Quantum Reference Frames and Their Applications to Thermodynamics
Sandu Popescu, Ana Belén Sainz, Anthony J. Short, Andreas Winter
aa r X i v : . [ qu a n t - ph ] A p r Quantum Reference Framesand Their Applications to Thermodynamics
Sandu Popescu , Ana Bel´en Sainz , Anthony J. Short , and Andreas Winter H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, United Kingdom Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5 ICREA—Instituci´o Catalana de Recerca i Estudis Avanc¸ats, Pg. Lluis Companys 23, 08010 Barcelona, Spain F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics, Departament de F´ısica,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona), SpainApril 12, 2018
We construct a quantum reference frame, which can be used to approximatelyimplement arbitrary unitary transformations on a system in the presence of anynumber of extensive conserved quantities, by absorbing any back action provided bythe conservation laws. Thus, the reference frame at the same time acts as a batteryfor the conserved quantities.Our construction features a physically intuitive, clear and implementation-friendlyrealisation. Indeed, the reference system is composed of the same types of subsys-tems as the original system and is finite for any desired accuracy. In addition, theinteraction with the reference frame can be broken down into two-body terms cou-pling the system to one of the reference frame subsystems at a time. We applythis construction to quantum thermodynamic setups with multiple, possibly non-commuting conserved quantities, which allows for the definition of explicit batteriesin such cases.
When considering what transformations of a quantum system are theoretically possible, we usu-ally imagine that we can implement any unitary transformation on the system. However, thismay not be consistent with conservation laws: Wigner [1] and Araki and Yanase [2] showed, inwhat is known as the WAY theorem, that an observer cannot measure any observable that doesnot commute with an additive conserved quantity, as such a measurement would necessarilychange this conserved quantity. From an information-theoretic perspective, WAY is a con-sequence of the no-programming theorem for projective measurements [3]. The situation wasrevisited, and clarified, by Aharonov and Susskind [4, 5], followed by Aharonov and Kaufherr [6]who pointed out its deep connection with the issue of quantum frames of reference: The con-servation laws always refer to a closed system, such as a freely floating spaceship, and there isa need to consider carefully the difference between observables defined in relation with frames
Sandu Popescu: [email protected] Bel´en Sainz: [email protected] J. Short: [email protected] Winter: [email protected] spoke on quantum thermodynamics at the Royal Society discussion meeting
Foundations of quantum mechanicsand their impact on contemporary society , 11-12 December 2017, and reported, among other things, on the contentof [10, 11]; the present paper addresses some of the issues highlighted there. f reference defined internally (i.e. inside the rocket) or externally. This enabled them to alsoclarify the issue of super-selection rules [7], another apparent constraint on quantum mechanics.Recently, there has been a renewed very strong interest in understanding quantum frames ofreference as well as their connection with conservation laws; see e.g. [8, 9].All the above-mentioned works address the fundamental question of which operations arephysically possible on a system under conservation laws. The present paper focuses on oneof the central issues in this area - how to implement a unitary transformation on the systemwith respect to an external reference frame. Previous approaches to this problem have eitherconsidered particular cases, or relied upon general group theoretic constructions for an internalreference frame. This latter approach requires the existence of a very abstract and complexancillary system to act as the reference frame, which interacts in a complex way with thesystem. Our approach, in contrast, gives a general construction for an internal reference framewhich is also easy to implement and understand physically. In particular, we use a singlefixed reference frame to perform an arbitrary unitary transformation with high precision onan unknown state of a system; the same reference frame will work for any extensive conservedquantities, and does not depend otherwise on the symmetries to which they are linked. Indeed,we could impose an arbitrary set of extensive conservation laws which are not even linked tosymmetries of nature, and the construction would still apply; this is for example the case in theapplications of the theory to thermodynamics, to be discussed later. Furthermore, the referenceframe (i) is composed of the same basic subsystems as the system on which it acts (and thusdoes not require the existence of a new type of system with special properties), (ii) is finitefor any desired precision, and (iii) the required transformations can be implemented by a seriesof interactions between two subsystems at a time, each step being a very short time evolutionof a simple Hamiltonian between the system and one of its replicas in the reference frame(Sec. 3). We also show that if the system itself is composed of smaller parts, the interactionwith the reference frame can be implemented by a quantum circuit in which each gate respectsthe conservation laws (Subsec. 3.1).Our motivation for the present work stems from recent research on quantum thermodynam-ics, and in particular by considering the thermodynamics of systems with multiple conservedquantities [10, 11]. Given access to a large generalised thermal state (composed of many identicalsubsystems with a small amount of assumed structure), and the ability to perform any unitary,it is possible to extract an arbitrary amount of any given conserved quantity by inputting anappropriate amount of the other conserved quantities [10]. A generalised second law restrictsthe rate at which different conserved quantities can be interconverted. In this picture, anychange in the conserved quantities of the system due to the action of the unitary is interpretedas work (implicitly stored in some battery). If the conserved quantities all commute, then theformalism can be extended to include explicit batteries, with the system+battery combinationobeying strict conservation laws. In [10] it was left as an open problem how to achieve a sim-ilar result for non-commuting conserved quantities. A similar problem was addressed in [11],where a reference frame was used as an explicit battery to implement such transformations.However, that work was based on interactions with a particular infinite dimensional choice of“perfect” reference system, which may not be available. Here we use our alternative referenceframe construction to present an explicit battery which is simply comprised of many copies ofthe subsystems on which it acts, in different initial states (Sec. 4), and which can operate witharbitrary effectiveness while being of finite size.2 Background
Consider a closed quantum system, such as the spaceship mentioned before, subjected to conser-vation laws. Consider a particle in that closed system, say a spin- particle. Suppose we wantto implement a given unitary evolution, defined in terms of an external frame of reference , saya rotation by an angle θ around the z -direction. Could this be achieved by apparatuses situated inside the rocket? The reason why this question received so much interest is that, apart fromthe problem itself and what it tells us about conservation laws and frames of reference, whencast in different set-ups, it has major implications for many other domains. For example, Kitaev et al. [12] considered it in the context of quantum cryptography protocols; more recently, it hasemerged as a crucial issue in quantum thermodynamics, cf. [13].The point is that conservation laws impose constraints to what can be done: The desiredunitary, U = exp( − iσ z θ ) in the above example, does not commute with the total angularmomentum along any other axis apart from z , say with J total x , hence it is impossible. Therefore,apparently there is no way to achieve our desired goal. It turns out, however, that there is a way:Following [6], the reason why a rotation along the external z axis is impossible in general is thatactions confined inside the closed system do not have access to the external frame of reference,so they have no knowledge of what the z axis actually is. All that internal apparatuses can dois to rotate the spin around some internally defined axis, say the direction from the floor to theceiling of the rocket: Such a rotation, relative to the internal degrees of freedom – the otherparts of the rocket – commutes with all components of total angular momentum so does notcontradict the angular momentum conservation law. If however, in advance of the request for therotation, the external observers align the internal direction – the “internal frame of reference”– with the external one, say the ‘floor to the ceiling’ direction to the external z -direction, andif rotating the spin does not produce too much of a back-reaction to the rocket to misalign it,the rotation along the internal direction coincides with a rotation around the z -axis.The central question is thus how to prepare and align the internal frame, which, for moregeneral unitaries, is not so obvious as in the case of spin rotations, and how to ensure it is robustenough, so that it stays aligned under the kick-back. Building up from the WAY theorem, newworks have further explored complex mathematical constructions to devise internal frames.For instance, a general version of this problem was also studied as the resource theory ofasymmetry [8, 14, 15]. Other approaches have further focused on the role of the particularsymmetry group G which, according to Noether’s theorem, comes with the conservation laws.As a consequence, the internal reference frame that one can define following this approach isheavily group dependent. This fact is clearly illustrated by the construction of the “model”reference frame L ( G ): the square integrable functions on the group G (under the invariantHaar measure) equipped with the left regular representation [8]. Hence, while this approachis mathematically very elegant, both in its definition and its interaction with the system, theconstruction is very complex, and so is the interaction required to implement the desired unitaryon the system. Hence one might wonder what sort of complex physical system this frame actuallyrepresents, and how to physically implement the necessary interaction.In the next section we show how one can construct instead a reference frame that is bothsimple to implement and physically intuitive, and which will work for any extensive conservedquantities. In this section, we construct a general quantum reference frame which will allow us to performan arbitrary unitary transformation U S with high precision on an unknown system state ρ S R , is composed of a large number of copies of the system, and isprepared in a particular state ρ R which is independent of U S or ρ S . The size of the referenceframe will depend on the precision to which we will attempt to simulate the unitary U S , andwill be made more precise in appendix D.We define an extensive conserved quantity in this context as one which can be written as A TOT = A S + P r ∈ R A r . In other words, as a sum of the same quantity A for the system andfor each subsystem r in the reference frame.We show below that any U S can be implemented on the system with high precision byan appropriately chosen joint unitary V on the combined system and reference frame whichcommutes with all conserved quantities. Theorem 1.
Let S be any finite dimensional quantum system. Then for every ǫ > , thereexists a reference frame (composed of a large number of copies of the system S , prepared insuitable states) with a fixed state ρ R , such that for every unitary U S on the system there existsa joint unitary V on the system and reference frame such that • V conserves all extensive conserved quantities: [ V, A
TOT ] = 0 . • V effectively implements U S on the system with ǫ precision, for any initial system state: (cid:13)(cid:13)(cid:13) tr R n V ρ S ⊗ ρ R V † o − U S ρ S U † S (cid:13)(cid:13)(cid:13) ≤ ǫ ∀ ρ S , (1) where ρ S and ρ R are density operators (i.e. ρ ≥ , tr ρ = 1 ), and k · k is the trace norm(which characterises how well two states can be distinguished).Proof. The proof consists of two parts. First, that infinitesimal rotations can be implemented,and then that from them a general unitary can be implemented without going above the precisionthreshold.For the first part of the proof, we fix a large N ≫ V α acting on thesystem and one copy of the system from the reference frame (denoted by R ) as follows: V α = exp (cid:18) − i αN SWAP (cid:19) , (2)where SWAP = P i,j | ij ih ji | . Since the conserved quantities are additive, and all subsystems areidentical, any function of the SWAP operation will conserve them. Therefore, [ V α , A TOT ] = 0for all α .Let σ be an arbitrary but fixed initial state for the reference frame particle. The effectiveaction on the system when applying V α on ρ S ⊗ σ is: ρ fS = tr R n V α ρ S ⊗ σ V † α o = tr R (cid:8) ρ S ⊗ σ − i αN SWAP ρ S ⊗ σ + i αN ρ S ⊗ σ SWAP (cid:9) + O (cid:18) N (cid:19) = ρ S − i αN [ σ, ρ S ] + O (cid:18) N (cid:19) , (3)where a Taylor expansion for small values of αN was used. The last equality follows fromtr B { SWAP ρ S ⊗ σ } = σ ρ S and tr B { ρ S ⊗ σ SWAP } = ρ S σ (see Appendix A). We will say that a trace class operator A is O (cid:0) N k (cid:1) if k A k ≤ c/N k for all sufficiently large N , where c is aconstant (which here depends on α ). We show this in detail in Appendix D. In other contexts, we will use thesame notation to refer to a number a satisfying | a | ≤ c/N k for all sufficiently large N . U σ ( α ) = exp {− i αN σ } we thus find that ρ fS = U σ ( α ) ρ S U † σ ( α ) + O (cid:18) N (cid:19) , i.e. (cid:13)(cid:13)(cid:13) tr B n V α ρ S ⊗ σ V † α o − U σ ( α ) ρ S U † σ ( α ) (cid:13)(cid:13)(cid:13) < O (cid:18) N (cid:19) . (4)The above procedure allows us to generate small rotations of a qubit about a particularstate σ . We now show how to generalise this to implement arbitrary small rotations.Let H S be the Hilbert space of the system, and { σ k } Dk =1 be a set of density operatorsdescribing states of the system such that B = { σ k } ∪ is an arbitrary but fixed operator basisof H S . Now consider the following state of R , composed of one copy of each of the states σ k : σ R = σ ⊗ σ ⊗ . . . σ D . (5)Suppose that we ultimately wish to implement an arbitary unitary U S = exp ( − iH ) on thesystem. As global phase factors in U S do not affect the evolution of the state, we can assumewithout loss of generality that H = P Dk =1 α k σ k . Additionally, as all values of exp( iθ ) can beobtained for θ ∈ ( − π, π ], we can choose H such that its eigenvalues lie within this range. As thesystem is finite dimensional, for any fixed operator basis B , this also implies that the parameters α k are also bounded by a constant (see appendix B).Let us now consider how to implement the small general rotation U H = exp (cid:16) − i HN (cid:17) . Denoteby V ( k ) α k the unitary V α k applied on the system and the particle from the reference frame thatis in state σ k . Now consider the action of the sequence of unitaries V seq = V ( D ) α D . . . V (2) α V (1) α onthe system ρ S and reference frame σ R . By keeping the first order terms in N , similarly to thecalculation in the infinitesimal-rotation case, we obtain: ρ fS = tr R ...R D n V seq ρ S ⊗ σ R V † seq o = tr R ...R D ( ρ S ⊗ σ R − i D X k =1 α k N SWAP
S,R k ( ρ S ⊗ σ R ) + i D X k =1 α k N ( ρ S ⊗ σ R ) SWAP S,R k ) + O (cid:18) N (cid:19) = ρ S − i N " D X k =1 α k σ k , ρ S + O (cid:18) N (cid:19) = ρ S − i N [ H, ρ S ] + O (cid:18) N (cid:19) = U H ρ S U † H + O (cid:18) N (cid:19) . (6)We are now in position to prove the general claim of the Theorem, where U S is an arbitraryunitary on S (which need not be close to the identity).Consider a reference frame composed of N copies of σ R : ρ R = σ ⊗ NR = ( σ ⊗ σ ⊗ . . . σ k ) ⊗ N . (7)The required value of N will depend on the precision to which we will attempt to simulate theunitary U S . Note that this implies that D = d − d is the Hilbert space dimension.
5o implement U S , we apply the unitary V seq on the system plus D reference particles N times, each time using a different copy of σ R in the reference frame. Each unitary in the sequencewill act on the system as the unitary U H up to an error O (cid:16) N (cid:17) . The upper bound on the totalerror after applying the sequence of V seq will be N O (cid:16) N (cid:17) ∼ O (cid:16) N (cid:17) (see appendix C). Hence (cid:13)(cid:13)(cid:13) tr R n V ρ S ⊗ ρ R V † o − U S ρ S U † S (cid:13)(cid:13)(cid:13) ≤ O (cid:18) N (cid:19) (8)Given any ǫ > U S , it is always possible to find a sufficiently large N suchthat O (cid:16) N (cid:17) ≤ ǫ . Hence a sufficiently large reference frame will allow us to take the systemto a final state U S ρ S U † S within an error upper bounded by ǫ as required. This is achieved byapplying a sequence of unitary transformations V α on the system plus reference frame particles,each of which commutes with (and hence conserves) all extensive quantities A TOT . Although the previous result will work for any system, including composite systems, the con-struction will in general require joint transformations involving all of the particles in the system.Particularly in the context of thermodynamics, it is common to consider systems composed ofmany particles, where this may pose practical difficulties. Here we show that our approach canbe adapted easily to allow for a ‘circuit’-model of transformations, where any unitary on thesystem can be implemented by a series of transformations involving a small number of particlesfrom the system and reference frame.If we have a composite system made up of some finite number of different types of subsys-tems, then our reference frame is simply the tensor product of the reference frames for eachtype of subsystem. Using our previous results, we can approximately implement any single-subsystem unitary. To implement any two-subsystem unitary between subsystems A and B ,we (i) consider those two subsystems as a composite system, and (ii) use the same procedureas before on this composite system and a similar composite system in the reference frame(hence involving only 4 primitive subsystems). For example, to implement exp (cid:18) − iα σ ( A )1 ⊗ σ ( B )1 N (cid:19) we would apply exp (cid:0) − i αN SWAP S A ,R A SWAP S B ,R B (cid:1) to the state ρ S A S B ⊗ (cid:16) σ ( A )1 ⊗ σ ( B )1 (cid:17) R A R B .Once we can perform any two-subsystem unitary, we can use standard techniques to constructa circuit to implement any unitary on the system up to the desired accuracy .If the subsystems are all of the same type, then an alternative approach would be to use(i) the fact that U np = √ SWAP is an entangling two-subsystem gate which commutes with allconserved quantities, and (ii) that such a gate plus all single subsystem unitaries is computa-tionally universal [17]. This would require only bipartite interactions, with the caveat of a morecomplicated construction.
Although the reference frame construction above could be of use in many different scenarios,one interesting application is to quantum thermodynamics with multiple conserved quantities[10, 11]. In this section, we briefly review the approach given in [10], and consider how ourreference frame can be thought of as a battery in such cases. In particular, we can apply the approach in [16] (generalised to arbitrary dimension) based on two-level uni-taries built from controlled unitary gates on two subsystems. We can then bound the total error by appropriatelybounding the error on each gate. τ = Z e − ( β A + ... + β k A k ) , where A i are an arbitrary set of conservedquantities, β i are generalised inverse temperatures associated with these conserved quantities and Z = tr { e } − ( β A + ... + β k A k ) is the generalised partition function. We will be particularlyinterested in cases in which the different conserved quantities do not commute.The simplest case to consider is a thermal bath with no additional system, in which thebattery is treated implicitly. In this scenario, one simply allows any unitary transformation onthe thermal state of the bath, and assumes that any change in the conserved quantities of thebath are accounted for by an equal and opposite amount of work of the corresponding type W A i = − ∆ h A bath i i , (9)in accordance with the first law of thermodynamics. A generalised second law of thermody-namics can then be derived stating that X i β i W A i ≤ , (10)where W A i is the the type A i work extracted from the bath, and β i the inverse temperatureconjugate to A i .In addition, if one now includes an additional non-thermal system in the picture, the amountof extracted work is bounded by X i β i W A i ≤ − ∆ F S , (11)where W A i = − ∆ h A sys i i − ∆ h A bath i i is the amount of A i type work extracted from the systemand bath, and F S = P i β i h A sys i i − S S is the free entropy of the system relative to the generalisedbath, with S S = − tr S { ρ S ln ρ S } the system’s entropy.Furthermore, given a small set of structural assumptions about the thermal bath, proto-cols can be constructed which can approximately implement any transformation satisfying thebounds given by eq. (10) and eq. (11). Therefore, we learn that in principle we can extract asmuch A i type work W A i as we want, as long as the other conserved quantities of the bath andsystem change as well to compensate.In the case in which the conserved quantities commute, and can in principle be stored sepa-rately from one another, one can extend these setups to include explicit quantum batteries [10].In particular, given a fixed initial state of a quantum battery, we can map every transforma-tion in the implicit battery scenario onto a global unitary which commutes with all conservedquantities A sys i + A bath i + A battery i and which achieves the same results to any desired accuracy.However, how to deal with explicit batteries in the presence of noncommmuting conservedquantities (under strict conservation) was left as an open question in [10], and treated using the L ( G ) construction in [11]. Here we resolve this question by showing how the reference framesintroduced earlier can be used as explicit batteries. Furthermore, because the same approachworks for any conserved quantities, this also shows how explicit batteries can be constructedwhen the conserved quantities cannot be separated from each other, or when a battery systemof the type in [10] or [11] is not available. E.g. for energy β i = k B T where k B is Boltzmann’s constant and T is the temperature, whereas for particlenumber conservation, β i = µ i k B T where µ i is the chemical potential. Analogous inverse temperatures can beassociated with other conserved quantities such as angular momentum.
7n particular, consider a bath and system composed of a number of subsystems, and areference frame composed of the same types of subsystems (as described in section 3.1) whichwill act as a battery. Any desired protocol on the system and bath in the implicit batteryscenario can be implemented approximately to any desired accuracy given access to a sufficientlylarge reference frame, whilst respecting strict conservation of all extensive conserved quantities.Because of these conservation laws∆ h A sys i i + ∆ h A bath i i + ∆ h A battery i i = 0 . (12)If the desired transformation on the system and bath would extract work W A i in the implicitbattery scenario, and this transformation is implemented with ǫ precision (in trace norm) usingthe reference frame, then | ∆ h A battery i i − W A i | ≤ ǫ (cid:13)(cid:13) A sysi + A bathi (cid:13)(cid:13) (13)where k · k represents the operator norm. Any deviation between the work extracted in the im-plicit battery scenario, and the conserved quantities stored in the explicit battery can thereforebe made as small as desired by taking a sufficiently large reference frame.Following this procedure, any changes in the average values of conserved quantities in thebattery correspond to stored or expended work. We have shown that it is possible to construct a simple reference frame, which allows us toapply any desired unitary on the system, and which moreover provides a physical intuition onthe nature of the operations that should be applied to achieve it. This reference frame can inaddition act as a battery of extensive conserved quantities. Previous general constructions ofquantum reference frames [8] have been based on the symmetry group associated with the con-served quantities via Noether’s theorem. This is mathematically elegant, but requires systemsand transformations which may be very difficult to find or implement physically. Our alternativeapproach works universally for any set of conserved quantities, and does not require Noether’stheorem in the sense that we do not need to first construct the group symmetry belonging tothe conservation law. Furthermore, it allows for a relatively simple physical implementationinvolving a ‘circuit’ model on multiple copies of the system in a fixed product state.The size of the frame in this “bottom-up” approach (compared to the “top-down” of L ( G )),determines the accuracy with which we can implement a given unitary. Here we have prioritisedthe simplicity of the construction, rather than minimising its dimension or some other indicatorof its complexity for a given error. We leave it as an open problem whether more intricateconstructions give a better accuracy for the same frame size. Acknowledgements
We thank Yakir Aharonov, Aram Harrow, Jonathan Oppenheim, Michalis Skontiniotis and RobSpekkens for various discussions on reference frames over some time, which have helped us fixour own ideas.This research was supported by Perimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government of Canada through the Department of In-novation, Science and Economic Development Canada and by the Province of Ontario throughthe Ministry of Research, Innovation and Science. SP acknowledges support from the ERC8dvanced Grant NLST and The Institute for Theoretical Studies, ETH Zurich. AW acknowl-edges support by the Spanish MINECO (project FIS2016-86681-P) with the support of FEDERfunds, and the Generalitat de Catalunya (project 2017-SGR-1127).
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A Partial trace relations involving SWAP
Here we present two helpful relations involving the partial trace and the SWAP = P i,j | ij ih ji | operation. tr B { A ⊗ B SWAP } = X ijk | i i A h ij | A ⊗ B SWAP | kj ih k | A = X ijk | i i A h ij | A ⊗ B | jk ih k | A = X ijk | i ih i | A | j ih j | B | k ih k | = AB, (14)tr B { SWAP A ⊗ B } = X ijk | i i A h ij | SWAP A ⊗ B | kj ih k | A = X ijk | i i A h ji | A ⊗ B | kj ih k | A = X ijk | i ih i | B | j ih j | A | k ih k | = BA. (15)
B Bound on parameters α k Here we show that the parameters α k appearing in H = P Dk =1 α k σ k (where k H k ≤ π ) arebounded by a constant. Consider the real vector space of Hermitian operators with Hibert-Schmidt inner product h A, B i = tr { AB } . Note that for any fixed operator basis { σ k } Dk =0 (where for simplicity we have defined σ = { ˜ σ l } Dk =0 such thattr { σ k ˜ σ l } = δ kl . It then follows that α k = tr { H ˜ σ k } and hence from the Cauchy-Schwarzinequality that | α k | ≤ k H k k ˜ σ k k ≤ √ Dπ k ˜ σ k k (16)where k H k = p tr { H } is the Hilbert-Schmidt norm. Hence α max = √ Dπ max k k ˜ σ k k (17)provides an upper bound for all parameters α k .10 Overall error bound due to product of small rotations