Quantum reference frames and their applications to thermodynamics

Abstract

We construct a quantum reference frame, which can be used to approximately implement arbitrary unitary transformations on a system in the presence of any number of extensive conserved quantities, by absorbing any back action provided by the conservation laws. Thus, the reference frame at the same time acts as a battery for the conserved quantities. Our construction features a physically intuitive, clear and implementation-friendly realization. Indeed, the reference system is composed of the same types of subsystems as the original system and is finite for any desired accuracy. In addition, the interaction with the reference frame can be broken down into two-body terms coupling the system to one of the reference frame subsystems at a time. We apply this construction to quantum thermodynamic set-ups with multiple, possibly non-commuting conserved quantities, which allows for the definition of explicit batteries in such cases.

This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

1. Introduction

When considering what transformations of a quantum system are theoretically possible, we usually imagine that we can implement any unitary transformation on the system. However, this may not be consistent with conservation laws: Wigner [1] and Araki & Yanase [2] showed, in what is known as the WAY theorem, that an observer cannot measure any observable that does not commute with an additive conserved quantity, and as such a measurement would necessarily change this conserved quantity. From an information-theoretic perspective, WAY is a consequence of the no-programming theorem for projective measurements [3]. The situation was revisited, and clarified, by Aharonov & Susskind [4,5], followed by Aharonov & Kaufherr [6], who pointed out its deep connection with the issue of quantum frames of reference: the conservation laws always refer to a closed system, such as a freely floating rocket, and there is a need to consider carefully the difference between observables defined in relation with frames of reference defined internally (i.e. inside the rocket) or externally. This enabled them to also clarify 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 of reference as well as their connection with conservation laws; see [8,9].

All the above-mentioned works address the fundamental question of which operations are physically possible on a system under conservation laws. This paper focuses on one of the central issues in this area—how to implement a unitary transformation on the system with respect to an external reference frame. Previous approaches to this problem have either considered particular cases, or relied upon general group theoretic constructions for an internal reference frame. This latter approach requires the existence of a very abstract and complex ancillary system to act as the reference frame, which interacts in a complex way with the system. Our approach, by contrast, gives a general construction for an internal reference frame which is also easy to implement and understand physically. In particular, we use a single fixed reference frame to perform an arbitrary unitary transformation with high precision on an unknown state of a system; the same reference frame will work for any extensive conserved quantities, 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 to symmetries of nature, and the construction would still apply; this is, for example, the case in the applications of the theory to thermodynamics, to be discussed later. Furthermore, the reference frame (i) is composed of the same basic subsystems as the system on which it acts (and thus does not require the existence of a new type of system with special properties) and (ii) is finite for any desired precision, and (iii) the required transformations can be implemented by a series of interactions between two subsystems at a time, each step being a very short time evolution of a simple Hamiltonian between the system and one of its replicas in the reference frame (§3). We also show that if the system itself is composed of smaller parts, the interaction with the reference frame can be implemented by a quantum circuit in which each gate respects the conservation laws (§3a).

Our motivation for this work stems from recent research on quantum thermodynamics, and in particular by considering the thermodynamics of systems with multiple conserved quantities 10; 11; 12; 13; 14; [15,16]. Given access to a large generalized thermal state (composed of many identical subsystems 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 an appropriate amount of the other conserved quantities 10; 11; 13; [15]. A generalized second law restricts the rate at which different conserved quantities can be interconverted. In this picture, any change in the conserved quantities of the system due to the action of the unitary is interpreted as work (implicitly stored in some battery). If the conserved quantities all commute, then the formalism can be extended to include explicit batteries, with the system+battery combination obeying strict conservation laws. In [15], it was left as an open problem how to achieve a similar result for non-commuting conserved quantities. A similar problem was addressed in [16], 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 a ‘perfect’ reference system, which may not be available. Here, we use our alternative reference frame construction to present an explicit battery which is simply composed of many copies of the subsystems on which it acts, in different initial states (§4), and which can operate with arbitrary effectiveness while being of finite size.

2. Background

Consider a closed quantum system, such as the rocket mentioned before, subjected to conservation laws. Consider a particle in that closed system, say a spin- particle. Suppose we want to implement a given unitary evolution, defined in terms of an external frame of reference, say a 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 from the problem itself and what it tells us about conservation laws and frames of reference, when cast in different set-ups, it has major implications for many other domains. For example, Kitaev et al. [17] considered it in the context of quantum cryptography protocols; more recently, it has emerged as a crucial issue in quantum thermodynamics; cf. [18].

The point is that conservation laws impose constraints to what can be done: the desired unitary, in the above example, does not commute with the total angular momentum along any other axis apart from z, say with , 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 that actions 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 do is to rotate the spin around some internally defined axis, say the direction from the floor to the ceiling of the rocket: such a rotation, relative to the internal degrees of freedom—the other parts of the rocket—commutes with all components of total angular momentum so does not contradict the angular momentum conservation law. If however, in advance of the request for the rotation, 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, and if 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 more general unitaries, is not so obvious as in the case of spin rotations, and how to ensure it is robust enough, so that it stays aligned under the kick-back. Building up from the WAY theorem, new works 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 of asymmetry [8,19,20]. Other approaches have further focused on the role of the particular symmetry 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 is heavily group dependent. This fact is clearly illustrated by the construction of the ‘model’ reference frame L²(G): the square-integrable functions on the group G (under the invariant Haar measure) equipped with the left regular representation [8]. Hence, while this approach is mathematically very elegant, both in its definition and its interaction with the system, the construction is very complex, and so is the interaction required to implement the desired unitary on the system. Hence, one might wonder what sort of complex physical system this frame actually represents, and how to physically implement the necessary interaction.

In the next section, we show how one can construct instead a reference frame that is both simple to implement and physically intuitive, and which will work for any extensive conserved quantities.

3. A general quantum reference frame

In this section, we construct a general quantum reference frame which will allow us to perform an arbitrary unitary transformation U_S with high precision on an unknown system state ρ_S despite arbitrary extensive conserved quantities.

The reference frame, R, is composed of a large number of copies of the system 21,¹ and is prepared in a particular state ρ_R which is independent of U_S or ρ_S. The size of the reference frame will depend on the precision to which we will attempt to simulate the unitary U_S, and will be made more precise in appendix D.

We define an extensive conserved quantity in this context as one which can be written as . In other words, as a sum of the same quantity A for the system and for each subsystem r in the reference frame.

We show below that any U_S can be implemented on the system with high precision by an appropriately chosen joint unitary V on the combined system and reference frame which commutes with all conserved quantities.

Theorem 3.1 —

Let S be any finite-dimensional quantum system. Then, for every ϵ>0, there exists a reference frame (composed of a large number of copies of the system S, prepared in suitable states) with a fixed state ρ_R, such that, for every unitary U_S on the system, there exists a 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 ϵ precision, for any initial system state:

  3.1

  where ρ_S and ρ_R are density operators (i.e. ρ≥0, tr ρ=1), and ∥⋅∥₁ is the trace norm (which z 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 precision threshold.

For the first part of the proof, we fix a large N≫1 and define the unitary V _α acting on the system and one copy of the system from the reference frame (denoted by R) as follows:

3.2

where . As the conserved quantities are additive, and all subsystems are identical, any function of the SWAP operation will conserve them. Therefore, [V _α,A^TOT]=0 for all α.

Let σ be an arbitrary but fixed initial state for the reference frame particle. The effective action on the system when applying V _α on ρ_S⊗σ is

3.3

where a Taylor expansion for small values of α/N was used.² The last equality follows from tr_B{SWAP ρ_S⊗σ}=σρ_S and tr_B{ρ_S⊗σ SWAP}=ρ_Sσ (see appendix A).

Defining , we thus find that

i.e.

3.4

The above procedure allows us to generate small rotations of a qubit system about a particular state σ. Note that this trick was used previously in 22 in a quantum algorithm for principal component analysis. We now show how to generalize this to implement arbitrary small rotations.

Let be the Hilbert space of the system, and be a set of density operators describing states of the system such that is an arbitrary but fixed operator basis of .³ Now consider the following state of R, composed of one copy of each of the states σ_k:

3.5

Suppose that we ultimately wish to implement an arbitary unitary on the system. As global phase factors in U_S do not affect the evolution of the state, we can assume without loss of generality that . Additionally, as all values of can be obtained for θ∈(−π,π], we can choose H such that its eigenvalues lie within this range. As the system is finite dimensional, for any fixed operator basis , 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 . Denote by the unitary V _αk applied on the system and the particle from the reference frame that is in state σ_k. Now consider the action of the sequence of unitaries on the system ρ_S and reference frame σ_R. By keeping the first-order terms in 1/N, similarly to the calculation in the infinitesimal-rotation case, we obtain

3.6

We are now in position to prove the general claim of the theorem, where U_S is an arbitrary unitary on S (which need not be close to the identity).

Consider a reference frame composed of N copies of σ_R:

3.7

The required value of N will depend on the precision to which we will attempt to simulate the unitary U_S.

To 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 sequence will act on the system as the unitary U_H up to an error . The upper bound on the total error after applying the sequence of V _seq will be (see appendix C). Hence

3.8

Given any ϵ>0 and a unitary U_S, it is always possible to find a sufficiently large N such that . Hence, a sufficiently large reference frame will allow us to take the system to a final state within an error upper-bounded by ϵ as required. This is achieved by applying 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. ▪

Interestingly, the synthesis of a unitary on a system by decomposing it into elementary rotations around directions in Hilbert space, has appeared recently and independently in the setting of learning and emulating an unknown unitary 23, further explored in 24. The work of ref. 23 (see footnote 4) even observes that the mechanism can be thought of as a reference frame. In contrast to our approach, which implements a known unitary U using a U-dependent interaction with a fixed reference system, the schemes of 23; 24 to implement an unknown unitary U are based on a fixed interaction with instead a U-dependent reference system. However, conservation laws were not discussed in 23; 24, and the schemes presented there will not generally respect them.

(a). Composite systems

Although the previous result will work for any system, including composite systems, the construction 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 of many particles, where this may pose practical difficulties. Here, we show that our approach can be adapted easily to allow for a ‘circuit’-model of transformations, where any unitary on the system can be implemented by a series of transformations involving a small number of particles from the system and reference frame.

If we have a composite system made up of some finite number of different types of subsystems, then our reference frame is simply the tensor product of the reference frames for each type 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 procedure as before on this composite system and a similar composite system in the reference frame (hence involving only four primitive subsystems). For example, to implement , we would apply to the state . Once we can perform any two-subsystem unitary, we can use standard techniques to construct a circuit to implement any unitary on the system up to the desired accuracy.⁴

If the subsystems are all of the same type, then an alternative approach would be to use (i) the fact that is an entangling two-subsystem gate which commutes with all conserved quantities, and (ii) that such a gate plus all single subsystem unitaries is computationally universal [26]. This would require only bipartite interactions, with the caveat of a more complicated construction.

4. An application to quantum thermodynamics

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; 12; 13; 14; [15,16]. In this section, we briefly review the approach given in [15], and consider how our reference frame can be thought of as a battery in such cases.

The general set-up consists of a thermal bath, an additional system out of equilibrium with respect to the bath, and a battery that stores any work extracted from the system and the bath. The thermal bath can be thought of as a collection of individual subsystems, each of which is in a generalized thermal state τ=(1/Z) e^{−(β₁A₁+⋯+β_kA_k)}, where A_i are an arbitrary set of conserved quantities, β_i are generalized inverse temperatures associated with these conserved quantities⁵ and Z=tr{e}^{−(β₁A₁+⋯+β_kA_k)} is the generalized partition function. We will be particularly interested 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 the battery is treated implicitly. In this scenario, one simply allows any unitary transformation on the thermal state of the bath, and assumes that any change in the conserved quantities of the bath are accounted for by an equal and opposite amount of work of the corresponding type

4.1

in accordance with the first law of thermodynamics. A generalized second law of thermodynamics can then be derived stating that

4.2

where W_Ai is the type A_i work extracted from the bath, and β_i the inverse temperature conjugate to A_i.

In addition, if one now includes an additional non-thermal system in the picture, the amount of extracted work is bounded by

4.3

where is the amount of A_i type work extracted from the system and the bath, and is the free entropy of the system relative to the generalized bath, with the system’s entropy.

Furthermore, given a small set of structural assumptions about the thermal bath, protocols can be constructed which can approximately implement any transformation satisfying the bounds given by equation (4.2) and equation (4.3). Therefore, we learn that in principle we can extract as much A_i type work W_Ai as we want, as long as the other conserved quantities of the bath and system also change to compensate.

In the case in which the conserved quantities commute, and can in principle be stored separately from one another, one can extend these set-ups to include explicit quantum batteries [15]. In particular, given a fixed initial state of a quantum battery, we can map every transformation in the implicit battery scenario onto a global unitary which commutes with all conserved quantities and which achieves the same results to any desired accuracy.

However, how to deal with explicit batteries in the presence of non-commmuting conserved quantities (under strict conservation) was left as an open question in [15], and treated using the L²(G) construction in [16]. Here, we resolve this question by showing how the reference frames introduced earlier can be used as explicit batteries. Furthermore, because the same approach works for any conserved quantities, this also shows how explicit batteries can be constructed when the conserved quantities cannot be separated from each other, or when a battery system of the type in [15] or [16] is not available.

In particular, consider a bath and system composed of a number of subsystems, and a reference frame composed of the same types of subsystems (as described in §3a) which will act as a battery. Any desired protocol on the system and the bath in the implicit battery scenario can be implemented approximately to any desired accuracy given access to a sufficiently large reference frame, while respecting strict conservation of all extensive conserved quantities. Because of these conservation laws,

4.4

If the desired transformation on the system and the bath would extract work W_Ai in the implicit battery scenario, and this transformation is implemented with ϵ precision (in trace norm) using the reference frame, then

4.5

where ∥⋅∥ represents the operator norm. Any deviation between the work extracted in the implicit battery scenario, and the conserved quantities stored in the explicit battery can therefore be 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 the battery correspond to stored or expended work.

5. Conclusion

We have shown that it is possible to construct a simple reference frame, which allows us to apply any desired unitary on the system, and which moreover provides a physical intuition on the nature of the operations that should be applied to achieve it. This reference frame can in addition act as a battery of extensive conserved quantities. Previous general constructions of quantum reference frames [8] have been based on the symmetry group associated with the conserved quantities via Noether’s theorem. This is mathematically elegant, but requires systems and transformations which may be very difficult to find or implement physically. Our alternative approach works universally for any set of conserved quantities, and does not require Noether’s theorem in the sense that we do not need to first construct the group symmetry belonging to the conservation law. Furthermore, it allows for a relatively simple physical implementation involving 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 L²(G)), determines the accuracy with which we can implement a given unitary. Here, we have prioritized the simplicity of the construction, rather than minimizing its dimension or some other indicator of its complexity for a given error. We leave it as an open problem whether more intricate constructions give a better accuracy for the same frame size.

Appendix A. Partial trace relations involving SWAP

Here, we present two helpful relations involving the partial trace and the operation.

A 1

and

A 2

Appendix B. Bound on parameters α_k

Here, we show that the parameters α_k appearing in (where ∥H∥≤π) are bounded by a constant. Consider the real vector space of Hermitian operators with Hibert–Schmidt inner product 〈A,B〉=tr{AB}. Note that, for any fixed operator basis (where for simplicity we have defined ), one can define a dual basis such that . It then follows that and hence from the Cauchy–Schwarz inequality that

B 1

where is the Hilbert–Schmidt norm. Hence,

B 2

provides an upper bound for all parameters α_k.

Appendix C. Overall error bound due to product of small rotations

Here, we prove that performing a sequence of N transformations involving the reference frame, each of which approximately implements up to error , will implement the overall transformation U_S=(U_H)^N up to error . Let us assume that after t steps, the system is in a state ρ^t_S which satisfies

C 1

where ∥⋅∥₁ denotes the trace norm. Hence,

Because at t=1 equation (3.6) implies , after the sequence of N applications of V _seq the error bound is , and the unitary effectively applied on the system is (U_H)^N=U_S.

Appendix D. Further details on the general and bounds

Here, we provide more detailed technical proof of our and bounds, showing that they are indeed bounded in the required sense, and giving an explicit bound in each case showing their dependence on the various constants. In the first case, we wish to show that

D 1

where . Expanding V _α and as power series in 1/N and using (3.3), we obtain

D 2

where we have used the fact that the trace norm satisfies ∥tr_B{O_AB}∥₁≤∥O_AB∥₁ [27] and ∥A∥₁=∥UAV ∥₁ for unitary U and V . For all N≥2α we therefore have

D 3

Hence, . Note that this is not the tightest provable bound, and a higher threshold for N can generate a smaller (yet more complicated) constant, but this is not the focus of this paper.

For the next stage of the proof, we must show that

D 4

Expanding the exponentials and proceeding almost identically to (D.7) and (D.8), for all N≥2α we obtain

D 5

Hence, . Combining these two results using the triangle inequality, we obtain

D 6

for all N≥2α. This is an explicit form of the bound given in (3.4).

We now consider the bounds for a general small rotation. First consider

D 7

Expanding each of the unitaries as a power series in 1/N and collecting terms in (1/N)ⁿ we note that the terms for n=0 and n=1 cancel (as in the proof in the main paper). The number of terms of order n is equal to the number of ways of distributing n indistinguishable balls in (2D) distinguishable boxes (as we can choose which of the 2D unitaries or generates each power of (1/N)). For simplicity, we upper bound this by the number of ways of distributing n distinguishable balls in (2D) distinguishable boxes, which is (2D)ⁿ. Each term also has a constant coefficient whose magnitude is upper bounded by where for all k and U_S. A derivation of the constant is given in appendix B.

Finally, each term is associated with an operator which is the trace over the reference frame of some sequence of SWAP operations before and after ρ_S⊗σ_R. Writing such a term as tr_R{Π₁ρ_S⊗σ_RΠ₂} where Π₁ and Π₂ are permutations, we see that

D 8

Combining all of these observations, we find that

D 9

As long as , this gives

D 10

Hence . Following an almost identical approach to (D.7), (D.8) and (D.10), one can also show that

D 11

where in the last line, we have used the fact that the dimension of the system space is , and that we can choose H such that its maximum eigenvalue is π. Combining this result with the result for η, we obtain

D 12

for sufficiently large N. Thus using the inductive argument in appendix C gives

D 13

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Authors' contributions

All authors contributed equally to the conception and writing of this paper. A.W. spoke on quantum thermodynamics at the Royal Society discussion meeting Foundations of quantum mechanics and their impact on contemporary society, 11–12 December 2017, and reported, among other things, on the content of [15,16]; this paper addresses some of the issues highlighted there.

Competing interests

We declare we have no competing interests.

Funding

This research was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. S.P. acknowledges support from the ERC Advanced Grant NLST and The Institute for Theoretical Studies, ETH Zurich. A.W. acknowledges support by the Spanish MINECO (project FIS2016-86681-P) with the support of FEDER funds, and the Generalitat de Catalunya (project 2017-SGR-1127).