arXiv:2407.05950

Quantum states as countable convex combination of pure states with bounded energy

Juan Pablo Lopez

Published Jul 2, 2024• 0 citations

Abstract

We give response to the question: in infinite dimension states,given a state with energy bounded by E, we can write the state as a countable convex combination of pure states with energy bounded by E. We review the Alicki-Fannes-Winter technique to obtain a uniform continuity bound for the von Neumann entropy in states that are a mix of pure states with bounded energy, using this bound we conclude that for a Hamiltonian satisfying the Gibb's hypothesis such states cannot exist.

DOI: 10.48550/arxiv.2407.05950

Related papers

On a conjectured property of the von Neumann entropy valid in the commutative case

Максим Евгеньевич Широков

arXiv:1201.1093Jan 5, 20120 citations

Extreme points of the set of quantum states with bounded energy

Stephan Weis, Максим Евгеньевич Широков

We show that for any energy observable every extreme point of the set of quantum states with bounded energy is a pure state. This allows us to write every state with bounded energy in terms of a continuous convex combination of pure states of bounded energy. Furthermore, we prove that any quantum state with finite energy can be represented as a continuous convex combination of pure states with the same energy. We discuss examples from quantum information theory.

arXiv:2002.03969Russian Mathematical SurveysFeb 1, 20218 citations

Quantum States as Countable Convex Combination of Pure States with Bounded Energy

Juan Pablo Lopez

Trends in mathematicsJan 1, 20260 citations

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Simon Becker, Nilanjana Datta, Michael G. Jabbour

arXiv:2104.02019IEEE Transactions on Information TheoryFeb 23, 202316 citations

Quantum States as Countable Convex Combination of Pure States with Bounded Energy

Juan Pablo Lopez

Jan 1, 20241 citation

Optimal continuity bound for the von Neumann entropy under energy constraints

Becker, S., Nilanjana Datta, Michael G. Jabbour +1 more

arXiv:2410.02686Oct 3, 20241 citation

arXiv:2407.05950

Quantum states as countable convex combination of pure states with bounded energy

Juan Pablo Lopez

Published Jul 2, 2024• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2407.05950

Related papers

On a conjectured property of the von Neumann entropy valid in the commutative case

Максим Евгеньевич Широков

It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian $H$ of the system satisfies the condition $\Tr\exp(-cH)0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian $H$ satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian $H$ of the system satisfies the condition $\Tr\exp(-cH)0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian $H$ satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.

arXiv:1201.1093Jan 5, 20120 citations

Extreme points of the set of quantum states with bounded energy

Stephan Weis, Максим Евгеньевич Широков

arXiv:2002.03969Russian Mathematical SurveysFeb 1, 20218 citations

Quantum States as Countable Convex Combination of Pure States with Bounded Energy

Juan Pablo Lopez

Trends in mathematicsJan 1, 20260 citations

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

Simon Becker, Nilanjana Datta, Michael G. Jabbour

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$ \alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

arXiv:2104.02019IEEE Transactions on Information TheoryFeb 23, 202316 citations

Quantum States as Countable Convex Combination of Pure States with Bounded Energy

Juan Pablo Lopez

Jan 1, 20241 citation

Optimal continuity bound for the von Neumann entropy under energy constraints

Becker, S., Nilanjana Datta, Michael G. Jabbour +1 more

Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].Using techniques proposed in [Sason, IEEE Trans. Inf. Th. 59, 7118 (2013)] and [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)], and based on the results from the latter, we construct a globally optimal continuity bound for the von Neumann entropy. This bound applies to any state under energy constraints imposed by arbitrary Hamiltonians that satisfy the Gibbs hypothesis. This completely solves the problem of finding an optimal continuity bound for the von Neumann entropy in this setting, previously known only for pairs of states that are sufficiently close to each other. Our main technical result, a globally optimal semicontinuity bound for the von Neumann entropy under general energy constraints, leads to this continuity bound. To prove it, we also derive an optimal Fano-type inequality for random variables with a countably infinite alphabet and a general constraint, as well as optimal semicontinuity and continuity bounds for the Shannon entropy in the same setting. In doing so, we improve the results derived in [Becker, Datta and Jabbour, IEEE Trans. Inf. Th. 69, 4128 (2023)].

arXiv:2410.02686Oct 3, 20241 citation

It is well known that the von Neumann entropy is continuous on a subset of quantum states with bounded energy provided the Hamiltonian

H

of the system satisfies the condition $\Tr\exp(-cH)0$. In this note we consider the following conjecture: every closed convex subset of quantum states, on which the von Neumann entropy is continuous, consists of states with bounded energy with respect to a particular Hamiltonian

H

satisfying the above condition. It is shown that the classical analog of this conjecture is valid (i.e. it is valid for the Shannon entropy). It is also shown that this conjecture holds for some types of subsets consisting of non-commuting states, but its validity for all subsets of quantum states remains an open question.

-Rényi and

\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.