Beyond density matrices: Geometric quantum states

Anzà, Fabio; Crutchfield, James P.

doi:10.1103/physreva.103.062218

Cited by 12 publications

(21 citation statements)

References 38 publications

(32 reference statements)

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“…The resulting formalism is equivalent to the standard one in familiar cases. However, it allows working with a new kind of quantum state, dubbed the geometric quantum state [1], that generalizes the familiar density matrix and provides more information about a quantum system's physical configuration.…”

Section: Discussionmentioning

confidence: 99%

“…The geometric formalism and the emergence of continuous mixed states, introduced in the prequel Ref. [1], suggest a new concept of Boltzmann entropy for quantum states-one markedly closer to that in classical statistical mechanics:…”

Section: Discussionmentioning

confidence: 99%

“…This builds on two companion works. The first argues that geometric quantum states, not density matrices, completely characterize the state of quantum systems [1]. While the second connects to experiments, introducing a method to estimate them, via a maximum entropy principle, starting from knowledge of the density matrix [2].…”

Section: Introductionmentioning

confidence: 99%

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Geometric Quantum Thermodynamics

Anzà¹,

Crutchfield²

2020

Preprint

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Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We develop both microcanonical and canonical ensembles, introducing continuous mixed states as distributions on the manifold of quantum states. We call out the experimental consequences for a gas of qudits. We define quantum heat and work in an intrinsic way, including single-trajectory work, and reformulate thermodynamic entropy in a way that accords with classical, quantum, and information-theoretic entropies. We give both the First and Second Laws of Thermodynamics and Jarzynki's Fluctuation Theorem. The result is a more transparent physics, than conventionally available, in which the mathematical structure and physical intuitions underlying classical and quantum dynamics are seen to be closely aligned.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Geometric Quantum Thermodynamics

Anzà¹,

Crutchfield²

2020

Preprint

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“…In CM, dynamical systems theory provides the correct set of tools to model and explain the emergent stochastic dynamics of both Hamiltonian and dissipative systems. Building on previous results [1] that leverage the geometric parallels between classical and quantum state spaces (both symplectic manifolds), in this manuscript we extend several tools of analysis for the out-of-equilibrium phenomena of classical systems to the quantum domain. This strengthens the parallels and provides a novel paradigm for investigating open quantum systems out of equilibrium.…”

Section: Introductionmentioning

confidence: 99%

Quantum Information Dimension and Geometric Entropy

Anzà¹,

Crutchfield²

2021

Preprint

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Geometric quantum mechanics, through its differential-geometric underpinning, provides additional tools of analysis and interpretation that bring quantum mechanics closer to classical mechanics: state spaces in both are equipped with symplectic geometry. This opens the door to revisiting foundational questions and issues, such as the nature of quantum entropy, from a geometric perspective. Central to this is the concept of geometric quantum state-the probability measure on a system's space of pure states. This space's continuity leads us to introduce two analysis tools, inspired by Renyi's information theory, to characterize and quantify fundamental properties of geometric quantum states: the quantum information dimension that is the rate of geometric quantum state compression and the dimensional geometric entropy that monitors information stored in quantum states. We recount their classical definitions, information-theoretic meanings, and physical interpretations, and adapt them to quantum systems via the geometric approach. We then explicitly compute them in various examples and classes of quantum system. We conclude commenting on future directions for information in geometric quantum mechanics.

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“…We immediately recognize that to proceed, we have to consider a generalization of the relative entropy to geometric quantum states. In complete analogy to the classical case, we need to guarantee that the geometric quantum states have the same support [51]. Hence, we introduce a geometric quantum generalization of the conditional distribution to include a generalized transition probability.…”

mentioning

confidence: 99%

Quantum and classical ergotropy from relative entropies

Sone,

Deffner

2021

Preprint

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The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. We prove that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state. This insight is exploited to define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both, quantum as well as classical scenarios, is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.

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Beyond density matrices: Geometric quantum states

Cited by 12 publications

References 38 publications

Geometric Quantum Thermodynamics

Geometric Quantum Thermodynamics

Quantum Information Dimension and Geometric Entropy

Quantum and classical ergotropy from relative entropies

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