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Decomposable Coherence and Quantum Fluctuation Relations

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Decomposable Coherence and Quantum Fluctuation Relations Decomposable coherence and quantum fluctuation relations Erick Hinds Mingo1 and David Jennings1,2,3 1Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK 2Department of Physics, University of Oxford, Oxford, OX1 3PU, UK 3School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK. November 11, 2019 In Newtonian mechanics, any closed-system measurement outcomes are random, the state |ψi is a dynamics of a composite system in a microstate perfectly sharp state (a pure state, or state of maximal will leave all its individual subsystems in dis- knowledge) and so should be viewed exactly on a par tinct microstates, however this fails dramatically with a classical state of well-defined location |xii, say. in quantum mechanics due to the existence of More precisely, the structure of the state space in any quantum entanglement. Here we introduce the physical theory determines many of the distinct char- notion of a ‘coherent work process’, and show acteristics of the particular theory. State spaces are that it is the direct extension of a work process always convex sets determined entirely by the extremal in classical mechanics into quantum theory. This points of the set – the pure states of the physical the- leads to the notion of ‘decomposable’ and ‘non- ory. The admissible measurements that the theory al- decomposable’ quantum coherence and gives a lows are defined in relation to this state space (see [5] for new perspective on recent results in the theory more details) and so are secondary theoretical ingredi- of asymmetry as well as early analysis in the ents. Therefore a comparison of classical and quantum theory of classical random variables. Within the mechanics at a fundamental level should equate sharp context of recent fluctuation relations, originally microstates with pure quantum states |ψi, as opposed framed in terms of quantum channels, we show to say comparing measurement statistics. that coherent work processes play the same role In classical mechanics a system with initial position as their classical counterparts, and so provide and conjugate momentum (x0, p0) evolves in time along a simple physical primitive for quantum coher- a well-defined trajectory (x(t), p(t)) in phase space de- ence in such systems. We also introduce a pure termined by the Hamiltonian of the system. This evo- state effective potential as a tool with which to lution extends into quantum mechanics where a quan- analyze the coherent component of these fluc- tum state evolves under a unitary transformation. The tuation relations, and which leads to a notion unitary dynamics of such a system S can also be un- of temperature-dependent mean coherence, pro- derstood in terms of paths, however it is now de- vides connections with multi-partite entangle- scribed in terms of a path integral [6] where the tran- ment, and gives a hierarchy of quantum correc- sition amplitudes are given by integrating the func- tions to the classical Crooks relation in powers i R tional exp[ dt(p(t)x ˙(t)−HS[x(t), p(t)])] over all paths ~ of inverse temperature. (x(t), p(t)) in phase space consistent with the bound- ary conditions, and where HS[x(t), p(t)] is the classi- cal Hamiltonian for the system S. At the operator arXiv:1812.08159v3 [quant-ph] 8 Nov 2019 level this dynamics is described a unitary transforma- 1 Introduction tion on states |ψi → U(t)|ψi, for some unitary operator The superposition principle is at the core of what makes U(t). Thus, in both classical and quantum mechanics quantum mechanics so special [1–4]. Classically a par- the Hamiltonian plays a key role in the time evolution ticle might be located in a microstate at any one of a of a system. Moreover in quantum mechanics, if the number of spatial sites with sharp momentum, however system is energetically closed then we have [U, HS] = 0 quantum mechanics allows a fundamentally new kind for the unitary evolution of the system. In both classi- of state – the particle being a superposition of multiple cal and quantum mechanics an initial sharp (pure) state different locations x , x , . , x . For this, the particle is evolved to a final sharp state. 1 2 √ n P iθk is in a state |ψi = k e pk|xki where θk are phase The path integral perspective describes this unitary angles, and pk is the probability of a measurement of evolution as a sum over all consistent paths, and so re- position returning the classical outcome xk. While the ceives contributions from trajectories that respect nei- Accepted in Quantum 2019-10-24, click title to verify 1 ther energy conservation nor the classical equations of and show that the classical notion of deterministic me- motion. However, in the “limit of ~ → 0” a stationary chanical work can be extended into quantum mechan- phase argument tells us that the dominant contributions ics in a way that exactly parallels the classical case, to the evolution come from those trajectories around and gives rise to connections with topics in contem- the classical phase space trajectory (xclass(t), pclass(t)), porary quantum physics. Firstly, it leads to a notion namely the one that obeys the Hamiltonian equations of decomposable and non-decomposable coherence and of motion [7] given by p˙ = −∂xHS, x˙ = ∂pHS. Thus we has an immediate description in terms of recent frame- recover classical mechanics in the limit. works for quantum coherence [8–11, 17, 18]. We then In the case of two quantum systems S and A, exactly extend these considerations in noisy quantum environ- the same formalism applies, and coherent dynamics of ments and show that this notion of a coherent work pro- the composite quantum system can be analysed without cess connects naturally with fluctuation relations [19– any further fuss – governed by a total Hamiltonian HSA 26]. The coherent fluctuation theorems we develop go for the joint system SA. However, we can now ask the beyond traditional relations, such as the Crooks rela- following question: tion, and allow the analysis of quantum coherence in a noisy thermal environment. Because quantum coher- What unitary transformations of a composite quantum ence is handled explicitly this also leads to connections system SA in a state |ψiSA are possible that (a) obey to many-body entanglement and our results provide a energy conservation over the system SA, but (b) give clean explanatory framework for recent experimental rise to a transformation of the system S from an proposals in trapped ion systems [25]. initial pure state and into a final pure state? It is clear that in the classical limit the answer to this 1.1 Summary of the core results trivial with sharp microstates – all classical dynamics This paper’s aim is to extend classical notions of work to that conserve the energy H give rise to deterministic SA quantum systems and provide novel tools for the anal- transformations of S. However for coherent quantum ysis of quantum thermodynamics and the role that co- systems, which can become entangled with each other, herence plays. To a large extent, quantum thermody- the answer is less obvious. Indeed, as we shall see, the namics has taken inspiration from statistical mechanics above is directly linked to central results from the re- when studying notions of ‘work’ [27–30]. In the present source theory of asymmetry [8–12] and foundational re- analysis, we instead revisit the deterministic Newto- sults from the 1930s on classical random variables [13– nian concept of work and propose a natural coherent 16]. extension into quantum mechanics. We then show how The physical motivation for this is given by consider- this coherent notion of work automatically arises in the ing the elementary notion of a mechanical work process non-deterministic context of recent fully quantum fluc- on a classical system along some trajectory. For this, tuation theorems [23] and provides a fully quantum- a mechanical system S initially in some definite phase mechanical account of thermodynamic processes. space state (x , p ) is evolved deterministically to some 0 0 In Section 2 we begin by defining a coherent work final state (x1, p1) and the mechanical work w for the R process process is computed from w = F · dx, where F (t) ω path |ψ i −→ |ψ i , (1) is the force exerted on the system along the particular 0 S 1 S path [7]. However, the system in this case is not en- which is a deterministic primitive to describe coherent ergetically isolated, and so the quantity of energy w is energy exchanges. The evolution is constrained to con- only meaningful because it corresponds to an equal and serve energy microscopically and preserve the statistical opposite change in energy of some external classical de- independence of the systems. In particular, the exclu- gree of freedom A that is used to induce the process. sion of entanglement generating processes in this defi- For example, we could imagine some external weight of nition is a choice required in order to be the quantum- mass m that is initialised at some definite height h0 = 0 mechanical equivalent of Newtonian work involving a and finishes deterministically in some final height h1. deterministic transition of a ‘weight’ system. Coher- In order to associate the height value h1 for A to w the ent work processes are found to separate into coher- work done in the process on S (via w = mgh1) it is nec- ently trivial and coherently non-trivial types, with the essary that energy is exactly conserved between the two distinction appearing as a consequence of the notion systems S and A.
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