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arXiv:2005.00005

Bistochastic operators and quantum random variables

Sarah Plosker, Christopher Ramsey

Published Apr 30, 2020• 1 citation
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Abstract

Given a positive operator-valued measure $ν$ acting on the Borel sets of a locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional) Hilbert space $\mathcal H$, one can consider $ν$-integrable functions $X\rightarrow \mathcal B(\mathcal H)$ that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.

DOI
10.48550/arxiv.2005.00005

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Given a positive operator-valued measure ννν acting on the Borel sets of a locally compact Hausdorff space XXX, with outcomes in the algebra B(H)\mathcal B(\mathcal H)B(H) of all bounded operators on a (possibly infinite-dimensional) Hilbert space H\mathcal HH, one can consider ννν-integrable functions X→B(H)X\rightarrow \mathcal B(\mathcal H)X→B(H) that are positive quantum random variables. We define a seminorm on the span of such functions which in the quotient leads to a Banach space. We consider bistochastic operators acting on this space and majorization of quantum random variables is then defined with respect to these operators. As in classical majorization theory, we relate majorization in this context to an inequality involving all possible convex functions of a certain type. Unlike the classical setting, continuity and convergence issues arise throughout the work.
It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
We consider positive operator valued measures whose image is the bounded operators acting on an infinite-dimensional Hilbert space, and we relax, when possible, the usual assumption of positivity of the operator valued measure seen in the quantum information theory literature. We define the Radon-Nikodym derivative of a positive operator valued measure with respect to a complex measure induced by a given quantum state; this derivative does not always exist when the Hilbert space is infinite dimensional in so much as its range may include unbounded operators. We define integrability of a positive quantum random variable with respect to a positive operator valued measure. Emphasis is put on the structure of operator valued measures, and we develop positive operator valued versions of the Lebesque decomposition theorem and Johnson's atomic and nonatomic decomposition theorem. Beyond these generalizations, we make connections between absolute continuity and the "cleanness" relation defined on positive operator valued measures as well as to the notion of atomic and nonatomic measures.
We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra B(H)\mathfrak{B}(\mathcal{H})B(H) of bounded operators on a separable Hilbert space H\mathcal{H}H ) to the space of all operators on H\mathcal{H}H relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation Φ\PhiΦ can be uniquely extended to a bounded linear operator on the Banach space of all G\sqrt{G}G​ -bounded operators on H\mathcal{H}H provided that the operation Φ\PhiΦ is GGG -limited: the predual operation Φ∗\Phi_{*}Φ∗​ maps the set of positive trace class operators ρ\rhoρ with finite value of TrρG\textrm{Tr}\rho GTrρG into itself. Assuming that GGG has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the GGG -limited property. Applications to the theory of Bosonic Gaussian channels are considered.
Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C∗C^{*}C∗ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S\mathscr {S}S S of a possibly infinite-dimensional, unital C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A is partitioned into the disjoint union of the orbits of an action of the group G\mathscr {G}G G of invertible elements of A\mathscr {A}A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H\mathcal {H}H H are smooth, homogeneous Banach manifolds of G=GL(H)\mathscr {G}=\mathcal {GL}(\mathcal {H})G=GL(H) G=GL(H) , and, when A\mathscr {A}A A admits a faithful tracial state τ\tau τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ\tau τ τ is a smooth, homogeneous Banach manifold for G\mathscr {G}G G .
We consider random linear operators Ω→L(Tp,Tp)\Omega\to\mathcal{L}(\mathcal{T}_{p},\mathcal{T}_{p})Ω→L(Tp​,Tp​) acting in a ppp th Schatten class Tp\mathcal{T}_{p}Tp​ in a separable Hilbert space H\mathcal{H}H for some 1⩽p<∞1\leqslant p<\infty1⩽p<∞ . Such a superoperator is called pre-channel since it is an extension of a quantum channel to a wider class of operators without requirements of trace-preserving and positivity. Instead of sum of i.i.d. variables there may be considered the composition of random semigroups eAit/ne^{A_{i}t/n}eAi​t/n in the Banach space Tp\mathcal{T}_{p}Tp​ . The law of large numbers is known in the case p=2p=2p=2 in the form of the usual law of large numbers for random operators in a Hilbert space. We obtain the law of large numbers for the case 1⩽p⩽21\leqslant p\leqslant 21⩽p⩽2 .
We introduce three metrics on the set of quantum probability measures over a compact Hausdorff space and characterize them in terms of the completely bounded norm of the corresponding unital completely positive maps.We extend the existing topological structures between scalar-valued information channels to operator-valued ones and associate them with topologies on the set of unital completely positive maps between a commutative C*-algebra and the C*-algebra of bounded weakly measurable operator-valued functions over a compact Hausdorff space.Given a measure on the input alphabet space, we introduce the notion of an almost everywhere defined operator-valued information channel and provide a characterization result.