examples. A. Clocks
Catalytic transformations described above provide anatural framework for understating how clocks are usedin a state preparation process. Suppose we want to pre-pare the system S with Hamiltonian H S in state σ S which contains coherence in the energy eigenbasis. Anysuch state is time-dependent, and therefore the abovestatement about the state of system S is ambiguous, un-less we say σ S is the state of system at certain time t withrespect to a clock R . The clock could be, for instance,a Harmonic Oscillator oscillating at a certain frequency,or a free particle moving on a line. Then, to prepare thesystem S in state σ S we need to directly or indirectlyinteract it with the clock R . But, there is a restrictionon the set of operations that we can implement on thesystem and clock: If the reference for time is defined justby the clock R and the rest of systems do not have anyinformation about how the time reference is defined, thenthe only operations we can perform on the system S andclock R are those which do not explicitly or implicitlydepend on the time reference. More precisely, these areoperations which are covariant with respect to the groupof time translations, i.e. they satisfy ∀ t ∈ R : E ( e − iH tot t ( · ) e iH tot t ) = e − iH tot t E ( · ) e iH tot t , (4)where H tot = H R + H S is the total Hamiltonian of systemand clock, and we have suppressed RS → RS superscript(We assume ~ = 1 throughout the paper).Therefore, in this context Eq.(1) has the following in-terpretation: to prepare the system S in state σ S wecouple it to a clock R , such that at the moment wherethe clock is in state τ R the system is in state σ S . Notethat ideally the clock should play the role of a catalyst,i.e. it should remain unaffected in the process.The restriction to the set of channels which are in-variant under time translations also arises in the con-text of quantum thermodynamics, where the only free unitaries and states are, respectively, energy-conservingunitaries and thermal states [6, 11, 12]. Since these uni-taries and states are all invariant under time-translations,it follows that the only free quantum operations in thisresource theory are those which are invariant under time-translations, i.e. satisfy Eq.(4). Again, in this contexta clock, i.e. a system in a state that breaks the time-translation symmetry can be used as an approximate cat-alyst to perform certain transformations [13]. B. Reference Frames
Similar to clocks, which are reference frames for time,other reference frames for other physical degrees of free-dom can be interpreted as catalysts. For instance, whenwe use a quantum gyroscope to prepare another systemin a direction defined by the gyroscope, we can only im-plement transformations which can be achieved usingisotropic operations, i.e. operations which are covari-ant with respect to the group of rotations G = SO(3).Therefore, the task of preparing system S in a directiondefined by the gyroscope R can be formally phrased as acatalytic transformation in the form of Eq.(3). Anotherexample, which is relevant in the context of quantum op-tics, is preparing states using a phase reference for thephase conjugate to the photon number operator in a par-ticular optical mode. Here, the relevant symmetry is thegroup of phase shifts, which is isomorphic to U(1) (See[5] for an overview of quantum reference frames). Note that in the resource theory of thermodynamics, as de-fined in [6], systems could be in arbitrary non-equilibrium (time-dependent) states. Then, some state transitions are forbidden inthis resource theory, simply because, e.g. the initial state is time-independent and the desired final state is time-dependent, andto prepare a time-dependent state one needs a clock, which is notallowed in this resource theory. Indeed, this resource theory canbe thought as the combination of two different resource theories:A resource theory for time-translation asymmetry (or unspeak-able coherence [14]), and a resource theory for energy and purity,where in the second resource theory all states are incoherent inthe energy eigenbasis. The latter resource theory seems to bemore related to the standard thermodynamics, where all statesare assumed to be close to equilibrium . C. Cloners
Another motivation for considering the catalytic trans-formations in Eq.(3) comes from the study of quantumcloning machines, or quantum cloners : Suppose we aregiven n ≥ ψ ∈ C d , andwe are interested to generate n + k copies of this state.The no-cloning theorem implies that for any k > ψ ⊗ n → ψ ⊗ ( n + k ) can be implemented onlyapproximately [15].Therefore, to compare the performance of differentcloners, we need to consider a figure of merit, such asfidelity of cloning. For most applications, we expect thata good cloner should act equally well on all pure states,and hence the figure of merit should care equally aboutall pure states. This natural requirement implies that theoptimal cloner E n → n + k from n copies to n + k copies canalways be chosen to satisfy the covariance condition E n → n + k ( U ⊗ n ( · ) U †⊗ n ) = U ⊗ ( n + k ) E n → n + k ( · ) U †⊗ ( n + k ) , (5)for all U ∈ U( d ) [7]. Note that if a cloner satisfiesthis symmetry then its performance on all pure statesis uniquely specified by its performance on one purestate, because all pure states can be transformed to eachother by unitary transformations. Therefore, assumingthis symmetry holds, to evaluate the performance of thecloner on an unknown pure state we can just focus onits action on one particular known pure state ψ . Then,in the desired transformation ψ ⊗ n → ψ ⊗ ( n + k ) , we caninterpret the given copies of state ψ as catalyst. In otherwords, choosing τ R = ψ ⊗ n and σ S = ψ ⊗ k in Eq.(3), thisequation describes the action of an ideal cloner. III. RELATIVE ENTROPY OF ASYMMETRY
A measure of asymmetry is a function that quantifiesthe amount of symmetry-breaking of states relative to agiven symmetry [4]. Formally, a function f from states toreal numbers is a measure of asymmetry with respect to asymmetry, if (i) it vanishes for all symmetric states, and(ii) it is non-increasing under G-covariant operations, i.e. f ( E ( ρ )) ≤ f ( ρ ) for any state ρ and G-covariant channel E . Note that to define measures of asymmetry on a givenHilbert space, we need to know the representation of thesymmetry on that space.A well-studied measures of asymmetry is the rel-ative entropy of asymmetry , defined by Γ( ρ ) ≡ inf ω ∈ sym( G ) S ( ρ k ω ), where S ( ρ k ρ ) = Tr( ρ (log ρ − log ρ )) is the relative entropy, and the minimization isover all states which are invariant under group G , i.e. U g ( ω ) = ω , for all g ∈ G (Throughout the paper weassume the base of logarithm is 2). Roughly speaking,this function quantifies the distance between state ρ andthe set of symmetric states, in terms of relative entropy.As shown in [16], it turns out that the relative entropy ofasymmetry is equal to the the entropy of the symmetrizedversion of state minus the entropy of the original state,that isΓ( ρ ) ≡ inf ω ∈ sym( G ) S ( ρ k ω ) = S ( ρ kG ( ρ )) = S ( G ( ρ )) − S ( ρ ) , (6)where S ( ρ ) = − Tr( ρ log ρ ) is the von-Neuman entropy .Here, the superoperator G is the uniform twirling over thegroup G , which projects its input to a symmetric state,and in the case of finite groups is given by G ( · ) = 1 | G | X g ∈ G U g ( · ) , (7)where | G | is the order of group, and we have suppressedthe superscript corresponding to the label of system. Forcontinuous groups, such as U(1) or SO(3), the sum inEq.(7) is replaced by the integral over the group withuniform (Haar) measure. Also, for the group of transla-tions generated by a Hamiltonian H , or more generallyany other observable, the uniform twirling can be definedas G ( · ) = lim T →∞ T Z T − T dt e − iHt ( · ) e iHt = X n Π n ( · )Π n , (8)where { Π n } n are projectors to the eigen-subspaces of H .Therefore, in this case the uniform twirling is equal tothe dephasing map, that is the map that dephases itsinput relative to the eigenbasis of H . Note that theuniform twirling G is a special case of the resource de-stroying maps recently introduced in [19], which leaveresource-free states (in this case symmetric states) un-changed and erase the resource (in this case asymmetry)in all other states.In addition to being a measure of asymmetry, rela-tive entropy of asymmetry has some other useful prop-erties: (i) It is convex, i.e. Γ( pρ + (1 − p ) ρ ) ≤ p Γ( ρ ) + (1 − p )Γ( ρ ) for all 0 ≤ p ≤
1. (ii) The factthat the representation of a group element g ∈ G on thejoint system X and X is given by U X g ⊗ U X g impliesthat for any group G and state ω in a finite-dimensionalHilbert space, Γ( ω ⊗ n ) increases at most logarithmicallywith n . (iii) For a finite group G , Γ is bounded by log | G | . IV. ACHIEVABLE ACCURACY IN CATALYTICTRANSFORMATIONS
In this section we find lower and upper bounds on theachievable accuracy in the catalytic transformations inEqs.(1,3), in terms of the increase in the relative entropyof asymmetry in the ideal transformation. See also [17] for an earlier study of function Γ( ρ ) = S ( G ( ρ )) − S ( ρ ). In this special case, the relative entropy of asymmetry, is calledthe relative entropy of coherence by [18].
A. Upper bound on accuracy
Using covariant channels we can only implement statetransitions which do not require increase of the relativeentropy of asymmetry. Furthermore, if the asymmetry ofthe desired output state is much larger than the asym-metry of the input, then the transformation cannot beimplemented, even approximately. In the SupplementaryMaterial, using the Fannes-Audenaert inequality [20], weprove the following: Suppose we want to transform state ρ X of system X to a state close to state ω Y of system Y .Then, for any arbitrary G-covariant channel E X → Y fromsystem X to system Y , let ǫ = kE X → Y ( ρ X ) − ω Y k bethe trace distance between the actual output E X → Y ( ρ X )and the desired output ω Y , where k · k is the sum of thesingular values of the operator. Then, either ǫ ≥
1, or ǫ × log D Y + 2 H ( ǫ ≥ ∆Γ , (9)where ∆Γ = Γ( ω Y ) − Γ( ρ X ) is the increase of the relativeentropy of asymmetry in the desired transition, D Y is therank of G ( ω Y ), and H ( x ) = − x log x − (1 − x ) log(1 − x )is the binary entropy function. Using the bound ǫ +2 H ( ǫ ) ≤ √ ǫ for ǫ ∈ (0 , q kE X → Y ( ρ X ) − ω Y k ≥ ∆Γ3 log D Y . (10)Eqs.(9) and (10) can be translated to an upper boundon the fidelity of E X → Y ( ρ X ) and ω Y , where the fidelityis defined by F ( ρ , ρ ) ≡ k√ ρ √ ρ k , and satisfies 2(1 − F ( ρ , ρ )) ≤ k ρ − ρ k . Also, note that by choosingsystems X = R and Y = RS , we can apply these boundsto the special case of catalytic transformations in theform of Eqs.(1,3). B. Lower bound on accuracy
We saw that if a state transformation requires a largeincrease of the relative entropy of asymmetry, then itcannot be implemented with high fidelity using covari-ant channels. On the other hand, even the transitionsin which this quantity does not increase may also be for-bidden under covariant operations. We will see exampleswhere Γ( ρ X ) is much larger than Γ( ω Y ) and still the tran-sition ρ X → ω Y cannot be implemented with covariantchannels, even approximately. This should be expectedbecause different states can break a given symmetry indifferent ways, i.e. they may break some subgroups butnot the others. However, a single measure of asymme-try cannot see this difference (Note that this is not theonly reason that a transition might be forbidden. In-deed, even for (cyclic) groups of prime order, which donot have non-trivial subgroups, a single measure of asym-metry does not contain enough information to determineif a state transition is possible or not.).However, perhaps surprisingly, it turns out that in thecase of catalytic transitions described in Eq.(3), or equiv-alently Eq.(1), just by looking to a single measure ofasymmetry, namely the relative entropy of asymmetry,we can determine if the transition can be approximatelyimplemented with covariant operations. To show this,we use some recent results on approximate recoverabil-ity [8] to find a lower bound on the achievable fidelityof implementing the catalytic transformations in Eq.(1)and Eq.(3). See also Sec.VI on reversibility, for a moregeneral proof of this lower bound.Consider the tripartite stateΣ CRS = 1 | G | X g ∈ G | g ih g | C ⊗ U Rg ( τ R ) ⊗ U Sg ( σ S ) , (11)where C , or the classical background reference frame, isa system with the Hilbert space spanned by the orthog-onal states {| g i : g ∈ G } . In the case of continuousgroups such as U(1) or SO(3) we replace the sum withan integral over group with uniform (Haar) measure. LetΣ CR = Tr S (Σ CRS ) and Σ RS = Tr C (Σ CRS ) be the re-duced state of the joint systems CR and RS , respectively.Then, it can be easily shown that the quantum mutualinformation between system R (or S ) and C is equal toΓ( τ R ) (or Γ( σ S )). Furthermore, the conditional mutualinformation I ( S : C | R ) Σ = S (Σ CR ) + S (Σ RS ) − S (Σ CRS ) − S (Σ R ) , (12)turns out to be equal to I ( C : S | R ) Σ = Γ( τ R ⊗ σ S ) − Γ( τ R ) ≡ ∆Γ , (13)that is the difference between the relative entropy ofasymmetry for states τ R ⊗ σ S and τ R .According to a recent result of Fawzi and Renner [8](See also [21–23]) if the conditional mutual information I ( C : S | R ) Σ is small then state Σ CRS can be approxi-mately reconstructed from the reduced state Σ CR in thefollowing sense: there exists a quantum channel R R → RS which maps system R to systems RS such thatF( R R → RS (Σ CR ) , Σ CRS ) ≥ − I ( C : S | R ) Σ , (14)where we have suppressed the identity super-operatorwhich acts on system C . The recovery map which satis-fies Eq.(14) can be chosen to be a rotated version of thePetz recovery map [8, 21, 22].Then, the reconstructed state has the following form R R → RS (Σ CR ) = 1 | G | X g ∈ G | g ih g | C ⊗ R R → RS ◦ U Rg ( τ R ) . (15)The fidelity of this state with state Σ CRS is given byF( R R → RS (Σ CR ) , Σ CRS )= 1 | G | X g ∈ G F ( R R → RS ◦ U Rg ( τ R ) , U RSg ( τ R ⊗ σ S ))= 1 | G | X g ∈ G F ( U RSg − ◦ R R → RS ◦ U Rg ( τ R ) , τ R ⊗ σ S ) , where we have used the notation U RSg = U Rg ⊗ U Sg .Here the first equality follows from the orthogonality ofstates {| g i : g ∈ G } , and the second equality followsfrom the invariance of fidelity under unitary transfor-mations. Next, using the concavity of fidelity [24], wefind that the average fidelity in the last line is less thanor equal to the fidelity of the averaged states. That isF( R R → RS (Σ CR ) , Σ CRS ) is less than or equal to F ( 1 | G | X g ∈ G U RSg − ◦ R R → RS ◦ U Rg ( τ R ) , τ R ⊗ σ S ) (17a)= F ( R R → RS sym ( τ R ) , τ R ⊗ σ S ) , (17b)where R R → RS sym ≡ | G | P g ∈ G U RSg − ◦ R R → RS ◦ U Rg is thesymmetrized version of R R → RS sym , which satisfies the co-variance condition in Eq.(2). Using this together withEq.(13) and Eq.(14) we conclude that Theorem 1
There exists a covariant channel E R → RS ,i.e. a channel satisfying Eq.(2), which transforms τ R toa state whose fidelity with the desired state τ R ⊗ σ S islower bounded by − ∆Γ2 , that is F ( E R → RS ( τ R ) , τ R ⊗ σ S ) ≥ − ∆Γ2 , (18) where ∆Γ = Γ( τ R ⊗ σ S ) − Γ( τ R ) is the increase in therelative entropy of asymmetry in the ideal process τ R → τ R ⊗ σ S . Indeed, as we explain in the Supplementary Material (SeeSec. A), it follows from the results of [21, 22] that thecovariant map E R → RS can be chosen to be the Petz Re-covery map, up to some covariant unitary rotations inthe input and output, i.e. E R → RS = V RS − t ◦ R R → RSP ◦ V Rt , (19)where R R → RSP is the Petz recovery map R R → RSP ( ω R ) = q G ( τ R ⊗ σ S )( 1 p G ( τ R ) ω R p G ( τ R ) ⊗ I S ) q G ( τ R ⊗ σ S ) . (20)Here I S is the identity operator on system S , and theinverse of p G ( τ R ) is defined only on the support of thisoperator. Furthermore, V Rt and V RS − t are unitary covari-ant channels acting on the input and output systems R and RS respectively, defined by V Rt ( ω R ) ≡ [ G ( τ R )] it ω R [ G ( τ R )] − it , (21a) V RS − t ( ω RS ) ≡ [ G ( τ RS )] − it ω RS [ G ( τ RS )] it (21b)for some unknown t ∈ R . Note that for any t ∈ R the support of state V Rt ( τ R ) is contained in the supportof p G ( τ R ), and therefore the action of R R → RSP is well-defined on this state.This theorem together with upper bound in Eq.(9) im-ply that the quantity ∆Γ mainly determines weather thecatalytic transition in Eq.(3) can be implemented approx-imately or not. Note that because of the monotonicityof fidelity under partial trace, after applying the map E R → RS the fidelity of the reduced state of system S withits desired state σ S is larger than or equal to both sidesof Eq.(18).In the following we consider some examples of applica-tions of this result. V. APPLICATIONSA. State preparation using quantum clocks
As a simple model for a clock we consider a harmonicoscillator with Hamiltonian H R = ω (1 / N ), where N = P ∞ n =0 n | n ih n | is the number operator, and ω is thefrequency of oscillation. Assume the system S is anotherharmonic oscillator with the same frequency, which isinitially in a time-independent state, such as the thermalstate. Suppose we want to prepare the system S in thetime-dependent state | + S i = ( | i + | i ) / √ R (Note that | i and | i are, respectively,the ground and the first excited states of the harmonicoscillator.).We consider two different initial states of clock, i.e. | ψ Rk i = 1 √ T T − X n =0 | kn i , k = 1 , . (22)Note that Γ( | ψ R , i ) = log T and Γ( | + S i ) = 1. There-fore, for large enough T , Γ( | ψ R , i ) could be arbitrarylarger than Γ( | + S i ). Despite this, it turns out that usingstate | ψ R i and operations which are covariant under timetranslations we can never prepare S in a state close to | + S i , that is a state whose fidelity with the desired state | + S i is larger than the fidelity of the time-independentstate | S i with this state. On the other hand, havingthe clock R in state | ψ R i we can prepare state | ψ R i| + S i with fidelity 1 − /T [25]. This can also be shown usingtheorem 1: first, note thatΓ( | ψ R i| + S i ) = log T + 1 T , Γ( | ψ R i| + S i ) = log T + 1 . Therefore, while in the transition | ψ R i → | ψ R i| + S i theincrease in Γ remains independent of T , that is ∆Γ = 1, This can be seen, e.g., using the fact that the input | ψ R i isinvariant under the time translation e − iH R π/ω , and thereforefor any time-invariant channel the output should also be invari-ant under this time translation. But under the time translation e iH S π/ω the state | + S i is mapped to an orthogonal state. Itfollows that regardless of how large is T , with a clock in state | ψ R i we can never prepare state | + S i , or something close to it. in the transition | ψ R i → | ψ R i| + S i the increase in Γ is1 /T , and hence vanishes for large T . Using theorem 1 weconclude that there exists a quantum channel which is in-variant under time translations and implements the lattertransition with fidelity larger than or equal to 2 − / (2 T ) ,which for T ≫ ≈ − / (2 T log e ). B. Petz recovery map as the optimal cloner
As the next example, we consider the application ofour result in the case of cloners. Consider a cloner thatreceives n copies of an unknown state ψ ∈ C d , and gen-erates an output in state close to n + k copies of thisstate. As we saw before, the optimal cloner can be cho-sen to satisfy the covariance condition in Eq.(5). There-fore, in this case the relevant symmetry is G = U( d ), thegroup of unitaries acting on C d . Then, using the fact thatfor any integer r , U ⊗ r acts irreducibly on the symmetricsubspace of ( C d ) ⊗ r , we find that G ( ψ ⊗ r ) = Π ( r )+ /d + ( r ),where Π ( r )+ is the projector to the symmetric subspace,whose dimension is d + ( r ) = (cid:0) r + d − d − (cid:1) . It follows that inthe desired transformation ψ ⊗ n → ψ ⊗ ( n + k ) , the increasein the relative entropy of asymmetry is ∆Γ = log d + ( n + k ) d + ( n ) .Therefore, by virtue of theorem 1, we find that this statetransition can be implemented using a covariant chan-nel, with fidelity larger than or equal to q d + ( n ) d + ( n + k ) . Thisis exactly the optimal fidelity of cloning, as shown byWerner [7].Furthermore, because G ( ψ ⊗ n ) and G ( ψ ⊗ ( n + k ) ) areboth completely mixed states on the symmetric sub-spaces of the input and output Hilbert spaces, it followsthat for any input state ψ ⊗ n , the unitary channels V Rt and V RS − t act trivially, and therefore the covariant chan-nel that achieves this fidelity is the Petz recovery map inEq.(20) itself, i.e. E n → n + k ( ψ ⊗ n ) = d + ( n ) d + ( n + k ) Π ( n + k )+ [ | ψ ih ψ | ⊗ n ⊗ I ⊗ k ]Π ( n + k )+ . Interestingly, this is exactly the optimal cloning mapwhich was originally found by Werner [7], and maximizesthe fidelity of cloning.
VI. REVERSIBILITY
The problem of catalytic transitions under covarianttransformations is indeed closely related to a more gen-eral problem, which is of independent interest: How wellwe can reverse the effect of a covariant channel on a givenstate using only covariant channels? Roughly speak-ing, one expects that because asymmetry is the resourcethat cannot be generated by covariant channels, if underthe effect of the first covariant channel the amount ofasymmetry of state does not decrease considerably, thenthe process should be approximately reversible. As weshow in the Supplementary Material, using the results of[8, 21, 22], this intuition is indeed correct:
Theorem 2
Suppose under the action of a covariantchannel F X → Y the relative entropy of asymmetry of agiven state ρ X drops by ∆Γ = Γ( ρ X ) − Γ( F X → Y ( ρ X )) .Then there exists a covariant channel R Y → X , such that F ( R Y → X ◦ F X → Y ( ρ X ) , ρ X ) ≥ − ∆Γ / . Note that choosing X = RS , Y = R , and the map F X → Y to be the partial trace over system S , we can obtain the-orem 1 as a special case. In the Supplementary Ma-terial we also present similar results in the context ofthe resource theory of (speakable) coherence, in termsof dephasing-covariant channels [14, 26]. See also [27]for similar results on reversibility in the context of theresource theory of thermodynamics. VII. CONCLUSION
One of the advantages of using the relatively abstractlanguage of symmetry to understand physical phenom-ena, is that it clarifies similarities in seemingly differentproblems in different contexts. Hence, intuitions fromone physical phenomenon can be applied to another prob-lem in another context. The resource theory of asymme-try, whose central question is what can and what cannot be done with symmetric time evolutions, inherits thisadvantage. Formalizing a result in the language of thisresource theory, can help to clarify its applications in awide range of problems, from clocks to cloners. [1] G. Gour and R. W. Spekkens, New Journal of Physics , 033023 (2008).[2] I. Marvian and R. W. Spekkens, arXiv:1104.0018 (2011).[3] I. Marvian and R. W. Spekkens, Physical Review A ,014102 (2014).[4] I. Marvian and R. W. Spekkens, Nat Commun (2014),URL http://dx.doi.org/10.1038/ncomms4821 .[5] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reviewsof Modern Physics , 555 (2007).[6] F. G. Brandao, M. Horodecki, J. Oppenheim, J. M.Renes, and R. W. Spekkens, Physical review letters ,250404 (2013).[7] R. F. Werner, Physical Review A , 1827 (1998).[8] O. Fawzi and R. Renner, Communications in Mathemat-ical Physics , 575 (2015).[9] D. Petz, The Quarterly Journal of Mathematics , 97(1988).[10] D. Petz, Communications in mathematical physics ,123 (1986).[11] M. Lostaglio, K. Korzekwa, D. Jennings, andT. Rudolph, Physical Review X , 021001 (2015).[12] M. Lostaglio, D. Jennings, and T. Rudolph, Nature com-munications (2015).[13] J. ˚A berg, Physical review letters ,150402 (2014), ISSN 1079-7114, URL http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.150402 .[14] I. Marvian and R. W. Spekkens, arXiv preprintarXiv:1602.08049 (2016).[15] W. K. Wootters and W. H. Zurek, Nature , 802 (1982).[16] G. Gour, I. Marvian, and R. W. Spekkens, Physical Re-view A , 012307 (2009).[17] J. A. Vaccaro, F. Anselmi, H. M. Wiseman, and K. Ja-cobs, Physical Review A , 032114 (2008).[18] T. Baumgratz, M. Cramer, and M. B. Ple-nio, Phys. Rev. Lett. , 140401 (2014), URL http://link.aps.org/doi/10.1103/PhysRevLett.113.140401 .[19] Z.-W. Liu, X. Hu, and S. Lloyd, arXiv preprintarXiv:1606.03723 (2016).[20] K.M.R. Audenaert, , 8127 (2007).[21] M. M. Wilde, in Proc. R. Soc. A (The Royal Society,2015), vol. 471, p. 20150338.[22] M. Berta, K. P. Seshadreesan, and M. M. Wilde, Journalof Mathematical Physics , 022205 (2015).[23] D. Sutter, M. Tomamichel, and A. W. Harrow, IEEETransactions on Information Theory , 2907 (2016).[24] M. Nielsen and I. Chuang, Quantum Computation andQuantum Information , Cambridge Series on Informationand the Natural Sciences (Cambridge University Press,2000), ISBN 9780521635035.[25] I. Marvian and R. Mann, Physical Review A , 022304(2008).[26] E. Chitambar and G. Gour, arXiv preprintarXiv:1602.06969 (2016).[27] S. Wehner, M. M. Wilde, and M. P. Woods, arXivpreprint arXiv:1506.08145 (2015). Supplementary Material
Appendix A: Reversibility under covariant operations
In this section we prove theorem 2, which indeed can be thought as a generalization of theorem 1. Theorem 2 isbasically a corollary of the following result from [21, 22], which is a generalization of Fawzi and Renner original result:
Theorem 3 (From [21, 22]) Consider a channel N and a pair of state ρ and κ where the support of ρ is containedin the support of κ . Then, there exists a recovery channel R κ which maps N ( κ ) to κ and S ( ρ k κ ) − S ( N ( ρ ) kN ( κ ) ≥ − F ( R κ ◦ N ( ρ ) , ρ ) , (A1) This cannel can be chosen to be R κ = V κ,t ◦ R κ,P ◦ V N ( κ ) , − t (A2) for some t ∈ R , where R κ,P is the Petz recovery map R κ,P ( ρ ) = √ κ N † (cid:0) p N ( κ ) ρ p N ( κ ) (cid:1) √ κ . (A3) and V ω,t ( · ) = ω it ( · ) ω − it . (A4)Note that N † is the adjoint of N defined via Tr( N † ( X ) Y ) = Tr( X N ( Y )). Proof. (Theorem 2)Suppose in theorem 3 we choose channel N to be F X → Y . In the following we suppress the superscripts X and Y .By assumption this channel is G-covariant, i.e. satisfies F ◦ U g = U g ◦ F : ∀ g ∈ G . (A5)Averaging over group G we find F ◦ G = G ◦ F . (A6)This implies S ( ρ kG ( ρ )) − S ( F ( ρ ) kF ◦ G ( ρ )) = S ( ρ kG ( ρ )) − S ( F ( ρ ) kG ◦ F ( ρ )) (A7)= Γ( ρ ) − Γ( F ( ρ )) ≡ ∆Γ , (A8)where the first equality follows from Eq.(A6). Therefore, from theorem 3 we know the recovery channel R κ definedby Eq.(A2) satisfies ∆Γ = S ( ρ kG ( ρ )) − S ( F ( ρ ) kF ◦ G ( ρ ) ≥ − F ( R κ ◦ F ( ρ ) , ρ ) , (A9)where κ = G ( ρ ). Next, we show that if F is G-Covariant then the recovery channel R κ = V κ,t ◦ R κ,P ◦ V F ( κ ) , − t , (A10)for κ = G ( ρ ) will also be G-Covariant. First, note that G -covariance of F implies G -covariance of F † . This can beseen by considering the adjoint of both sides of Eq.(A5), which implies U g − ◦ F † = F † ◦ U g − : ∀ g ∈ G , (A11) In the case of the group of translations { e − iHt : t ∈ R } we can average over a finite interval [ − T, T ], and then look at the limit of T → ∞ . and hence U g ◦ F † = F † ◦ U g for all g ∈ G . Then, we note that both states κ = G ( ρ ) and F ( κ ) = F ◦ G ( ρ ) = G ◦ F ( ρ )are G-invariant. It follows that all the channels V κ,t ( ω ) = [ G ( ρ )] it ( ω )[ G ( ρ )] − it (A12)and V F ( κ ) , − t ( ω ) = [ F ◦ G ( ρ )] − it ( ω )[ F ◦ G ( ρ )] it (A13)and R κ,P ( ω ) = p G ( ρ ) F † (cid:0) p G ◦ F ( ρ ) ω p G ◦ F ( ρ ) (cid:1)p G ( ρ ) (A14)are G-covariant, and so is their combination R κ = V κ,t ◦ R κ,P ◦ V F ( κ ) , − t . (A15)This completes the proof. Appendix B: General upper bound on accuracy (proof of Eq.(9))
Recall the Fannes-Audenaert [20] inequality | S ( ρ ) − S ( ρ ) | ≤ log D × k ρ − ρ k H ( k ρ − ρ k , (B1)where ρ and ρ are arbitrary pair of density operators in a D -dimensional Hilbert space, and H ( x ) = − x log x − (1 − x ) log(1 − x ). Using the fact the uniform twirling G is a quantum channel, and the trace norm is non-increasing underquantum channels, we have kG ( ρ ) − G ( ρ ) k ≤ k ρ − ρ k . (B2)Then, assuming k ρ − ρ k ≤
1, and using the fact that function H is monotonically increasing in the interval (0 , / | S ( G ( ρ )) − S ( G ( ρ )) | ≤ log D × kG ( ρ ) − G ( ρ ) k H ( kG ( ρ ) − G ( ρ ) k ≤ log D × k ρ − ρ k H ( k ρ − ρ k . (B3b)This bound together with bound (B1) and the triangle inequality implyΓ( ρ ) − Γ( ρ ) = [ S ( G ( ρ )) − S ( ρ )] − [ S ( G ( ρ )) − S ( ρ )] (B4a) ≤ | S ( ρ ) − S ( ρ ) | + | S ( G ( ρ )) − S ( G ( ρ )) | (B4b) ≤ log D × k ρ − ρ k + 2 H ( k ρ − ρ k ρ = ω Y and ρ = E X → Y ( ρ X ), and using the fact that since E X → Y is G-Covariant thenΓ( E X → Y ( ρ X )) ≤ Γ( ρ X ), we findΓ( ω Y ) − Γ( ρ X ) ≤ Γ( ω Y ) − Γ( E X → Y ( ρ X ) ≤ log D ∗ Y × k ω Y − E X → Y ( ρ X ) k + 2 H ( k ω Y − E X → Y ( ρ X ) k , (B5)where D ∗ Y is the dimension of system Y . This is a weaker version of Eq.(9), because in general, D Y ≤ D ∗ Y , that isthe rank of G ( ω Y ) is less than or equal to the dimension of system Y . Next, we prove bound (9).Let Π be the projector to the support of G ( ω Y ). Using the fact that G ( τ Y ) commutes with U Y ( g ), for all g ∈ G ,we find that Π also commutes with U Y ( g ), for all g ∈ G . Define the channel L Y → Y ( ω Y ) = Π ω Y Π + Tr(Π ω Y ) ΠTr(Π) . (B6)0This is the quantum channel which projects any input state to a state restricted to the support of G ( ω Y ), and if theinput state is found to be outside this subspace, then it prepares the totally mixed state in the support of G ( ω Y ).Using the fact that Π commutes with U Y ( g ) for all g ∈ G we can easily see that L Y → Y is a covariant channel. Thisimplies Γ( ω Y ) − Γ( ρ X ) ≤ Γ( ω Y ) − Γ( E X → Y ( ρ X )) ≤ Γ( ω Y ) − Γ( L Y → Y ◦ E X → Y ( ρ X )) , where both bounds follow from the monotonicity of Γ under covariant channels. Next, note that the support of bothdensity operators ω Y and L Y → Y ◦ E X → Y ( ρ X ) are contained in the support of E X → Y ( ρ X ), whose dimension is denotedby D Y . Therefore, we can use Eq.(B4) with D = D Y . Then, we find∆Γ ≡ Γ( ω Y ) − Γ( ρ X ) (B7a) ≤ Γ( ω Y ) − Γ( E X → Y ( ρ X )) (B7b) ≤ Γ( ω Y ) − Γ( L Y → Y ◦ E X → Y ( ρ X )) (B7c) ≤ log D Y × k ω Y − L Y → Y ◦ E X → Y ( ρ X ) k + 2 H (1 / × k ω Y − L Y → Y ◦ E X → Y ( ρ X )) (B7d)= log D Y × kL Y → Y ( ω Y ) − L Y → Y ◦ E X → Y ( ρ X ) k + 2 H (1 / × kL Y → Y ◦ ( ω Y ) − L Y → Y ◦ E X → Y ( ρ X ) k ) (B7e) ≤ log D Y × k ω Y − E X → Y ( ρ X ) k + 2 H (1 / × k ω Y − E X → Y ( ρ X ) k ) , (B7f)where to get the fourth line we have used Eq.(B4), and to get the fifth line we have used the fact that the supportof ω Y is contained in the support of G ( ω Y ), and therefore the channel L Y → Y leaves ω Y invariant, and to get the lastline we have used monotonicity of the trace distance under quantum channels. This completes the proof of Eq.(9). Appendix C: Recoverability for dephasing-covariant channels
In section A, we used theorem 3 to prove that if under a covariant channel the relative entropy of asymmetry doesnot drop considerably, then the process can be approximately reversed using a covariant channel. In this section weuse theorem 3 again to show that a similar theorem holds in the case of dephasing-covariant channels [14, 26].Consider a complete set of orthogonal projectors { P j } j . Then, the dephasing map D is defined by D ( · ) = X j P j ( · ) P j . (C1)A quantum channel F is called dephasing-covariant [14, 26], if it satisfies F ◦ D = D ◦ F . (C2)Note that here we are only considering quantum channels whose input and output spaces are the same.The relative entropy of coherence [18] is defined byΛ( ρ ) = inf ω ∈I S ( ρ k ω ) = S ( ρ kD ( ρ )) = S ( D ( ρ )) − S ( ρ ) , (C3)where I is the set of incoherent states relative this basis, i.e. states that can be written as P j p j P j for a probabilitydistribution p j . It can be easily shown that the relative entropy of coherence is non-increasing under dephasing-covariant channels [14, 26]. Theorem 4
Suppose, under the action of dephasing-covariant channel F the relative entropy of coherence of state ρ drops by ∆Λ , i.e. ∆Λ = Λ( ρ ) − Λ( F ( ρ )) . (C4) Then, there exists a dephasing-covariant channel R such that F ( R ◦ F ( ρ ) , ρ ) ≥ − ∆Λ / . (C5)1 Proof.
The result follows from theorem 3 for the special case where N = F is a dephasing-covaraint channel, and κ = D ( ρ ).First, note that S ( ρ kD ( ρ )) − S ( F ( ρ ) kF ◦ D ( ρ )) = S ( ρ kD ( ρ )) − S ( F ( ρ ) kD ◦ F ( ρ )) = ∆Λ , (C6)where the first equality follows from the fact that channel F is Dephasing-covariant.Therefore, from theorem 3 we know the recovery channel R κ defined by Eq.(A2) satisfies∆Λ = S ( ρ kD ( ρ )) − S ( F ( ρ ) kF ◦ D ( ρ ) ≥ − F ( R κ ◦ F ( ρ ) , ρ ) , (C7)where κ = D ( ρ ). Next, we show that if we choose κ = D ( ρ ) and N = F , then the fact that F is dephasing-covariantimplies that the recovery channel R κ is also Dephasing-covariant.First, by looking to the adjoint of Eq.(C2) we find that if F is dephasing-covaraint, then F † is also dephasing-covariant (Note that D † = D ). Combining this with the fact that states D ( ρ ), and F ◦ D ( ρ ) = D ◦ F ( ρ ) are bothincoherent we find that the Petz recovery channel, defined by Eq.(A3) is also dephasing-covariant.Finally, using the fact that both states D ( ρ ), and F ◦ D ( ρ ) = D ◦ F ( ρ ) are incoherent, we find that for all t ∈ R ,the operations V D ( ρ ) ,t and V F◦D ( ρ ) ,t defined by Eq.(A12) and Eq.(A13) are also dephasing-covariant unitary channels.Because the combination of dephasing-covaraint channels is a dephasing-covaraint channel, we conclude that therecovery channel V D ( ρ ) ,t ◦ R D ( ρ ) ,P ◦ V F◦D ( ρ ) , − tt