Imperial College London

DrDavidJennings

Faculty of Natural SciencesDepartment of Physics

Research Associate
 
 
 
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Contact

 

+44 (0)20 7594 0971d.jennings Website

 
 
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Location

 

Electrical EngineeringSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
to

49 results found

Alexander R, Gvirtz-Chen S, Koukoulekidis N, Jennings Det al., 2023, General entropic constraints on CSS codes within magic distillation protocols, PRX Quantum, Vol: 4, ISSN: 2691-3399

Magic states are fundamental building blocks on the road to fault-tolerant quantum computing. Calderbank-Shor-Steane (CSS) codes play a crucial role in the construction of magic distillation protocols. Previous work has cast quantum computing with magic states for odd dimension d within a phase-space setting in which universal quantum computing is described by the statistical mechanics of quasiprobability distributions. Here we extend this framework to the important d=2 qubit case and show that we can exploit common structures in CSS circuits to obtain distillation bounds capable of outperforming previous monotone bounds in regimes of practical interest. Moreover, in the case of CSS-code projections, we arrive at a novel cutoff result on the code length n of the CSS code in terms of parameters characterizing a desired distillation, which implies that for fixed target error rate and acceptance probability, one needs to consider only CSS codes below a threshold number of qubits. These entropic constraints are not due simply to the data-processing inequality but rely explicitly on the stochastic representation of such protocols.

Journal article

Koukoulekidis N, Kwon H, Jee HH, Jennings D, Kim MSet al., 2022, Faster Born probability estimation via gate merging and frame optimisation, Quantum, Vol: 6, Pages: 838-838, ISSN: 2521-327X

Outcome probability estimation via classical methods is an important task for validating quantum computing devices. Outcome probabilities of any quantum circuit can be estimated using Monte Carlo sampling, where the amount of negativity present in the circuit frame representation quantifies the overhead on the number of samples required to achieve a certain precision. In this paper, we propose two classical sub-routines: circuit gate merging and frame optimisation, which optimise the circuit representation to reduce the sampling overhead. We show that the runtimes of both sub-routines scale polynomially in circuit size and gate depth. Our methods are applicable to general circuits, regardless of generating gate sets, qudit dimensions and the chosen frame representations for the circuit components. We numerically demonstrate that our methods provide improved scaling in the negativity overhead for all tested cases of random circuits with Clifford+T and Haar-random gates, and that the performance of our methods compares favourably with prior quasi-probability simulators as the number of non-Clifford gates increases.

Journal article

Alexander R, Gvirtz-Chen S, Jennings D, 2022, Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system, New Journal of Physics, Vol: 24, ISSN: 1367-2630

Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.

Journal article

Koukoulekidis N, Jennings D, 2022, Constraints on magic state protocols from the statistical mechanics of Wigner negativity, NPJ QUANTUM INFORMATION, Vol: 8

Journal article

Girling M, Cirstoiu C, Jennings D, 2022, Estimation of correlations and non-separability in quantum channels via unitarity benchmarking, Physical Review Research, Vol: 4, ISSN: 2643-1564

The ability to transfer quantum information between systems is a fundamental component of quantum technologies and leads to correlations within the global quantum process. However, correlation structures in quantum channels are less studied than those in quantum states. Motivated by recent techniques in randomized benchmarking, we develop a range of results for efficient estimation of correlations within a bipartite quantum channel. We introduce subunitarity measures that are invariant under local changes of basis, generalize the unitarity of a channel, and allow for the analysis of quantum information exchange within channels. Using these, we show that unitarity is monogamous, and we provide an information-disturbance relation. We then define a notion of correlated unitarity that quantifies the correlations within a given channel. Crucially, we show that this measure is strictly bounded on the set of separable channels and therefore provides a witness of nonseparability. Finally, we describe how such measures for effective noise channels can be efficiently estimated within different randomized benchmarking protocols. We find that the correlated unitarity can be estimated in a SPAM-robust manner for any separable quantum channel, and we show that a benchmarking/tomography protocol with mid-circuit resets can reliably witness nonseparability for sufficiently small reset errors. The tools we develop provide information beyond that obtained via simultaneous randomized benchmarking and so could find application in the analysis of cross-talk errors in quantum devices.

Journal article

Koukoulekidis N, Jennings D, 2022, Constraints on magic state protocols from the statistical mechanics of Wigner negativity, Publisher: Nature Research

Magic states are key ingredients in schemes to realize universalfault-tolerant quantum computation. Theories of magic states attempt toquantify this computational element via monotones and determine how thesestates may be efficiently transformed into useful forms. Here, we develop astatistical mechanical framework based on majorization to describe Wignernegative magic states for qudits of odd prime dimension processed underClifford circuits. We show that majorization allows us to both quantifydisorder in the Wigner representation and derive upper bounds for magicdistillation. These bounds are shown to be tighter than other bounds, such asfrom mana and thauma, and can be used to incorporate hardware physics, such astemperature dependence and system Hamiltonians. We also show that a subset ofsingle-shot R\'{e}nyi entropies remain well-defined on quasi-distributions, arefully meaningful in terms of data processing and can acquire negative valuesthat signal magic. We find that the mana of a magic state is the measure ofdivergence of these R\'{e}nyi entropies as one approaches the Shannon entropyfor Wigner distributions, and discuss how distillation lower bounds could beobtained in this setting. This use of majorization for quasi-distributionscould find application in other studies of non-classicality, and raises novelquestions in the context of classical statistical mechanics.

Working paper

Walton A, Ghesquiere A, Brumpton G, Jennings D, Varcoe Bet al., 2021, Thermal state quantum key distribution, Journal of Physics B: Atomic, Molecular and Optical Physics, Vol: 54, ISSN: 0953-4075

We analyse a central broadcast continuous variable quantum key distribution protocol in which a beam produced by a thermal source is used to create a secret key between two parties, Alice and Bob. A beam splitter divides the initial beam into a pair of output beams, which are sent to Alice and Bob, with Eve intercepting Bob's beam. We investigate the protocol in detail, calculating mutual informations through a pair of analytic methods and comparing the results to the outputs of a Monte Carlo simulation of the protocol. In a lossless system, we find that a lower bound on the key rate remains positive in the protocol under a beam splitter attack, provided Bob receives a nonzero proportion of the beam initially sent to him. This suggests that the thermal state protocol could be used experimentally to produce secure keys.

Journal article

Koukoulekidis N, Alexander R, Hebdige T, Jennings Det al., 2021, The geometry of passivity for quantum states and a novel elementary derivation of the Gibbs state, Quantum: the open journal for quantum science, Vol: 5, Pages: 1-24, ISSN: 2521-327X

Passivity is a fundamental concept that constitutes a necessary condition for any quantum system to attain thermodynamic equilibrium, and for a notion of temperature to emerge. While extensive work has been done that exploits this, the transition from passivity at a single-shot level to the completely passive Gibbs state is technically clear but lacks a good over-arching intuition. Here, we reformulate passivity for quantum systems in purely geometric terms. This description makes the emergence of the Gibbs state from passive states entirely transparent. Beyond clarifying existing results, it also provides novel analysis for non-equilibrium quantum systems. We show that, to every passive state, one can associate a simple convex shape in a 2-dimensional plane, and that the area of this shape measures the degree to which the system deviates from the manifold of equilibrium states. This provides a novel geometric measure of athermality with relations to both ergotropy and β--athermality.

Journal article

Jennings D, Cirstoiu C, Korzekwa K, 2020, Robustness of Noether's principle: maximal disconnects between conservation laws and symmetries in quantum theory, Physical Review X, Vol: 10, Pages: 041035 – 1-041035 – 41, ISSN: 2160-3308

To what extent does Noether’s principle apply to quantum channels? Here, we quantify the degreeto which imposing a symmetry constraint on quantum channels implies a conservation law, and showthat this relates to physically impossible transformations in quantum theory, such as time-reversaland spin-inversion. In this analysis, the convex structure and extremal points of the set of quantumchannels symmetric under the action of a Lie group G becomes essential. It allows us to derivebounds on the deviation from conservation laws under any symmetric quantum channel in terms ofthe deviation from closed dynamics as measured by the unitarity of the channel E. In particular,we investigate in detail the U(1) and SU(2) symmetries related to energy and angular momentumconservation laws. In the latter case, we provide fundamental limits on how much a spin-jA systemcan be used to polarise a larger spin-jB system, and on how much one can invert spin polarisationusing a rotationally-symmetric operation. Finally, we also establish novel links between unitarity,complementary channels and purity that are of independent interest.

Journal article

Mingo EH, Jennings D, 2019, Decomposable coherence and quantum fluctuation relations, Quantum, Vol: 3, Pages: 202-202, ISSN: 2521-327X

In Newtonian mechanics, any closed-system dynamics of a composite system in a microstate will leave all its individual subsystems in distinct microstates, however this fails dramatically in quantum mechanics due to the existence of quantum entanglement. Here we introduce the notion of a `coherent work process', and show that it is the direct extension of a work process in classical mechanics into quantum theory. This leads to the notion of `decomposable' and `non-decomposable' quantum coherence and gives a new perspective on recent results in the theory of asymmetry as well as early analysis in the theory of classical random variables. Within the context of recent fluctuation relations, originally framed in terms of quantum channels, we show that coherent work processes play the same role as their classical counterparts, and so provide a simple physical primitive for quantum coherence in such systems. We also introduce a pure state effective potential as a tool with which to analyze the coherent component of these fluctuation relations, and which leads to a notion of temperature-dependent mean coherence, provides connections with multi-partite entanglement, and gives a hierarchy of quantum corrections to the classical Crooks relation in powers of inverse temperature.

Journal article

Holmes Z, Weidt S, Jennings D, Anders J, Mintert Fet al., 2019, Coherent fluctuation relations: from the abstract to the concrete, Quantum, Vol: 3, ISSN: 2521-327X

Recent studies using the quantum information theoretic approach to thermodynamics show that the presence of coherence in quantum systems generates corrections to classical fluctuation theorems. To explicate the physical origins and implications of such corrections, we here convert an abstract framework of an autonomous quantum Crooks relation into quantum Crooks equalities for well-known coherent, squeezed and cat states. We further provide a proposal for a concrete experimental scenario to test these equalities. Our scheme consists of the autonomous evolution of a trapped ion and uses a position dependent AC Stark shift.

Journal article

Hebdige T, Jennings D, 2019, On the classification of two-qubit group orbits and the use of coarse-grained 'shape' as a superselection property, Quantum, Vol: 3, ISSN: 2521-327X

Recently a complete set of entropic conditions has been derived for the interconversion structure of states under quantum operations that respect a specified symmetry action, however the core structure of these conditions is still only partially understood. Here we develop a coarse-grained description with the aim of shedding light on both the structure and the complexity of this general problem. Specifically, we consider the degree to which one can associate a basic `shape' property to a quantum state or channel that captures coarse-grained data either for state interconversion or for the use of a state within a simulation protocol. We provide a complete solution for the two-qubit case under the rotation group, give analysis for the more general case and discuss possible extensions of the approach.

Journal article

Gour G, Jennings D, Buscemi F, Duan R, Marvian Iet al., 2018, Quantum majorization and a complete set of entropic conditions for quantum thermodynamics, Nature Communications, Vol: 9, ISSN: 2041-1723

What does it mean for one quantum process to be more disordered than another? Interestingly,this apparently abstract question arises naturally in a wide range of areas such as information the-ory, thermodynamics, quantum reference frames and the resource theory of asymmetry. Here weuse a quantum-mechanical generalization of majorization to develop a framework for answering thisquestion, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. Wealso investigate some of the applications of this framework, and remarkably find that, in the contextof quantum thermodynamics it provides the first complete set of necessary and sufficient conditionsfor arbitrary quantum state transformations under thermodynamic processes, which rigorously ac-counts for quantum-mechanical properties, such as coherence. Our framework of generalized thermalprocesses extends thermal operations, and is based on natural physical principles, namely, energyconservation, the existence of equilibrium states, and the requirement that quantum coherence beaccounted for thermodynamically.

Journal article

Kwon H, Jeong H, Jennings D, Yadin B, Kim MSet al., 2018, Clock-work trade-off relation for coherence in quantum thermodynamics, Physical Review Letters, Vol: 120, ISSN: 0031-9007

In thermodynamics, quantum coherences—superpositions between energy eigenstates—behave in distinctly nonclassical ways. Here we describe how thermodynamic coherence splits into two kinds—“internal” coherence that admits an energetic value in terms of thermodynamic work, and “external” coherence that does not have energetic value, but instead corresponds to the functioning of the system as a quantum clock. For the latter form of coherence, we provide dynamical constraints that relate to quantum metrology and macroscopicity, while for the former, we show that quantum states exist that have finite internal coherence yet with zero deterministic work value. Finally, under minimal thermodynamic assumptions, we establish a clock–work trade-off relation between these two types of coherences. This can be viewed as a form of time-energy conjugate relation within quantum thermodynamics that bounds the total maximum of clock and work resources for a given system.

Journal article

Hinds Mingo E, Guryanova Y, Faist P, Jennings Det al., 2018, Quantum Thermodynamics with Multiple Conserved Quantities, Vol: 195, Pages: 751-771, ISSN: 0168-1222

In this chapter we address the topic of quantum thermodynamics in the presence of additional observables beyond the energy of the system. In particular we discuss the special role that the generalized Gibbs ensemble plays in this theory, and derive this state from the perspectives of a micro-canonical ensemble, dynamical typicality and a resource-theory formulation. A notable obstacle occurs when some of the observables do not commute, and so it is impossible for the observables to simultaneously take on sharp microscopic values. We show how this can be circumvented, discuss information-theoretic aspects of the setting, and explain how thermodynamic costs can be traded between the different observables. Finally, we discuss open problems and future directions for the topic.

Journal article

Sparaciari C, Jennings D, Oppenheim J, 2017, Energetic instability of passive states in thermodynamics, NATURE COMMUNICATIONS, Vol: 8, ISSN: 2041-1723

Passivity is a fundamental concept in thermodynamics that demands a quantum system’s energy cannot be lowered by any reversible, unitary process acting on the system. In the limit of many such systems, passivity leads in turn to the concept of complete passivity, thermal states and the emergence of a thermodynamic temperature. Here we only consider a single system and show that every passive state except the thermal state is unstable under a weaker form of reversibility. Indeed, we show that given a single copy of any athermal quantum state, an optimal amount of energy can be extracted from it when we utilise a machine that operates in a reversible cycle. This means that for individual systems, the only form of passivity that is stable under general reversible processes is complete passivity, and thus provides a physically motivated identification of thermal states when we are not operating in the thermodynamic limit.

Journal article

Lostaglio M, Jennings D, Rudolph T, 2017, Thermodynamic resource theories, non-commutativity and maximum entropy principles, NEW JOURNAL OF PHYSICS, Vol: 19, ISSN: 1367-2630

We discuss some features of thermodynamics in the presence of multiple conserved quantities. We prove a generalisation of Landauer principle illustrating tradeoffs between the erasure costs paid in different 'currencies'. We then show how the maximum entropy and complete passivity approaches give different answers in the presence of multiple observables. We discuss how this seems to prevent current resource theories from fully capturing thermodynamic aspects of non-commutativity.

Journal article

Korzekwa K, Lostaglio M, Jennings D, Oppenheim Jet al., 2016, The extraction of work from quantum coherence, New Journal of Physics, Vol: 18, ISSN: 1367-2630

The interplay between quantum-mechanical properties, such as coherence,and classical notions, such as energy, is a subtle topic at the forefront of quantumthermodynamics. The traditional Carnot argument limits the conversion of heat towork; here we critically assess the problem of converting coherence to work. Through acareful account of all resources involved in the thermodynamic transformations withina fully quantum-mechanical treatment, we show that there exist thermal machinesextracting work from coherence arbitrarily well. Such machines only need to act onindividual copies of a state and can be reused. On the other hand, we show thatfor any thermal machine with finite resources not all the coherence of a state can beextracted as work. However, even bounded thermal machines can be reused infinitelymany times in the process of work extraction from coherence.

Journal article

Frenzel MF, Jennings D, Rudolph T, 2016, Quasi-autonomous quantum thermal machines and quantum to classical energy flow, New Journal of Physics, Vol: 18, ISSN: 1367-2630

There are both practical and foundational motivations to consider the thermodynamics of quantumsystems at small scales. Here we address the issue of autonomous quantum thermal machinesthat are tailored to achieve some specific thermodynamic primitive, such as work extraction in thepresence of a thermal environment, while having minimal or no control from the macroscopic regime.Beyond experimental implementations, this provides an arena in which to address certain foundationalaspects such as the role of coherence in thermodynamics, the use of clock degrees of freedomand the simulation of local time-dependent Hamiltonians in a particular quantum subsystem. Forsmall-scale systems additional issues arise. Firstly, it is not clear to what degree genuine orderedthermodynamic work has been extracted, and secondly non-trivial back-actions on the thermal machinemust be accounted for. We find that both these aspects can be resolved through a judiciouschoice of quantum measurements that magnify thermodynamic properties up the ladder of lengthscales,while simultaneously stabilizing the quantum thermal machine. Within this framework weshow that thermodynamic reversibility is obtained in a particular Zeno limit, and finally illustratethese concepts with a concrete example involving spin-systems.

Journal article

Jevtic S, Jennings D, Rudolph T, Hirono Y, Nakayama S, Murao Met al., 2015, Exchange Fluctuation Theorem for correlated quantum systems, Physical Review E, Vol: 92, Pages: 042113-1-042113-12, ISSN: 1539-3755

We extend the Exchange Fluctuation Theorem for energy exchange betweenthermal quantum systems beyond the assumption of molecular chaos, and describethe non-equilibrium exchange dynamics of correlated quantum states. Therelation quantifies how the tendency for systems to equilibrate is modified inhigh-correlation environments. Our results elucidate the role of measurementdisturbance for such scenarios. We show a simple application by finding asemi-classical maximum work theorem in the presence of correlations.

Journal article

Dale H, Jennings D, Rudolph T, 2015, Provable quantum advantage in randomness processing, Nature Communications, Vol: 6, ISSN: 2041-1723

Quantum advantage is notoriously hard to find and even harder to prove. For example the class of functions computable with classical physics exactly coincides with the class computable quantum mechanically. It is strongly believed, but not proven, that quantum computing provides exponential speed-up for a range of problems, such as factoring. Here we address a computational scenario of randomness processing in which quantum theory provably yields, not only resource reduction over classical stochastic physics, but a strictly larger class of problems which can be solved. Beyond new foundational insights into the nature and malleability of randomness, and the distinction between quantum and classical information, these results also offer the potential of developing classically intractable simulations with currently accessible quantum technologies.

Journal article

Jennings D, Leifer M, 2015, No return to classical reality, Contemporary Physics, Vol: 57, Pages: 60-82, ISSN: 0010-7514

At a fundamental level, the classical picture of the world is dead, and has been dead now for almost a century. Pinning down exactly which quantum phenomena are responsible for this has proved to be a tricky and controversial question, but a lot of progress has been made in the past few decades. We now have a range of precise statements showing that whatever the ultimate laws of nature are, they cannot be classical. In this article, we review results on the fundamental phenomena of quantum theory that cannot be understood in classical terms. We proceed by first granting quite a broad notion of classicality, describe a range of quantum phenomena (such as randomness, discreteness, the indistinguishability of states, measurement-uncertainty, measurement-disturbance, complementarity, non-commutativity, interference, the no-cloning theorem and the collapse of the wave-packet) that do fall under its liberal scope, and then finally describe some aspects of quantum physics that can never admit a classical understanding – the intrinsically quantum mechanical aspects of nature. The most famous of these is Bell’s theorem, but we also review two more recent results in this area. Firstly, Hardy’s theorem shows that even a finite-dimensional quantum system must contain an infinite amount of information, and secondly, the Pusey–Barrett–Rudolph theorem shows that the wave function must be an objective property of an individual quantum system. Besides being of foundational interest, results of this sort now find surprising practical applications in areas such as quantum information science and the simulation of quantum systems.

Journal article

Milne A, Jennings D, Rudolph T, 2015, Geometric representation of two-qubit entanglement witnesses, PHYSICAL REVIEW A, Vol: 92, ISSN: 1050-2947

Journal article

Jennings D, Brockt C, Osborne T, Haegeman J, Verstraete Fet al., 2015, Continuum tensor network field states, path integral representations and spatial symmetries, New Journal of Physics, Vol: 17, ISSN: 1367-2630

Journal article

Jennings E, Jennings D, 2015, Non-linear stochastic growth rates and redshift space distortions, MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Vol: 449, Pages: 3407-3419, ISSN: 0035-8711

Journal article

Lostaglio M, Korzekwa K, Jennings D, Rudolph Tet al., 2015, Quantum coherence, time-translation symmetry, and thermodynamics, Physical Review X, Vol: 5, ISSN: 2160-3308

The first law of thermodynamics imposes not just a constraint on the energy content of systems in extreme quantum regimes but also symmetry constraints related to the thermodynamic processing of quantum coherence. We show that this thermodynamic symmetry decomposes any quantum state into mode operators that quantify the coherence present in the state. We then establish general upper and lower bounds for the evolution of quantum coherence under arbitrary thermal operations, valid for any temperature. We identify primitive coherence manipulations and show that the transfer of coherence between energy levels manifests irreversibility not captured by free energy. Moreover, the recently developed thermomajorization relations on block-diagonal quantum states are observed to be special cases of this symmetry analysis.

Journal article

Lostaglio R, Jennings D, Rudolph T, 2015, Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nature Communications, Vol: 6, ISSN: 2041-1723

Recent studies have developed fundamental limitations on nanoscale thermodynamics, in terms of a set of independent free energy relations. Here we show that free energy relations cannot properly describe quantum coherence in thermodynamic processes. By casting time-asymmetry as a quantifiable, fundamental resource of a quantum state, we arrive at an additional, independent set of thermodynamic constraints that naturally extend the existing ones. These asymmetry relations reveal that the traditional Szilárd engine argument does not extend automatically to quantum coherences, but instead only relational coherences in a multipartite scenario can contribute to thermodynamic work. We find that coherence transformations are always irreversible. Our results also reveal additional structural parallels between thermodynamics and the theory of entanglement.

Journal article

Milne A, Jevtic S, Jennings D, Wiseman H, Rudolph Tet al., 2015, Corrigendum: Quantum steering ellipsoids, extremal physical states and monogamy (2014 New J. Phys. 16 083017), New Journal of Physics, Vol: 17, ISSN: 1367-2630

Journal article

Frenzel MF, Jennings D, Rudolph T, 2014, Reexamination of pure qubit work extraction, Physical Review E, Vol: 90, ISSN: 1539-3755

Many work extraction or information erasure processes in the literature involve the raising and loweringof energy levels via external fields. But even if the actual system is treated quantum mechanically, the fieldis assumed to be classical and of infinite strength, hence not developing any correlations with the system orexperiencing back-actions. We extend these considerations to a fully quantum mechanical treatment by studyinga spin-1/2 particle coupled to a finite-sized directional quantum reference frame, a spin-l system, which modelsan external field. With this concrete model together with a bosonic thermal bath, we analyze the back-actiona finite-size field suffers during a quantum-mechanical work extraction process and the effect this has on theextractable work and highlight a range of assumptions commonly made when considering such processes. Thewell-known semiclassical treatment of work extraction from a pure qubit predicts a maximum extractable workW = kT log 2 for a quasistatic process, which holds as a strict upper bound in the fully quantum mechanical caseand is attained only in the classical limit. We also address the problem of emergent local time dependence in ajoint system with a globally fixed Hamiltonian.

Journal article

Milne A, Jennings D, Jevtic S, Rudolph Tet al., 2014, Quantum correlations of two-qubit states with one maximally mixed marginal, PHYSICAL REVIEW A, Vol: 90, ISSN: 1050-2947

Journal article

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