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arXiv:1912.02099 · Quantum

Energy upper bound for structurally stable N -passive states

Raffaele Salvia, Vittorio Giovannetti

Published May 28, 2020• 9 citations
arXivHTMLPDF

Abstract

Passive states are special configurations of a quantum system which exhibit no energy decrement at the end of an arbitrary cyclic driving of the model Hamiltonian. When applied to an increasing number of copies of the initial density matrix, the requirement of passivity induces a hierarchical ordering which, in the asymptotic limit of infinitely many elements, pinpoints ground states and thermal Gibbs states. In particular, for large values of N the energy content of a N -passive state which is also structurally stable (i.e. capable to maintain its passivity status under small perturbations of the model Hamiltonian), is expected to be close to the corresponding value of the thermal Gibbs state which has the same entropy. In the present paper we provide a quantitative assessment of this fact, by producing an upper bound for the energy of an arbitrary N -passive, structurally stable state which only depends on the spectral properties of the Hamiltonian of the system. We also show the condition under which our inequality can be saturated. A generalization of the bound is finally presented that, for sufficiently large N , applies to states which are N -passive, but not necessarily structurally stable.

DOI
10.22331/q-2020-05-28-274

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Passive states are special configurations of a quantum system which exhibit no energy decrement at the end of an arbitrary cyclic driving of the model Hamiltonian. When applied to an increasing number of copies of the initial density matrix, the requirement of passivity induces a hierarchical ordering which, in the asymptotic limit of infinitely many elements, pinpoints ground states and thermal Gibbs states. In particular, for large values of N the energy content of a N -passive state which is also structurally stable (i.e. capable to maintain its passivity status under small perturbations of the model Hamiltonian), is expected to be close to the corresponding value of the thermal Gibbs state which has the same entropy. In the present paper we provide a quantitative assessment of this fact, by producing an upper bound for the energy of an arbitrary N -passive, structurally stable state which only depends on the spectral properties of the Hamiltonian of the system. We also show the condition under which our inequality can be saturated. A generalization of the bound is finally presented that, for sufficiently large N , applies to states which are N -passive, but not necessarily structurally stable.
Passive states are defined as those states that do not allow for work extraction in a cyclic (unitary) process. Within the set of passive states, thermal states are the most stable ones: they maximize the entropy for a given energy, and similarly they minimize the energy for a given entropy. Here we find the passive states lying in the other extreme, i.e., those that maximize the energy for a given entropy, which we show also minimize the entropy when the energy is fixed. These extremal properties make these states useful to obtain fundamental bounds for the thermodynamics of finite-dimensional quantum systems, which we show in several scenarios.
Passive states of quantum systems are states from which no system energy can be extracted by any cyclic (unitary) process. Gibbs states of all temperatures are passive. Strong local (SL) passive states are defined to allow any general quantum operation, but the operation is required to be local, being applied only to a specific subsystem. Any mixture of eigenstates in a system-dependent neighborhood of a nondegenerate entangled ground state is found to be SL passive. In particular, Gibbs states are SL passive with respect to a subsystem only at or below a critical system-dependent temperature. SL passivity is associated in many-body systems with the presence of ground state entanglement in a way suggestive of collective quantum phenomena such as quantum phase transitions, superconductivity, and the quantum Hall effect. The presence of SL passivity is detailed for some simple spin systems where it is found that SL passivity is neither confined to systems of only a few particles nor limited to the near vicinity of the ground state.
In this article, in addition to the characterization of geometrical state spaces for the passive states, an operational approach has been introduced to distinguish them on their charging capabilities of a quantum battery. Unlike the thermal states, the structural instability of passive states assures the existence of a natural number n, for which n+1 copies of the state can charge a quantum battery while n copies cannot. This phenomenon can be presented in an n copy resource-theoretic approach, for which the free states are unable to charge the battery in n copies. Here we have exhibited the single copy scenario explicitly. We also show that general ordering of the passive states on the basis of their charging capabilities is not possible and even the macroscopic entities (viz. energy and entropy) are unable to order them precisely. Interestingly, for some of the passive states, the majorization criterion gives sufficient order to the charging and discharging capabilities. However, the charging capacity for the set of thermal states (for which charging is possible) is directly proportional to their temperature.
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.
Passive states, i.e., those states from which no work can be extracted via unitary operations, play an important role in the foundations and applications of quantum thermodynamics. They generalize the familiar Gibbs thermal states, which are the sole passive states being stable under tensor product. Here, we introduce a partial order on the set of passive states that captures the idea of a passive state being virtually cooler than another one. This partial order, which we build by defining the notion of relative passivity, offers a fine-grained comparison between passive states based on virtual temperatures (just like thermal states are compared based on their temperatures). We then characterize the quantum operations that are closed on the set of virtually cooler states with respect to some fixed input and output passive states. Viewing the activity, i.e., nonpassivity, of a state as a resource, our main result is then a necessary and sufficient condition on the transformation of a class of pure active states under these relative passivity-preserving operations. This condition gives a quantum thermodynamical meaning to the majorization relation on the set of nonincreasing vectors due to Hoffman. The maximum extractable work under relative passivity-preserving operations is then shown to be equal to the ergotropy of these pure active states. Finally, we are able to fully characterize passivity-preserving operations in the simpler case of qubit systems, and hence to derive a state interconversion condition under passivity-preserving qubit operations. The prospect of this work is a general resource-theoretical framework for the extractable work via quantum operations going beyond thermal operations.
Abstract 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.