arXiv subjects

Philippe Faist

Publications and source records attributed to Philippe Faist.

29 records · Page 2Linked to original sources

arXiv2019

Continuous symmetries and approximate quantum error correction

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is covariant with respect to a symmetry group $G$ if a $G$ transformation on the logical system can be realized by performing transformations on the individual subsystems. For a $G$-covariant code with $G$ a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems $n$ or the dimension $d$ of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with $n$ or $d$ as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large $d$ (using random codes) or $n$ (using codes based on $W$-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

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arXiv2018

Practical and reliable error bars for quantum process tomography

Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure-of-merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package---QPtomographer---implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of e.g. error correction codes, implementations of unitary gates or more generally any noise process affecting a quantum system.

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arXiv2018

Smooth entropy in axiomatic thermodynamics

Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including microscopic systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy.

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arXiv2018

Quantum thermodynamics with multiple conserved quantities

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.

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arXiv2018

Fundamental work cost of quantum processes

Information-theoretic approaches provide a promising avenue for extending the laws of thermodynamics to the nanoscale. Here, we provide a general fundamental lower limit, valid for systems with an arbitrary Hamiltonian and in contact with any thermodynamic bath, on the work cost for the implementation of any logical process. This limit is given by a new information measure---the coherent relative entropy---which accounts for the Gibbs weight of each microstate. The coherent relative entropy enjoys a collection of natural properties justifying its interpretation as a measure of information, and can be understood as a generalization of a quantum relative entropy difference. As an application, we show that the standard first and second laws of thermodynamics emerge from our microscopic picture in the macroscopic limit. Finally, our results have an impact on understanding the role of the observer in thermodynamics: Our approach may be applied at any level of knowledge---for instance at the microscopic, mesoscopic or macroscopic scales---thus providing a formulation of thermodynamics that is inherently relative to the observer. We obtain a precise criterion for when the laws of thermodynamics can be applied, thus making a step forward in determining the exact extent of the universality of thermodynamics and enabling a systematic treatment of Maxwell-demon-like situations.

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arXiv2017

Axiomatic relation between thermodynamic and information-theoretic entropies

Thermodynamic entropy, as defined by Clausius, characterizes macroscopic observations of a system based on phenomenological quantities such as temperature and heat. In contrast, information-theoretic entropy, introduced by Shannon, is a measure of uncertainty. In this Letter, we connect these two notions of entropy, using an axiomatic framework for thermodynamics [Lieb, Yngvason, Proc. Roy. Soc.(2013)]. In particular, we obtain a direct relation between the Clausius entropy and the Shannon entropy, or its generalisation to quantum systems, the von Neumann entropy. More generally, we find that entropy measures relevant in non-equilibrium thermodynamics correspond to entropies used in one-shot information theory.

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arXiv2016

Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges

The grand canonical ensemble lies at the core of quantum and classical statistical mechanics. A small system thermalizes to this ensemble while exchanging heat and particles with a bath. A quantum system may exchange quantities represented by operators that fail to commute. Whether such a system thermalizes and what form the thermal state has are questions about truly quantum thermodynamics. Here we investigate this thermal state from three perspectives. First, we introduce an approximate microcanonical ensemble. If this ensemble characterizes the system-and-bath composite, tracing out the bath yields the system's thermal state. This state is expected to be the equilibrium point, we argue, of typical dynamics. Finally, we define a resource-theory model for thermodynamic exchanges of noncommuting observables. Complete passivity---the inability to extract work from equilibrium states---implies the thermal state's form, too. Our work opens new avenues into equilibrium in the presence of quantum noncommutation.

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arXiv2016

Quantum Coarse-Graining: An Information-Theoretic Approach to Thermodynamics

We investigate fundamental connections between thermodynamics and quantum information theory. First, we show that the operational framework of thermal operations is nonequivalent to the framework of Gibbs-preserving maps, and we comment on this gap. We then introduce a fully information-theoretic framework generalizing the above by making further abstraction of physical quantities such as energy. It is technically convenient to work with and reproduces known results for finite-size quantum thermodynamics. With our framework we may determine the minimal work cost of implementing any logical process. In the case of information processing on memory registers with a degenerate Hamiltonian, the answer is given by the max-entropy, a measure of information known from quantum information theory. In the general case, we obtain a new information measure, the "coherent relative entropy", which generalizes both the conditional entropy and the relative entropy. It satisfies a collection of properties which justifies its interpretation as an entropy measure and which connects it to known quantities. We then present how, from our framework, macroscopic thermodynamics emerges by typicality, after singling out an appropriate class of thermodynamic states possessing some suitable reversibility property. A natural thermodynamic potential emerges, dictating possible state transformations, and whose differential describes the physics of the system. The textbook thermodynamics of a gas is recovered as well as the form of the second law relating thermodynamic entropy and heat exchange. Finally, noting that quantum states are relative to the observer, we see that the procedure above gives rise to a natural form of coarse-graining in quantum mechanics: Each observer can consistently apply the formalism of quantum information according to their own fundamental unit of information.

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arXiv2016

Practical, Reliable Error Bars in Quantum Tomography

Precise characterization of quantum devices is usually achieved with quantum tomography. However, most methods which are currently widely used in experiments, such as maximum likelihood estimation, lack a well-justified error analysis. Promising recent methods based on confidence regions are difficult to apply in practice or yield error bars which are unnecessarily large. Here, we propose a practical yet robust method for obtaining error bars. We do so by introducing a novel representation of the output of the tomography procedure, the "quantum error bars". This representation is (i) concise, being given in terms of few parameters, (ii) intuitive, providing a fair idea of the "spread" of the error, and (iii) useful, containing the necessary information for constructing confidence regions. The statements resulting from our method are formulated in terms of a figure of merit, such as the fidelity to a reference state. We present an algorithm for computing this representation and provide ready-to-use software. Our procedure is applied to actual experimental data obtained from two superconducting qubits in an entangled state, demonstrating the applicability of our method.

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arXiv2015

The Minimal Work Cost of Information Processing

Irreversible information processing cannot be carried out without some inevitable thermodynamical work cost. This fundamental restriction, known as Landauer's principle, is increasingly relevant today, as the energy dissipation of computing devices impedes the development of their performance. Here we determine the minimal work required to carry out any logical process, for instance a computation. It is given by the entropy of the discarded information conditional to the output of the computation. Our formula takes precisely into account the statistically fluctuating work requirement of the logical process. It enables the explicit calculation of practical scenarios, such as computational circuits or quantum measurements. On the conceptual level, our result gives a precise and operationally justified connection between thermodynamic and information entropy, and explains the emergence of the entropy state function in macroscopic thermodynamics.

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arXiv2015

Gibbs-Preserving Maps outperform Thermal Operations in the quantum regime

In this brief note, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers Thermal Operations---unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal Operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true---Gibbs-Preserving Maps are no more powerful than Thermal Operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-Preserving Map can generate coherent superpositions of energy levels while Thermal Operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.

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