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Proceedings of the Steklov Institute of Mathematics

On Extreme Points of Sets in Operator Spaces and State Spaces

Grigori Gennadievich Amosov, Айрат Мидхатович Бикчентаев, Vsevolod Zhanovich Sakbaev

Published Mar 1, 2024• 3 citations
Publisher

Abstract

We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere $$S_1(\mathcal H)$$ of a Hilbert space $$\mathcal H$$ . For a measure on $$S_1(\mathcal H)$$ , we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital $$\mathrm C^*$$ -algebra are characterized in terms of extreme points. We also study extreme points $$\mathrm{extr}(\mathcal E^1)$$ of the unit ball $$\mathcal E^1$$ of a normed ideal operator space $$\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle$$ on $$\mathcal H$$ . If $$U\in\mathrm{extr}(\mathcal E^1)$$ for some unitary operator $$U\in\mathcal{B}(\mathcal H)$$ , then $$V\in\mathrm{extr}(\mathcal E^1)$$ for all unitary operators $$V\in\mathcal{B}(\mathcal H)$$ . In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.

DOI
10.1134/s0081543824010024

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We obtain a representation of the set of quantum states in terms of barycenters of nonnegative normalized finitely additive measures on the unit sphere S1(H)S_1(\mathcal H)S1​(H) of a Hilbert space H\mathcal HH . For a measure on S1(H)S_1(\mathcal H)S1​(H) , we find conditions in terms of its properties under which the barycenter of this measure belongs to the set of extreme points of the family of quantum states and to the set of normal states. The unitary elements of a unital C∗\mathrm C^*C∗ -algebra are characterized in terms of extreme points. We also study extreme points extr(E1)\mathrm{extr}(\mathcal E^1)extr(E1) of the unit ball E1\mathcal E^1E1 of a normed ideal operator space ⟨E,∥⋅∥E⟩\langle\mathcal E,\|\kern1pt{\cdot}\kern1pt\|_{\mathcal E}\rangle⟨E,∥⋅∥E​⟩ on H\mathcal HH . If U∈extr(E1)U\in\mathrm{extr}(\mathcal E^1)U∈extr(E1) for some unitary operator U∈B(H)U\in\mathcal{B}(\mathcal H)U∈B(H) , then V∈extr(E1)V\in\mathrm{extr}(\mathcal E^1)V∈extr(E1) for all unitary operators V∈B(H)V\in\mathcal{B}(\mathcal H)V∈B(H) . In addition, we construct quantum correlations corresponding to singular states on the algebra of all bounded operators in a Hilbert space.
We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k≤d2 points of the outcome space, d<∞ being the dimension of the Hilbert space. We prove that for second-countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein–Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k≤d2 points of the outcome space.
We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k≤d2 points of the outcome space, d<∞ being the dimension of the Hilbert space. We prove that for second-countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein–Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k≤d2 points of the outcome space.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.
Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.
The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.