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Mathematical Notes

On analogs of spectral decomposition of a quantum state

Grigori Gennadievich Amosov, Vsevolod Zhanovich Sakbaev

Published Mar 1, 2013• 8 citations
Publisher

Abstract

The set of quantum states in a Hilbert space is considered. The structure of the set of extreme points of the set of states is investigated and an arbitrary state is represented as the Pettis integral over a finitely additive measure on the set of vector states, which is a generalization of the spectral decomposition of a normal state.
DOI
10.1134/s0001434613030012

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In this paper we consider the set of quantum states and passages to the limit for sequences of quantum dynamic semigroups in the mentioned set. We study the structure of the set of extreme points of the quantum state set and represent an arbitrary state as an integral over the set of one-dimensional orthogonal projectors; the obtained representation is similar to the spectral decomposition of a normal state. We apply the obtained results to the analysis of sequences of quantum dynamic semigroups which occur in the regularization of a degenerate Hamiltonian.
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.
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.
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.
1. Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.