Abstract
It is shown that a number of optimization problems in quantum information theory: the \(\chi\)-capacity (called the Holevo capacity in literature) of a quantum channel; the classical capacity of quantum observable; entanglement of formation—can be recast as a generalization of a Bayes problem over the set of all quantum states. This allows us to consider it as a convex programming problem for which necessary and sufficient optimality conditions along with the dual problem can be formulated.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Let \(\mathcal{H}\) be a finite-dimensional Hilbert space, \(\mathfrak{T}(\mathcal{H})\) the Banach space of operators on \(\mathcal{H}\) equipped with trace norm, \(\mathfrak{S}=\mathfrak{S}(\mathcal{H})\) the compact convex set of quantum states equipped with trace-norm distance. An ensemble is a probability measure \(\pi(dS)\) on the set \(\mathfrak{S}\), cf. [2], and the set of all such measures is denoted \(\mathcal{P}(\mathfrak{S})\).
Let \(f\) be a continuous concave function on the compact convex set \(\mathfrak{S}\). The proof of theorem below uses very little of other special properties of \(f\) and \(\mathfrak{S}\). Consider the functional
on \(\mathcal{P}(\mathfrak{S})\). From the definition, it is a continuous affine functional. We are interested in minimization of this functional on the closed convex subset \(\mathcal{P}_{\overline{S}}\) of probability measures \(\pi\) with the fixed given barycenter
Under mild additional conditions the functional \(F(\pi)\) attains its minimum \(\mathcal{F}(\overline{S})\) on the compact set \(\mathcal{P}_{\overline{S}}\). The resulting function \(\mathcal{F}(S),S\in\mathfrak{S},\) is convex, in fact it is equal to the convex closure of \(f(S)\) i.e. the greatest lower semicontinuous convex function majorized by \(f(S)\) [7]. Concavity of \(f\) implies then that we can choose the minimizing measure \(\pi\) to be supported by the pure states since we can always make spectral decompositions of all the density operators \(S\) into pure states without changing the barycenter and without increasing the value \(F(\pi)\).
This problem is relevant to a number of issues in quantum information theory.
1. Computation of the \(\chi\)-capacity [2] of a quantum channel \(\Phi\) defined as
Here the second term in the squared brackets is the convex closure of the channel output entropy, with
2. A similar case is the classical capacity of quantum-classical channel (observable) given by the map \(M:S\rightarrow p_{S}(z)=\textrm{Tr\,}Sm(z),\) where \(m(z)\) is a uniformly bounded positive-operator-valued function of \(z\in Z,\) such that \(\int m(z)dz=I\) (the unit operator). Here \((Z,dz)\) is a measure space. Then \(p_{S}(z)\) is the probability density of the outcomes of the quantum measurement described by \(m(z)\). In this case the classical capacity of the channel is equal to the \(\chi\)-capacity (the channel is entanglement-breaking) and
where \(h\left(p_{S}\right)=-\int p_{S}(z)\log p_{S}(z)dz\) is the differential entropy of this probability density and
3. Another case of interest described e.g. in Sec. 7.5 of [1] is the Entanglement of Formation. Let \(S_{12}\) be a state in the tensor product of two Hilbert spaces \(\mathcal{H}_{1}\otimes\mathcal{H}_{2}.\) Entanglement of Formation of the state \(S_{12}\) is the convex closure of \(H(S_{1})\), where \(S_{1}=\) \(\textrm{Tr}_{2}S_{12}\) is the partial state:
minimization is over all probability measures \(\pi(dS)\) on \(\mathfrak{S}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})\) satisfying \(\int_{\mathfrak{S}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})}S\,\pi(dS)=S_{12}.\) In that case \(f(S_{12})=H(S_{1}),K(S_{12})=-\left(\log S_{1}\otimes I_{2}\right).\)
In all these cases we are looking for the solution of the convex programming problem
where \(\overline{S}\) is a fixed density operator. We will give the duality relation and necessary and sufficient conditions for optimality basing on the results obtained in the monograph [5].
We start by introducing an equivalent but more convenient definition: we now call ensemble a measure \(\Pi(dS)\) on \(\mathfrak{S}\) with values in the positive cone of \(\mathfrak{T}(\mathcal{H}),\) such that \(\Pi(\mathfrak{S})\in\mathfrak{S}.\) We call \(\Pi(\mathfrak{S})\equiv\overline{S}_{\Pi}\) the average state of the ensemble. The equivalence with the initial definition is established by the relations
where \(\pi(A)=\textrm{Tr\,}\Pi(A),\,A\) is any Borel subset of \(\mathfrak{S}.\) Then the minimized functional can be rewritten as the scalar integral of the operator-valued function \(K(S)\) with respect to the operator-valued measure \(\Pi(dS),\) the construction of which was given in the infinite-dimensional case in Ch. I of [5] (see also [4]):
In what follows we assume that \(\mathcal{H}\) is finite-dimensional. In that case case we can just assume that \(K(S)\) is measurable function with values in the cone of positive operators on \(\mathcal{H}\) and the scalar integral can be understood via the expression \(\int_{\mathfrak{S}}\textrm{Tr\,}\,K(S)S\,\pi(dS)\). Then the optimization problem
becomes similar to a generalization of the Bayes problem [4, 5]. Combination of Theorem 2.1 and Theorem 2.2 from Ch. II of [5] implies
Theorem. The problem dual to (3) is
The following statements are equivalent:
(i) \(\Pi_{0}(dS)\) is the solution of the problem (3); \(\Lambda_{0}\) is the solution of the problem (4);
(ii) a. \(\Lambda_{0}\leq K(S)\) for all \(S\in\mathfrak{S};\)
b. \(\int_{A}\left[K(S)-\Lambda_{0}\right]\,\Pi_{0}(dS)=0\) for any Borel subset \(A\subseteq\mathfrak{S}.\)
The condition (ii.b) can be rewritten as
which means that the equality holds a.e. with respect to the measure \(\pi_{0}.\) By integrating, we obtain
which gives equation for determination of \(\Lambda_{0}\). Note that \(\Lambda_{0}\) must be Hermitean operator.
In the case of measurement channel, this equation reduces to
For completeness, we give the proof of the Theorem, taking into account simplifications due to finite dimensionality of \(\mathcal{H}.\) An infinite-dimensional generalization of the Theorem would have important applications to ensemble optimization problems.
Proof. In what follows we use the notation \(\left\langle X,Y\right\rangle=\textrm{Tr\,}X\,Y.\) Let us fix \(S_{0}\in\mathfrak{S}\) and show that
It is sufficient to show that
Let \(\Pi\) be such that \(\Pi(\mathfrak{S})\leq\overline{S}.\) Defining
we get \(\overline{\Pi}(\mathfrak{S})=\overline{S}\) and equality in (5).
Denote \(G(\Pi)=\Pi(\mathfrak{S})-\overline{S}\) and consider the problem of minimizing the functional
over the convex set of \(\Pi\)’s satisfying \(G(\Pi)\leq 0.\) For this we compute the dual functional
where the infimum is over the set of all positive \(\mathfrak{S}-\)valued measures. We show that
Let \(S\) be such that for some \(S^{\prime}\) and some \(X\geq 0\)
Defining \(\Pi_{n}(A)=nX\,1_{A}(S^{\prime}),\) we have
whence \(\varphi(S)=-\infty.\)
Let now
By letting \(\Pi(A)\equiv 0,\) we obtain \(\varphi(S)\leq\left\langle K(S_{0}),\overline{S}\right\rangle-\left\langle S,\overline{S}\right\rangle.\) The converse inequality follows from
which is obtained from (7) by integration (see Lemma 2.1 in Ch. I of [5]). This implies the first line in (6).
According to the general Lagrange duality theorem (see Appendix)
i.e. taking into account (5), (6)
Denoting \(\Lambda=K(S_{0})-S,\) we come to (4).
Let (i) be fulfilled, (4) implies
The inequality (iia): \(\Lambda_{0}\leq K(S)\), \(S\in\mathfrak{S}\) holds by (4). It follows for any Borel \(A\subseteq\mathfrak{S}:\)
By (9) it should be equality here, i.e. \(\int_{A}\left\langle K(S)-\Lambda_{0},\Pi_{0}(dS)\right\rangle=0.\) But since \(K(S)-\Lambda_{0}\geq 0,\) this implies (iib) (for detail see Proposition 3.3 from Ch. I of [5]).
Conversely, for arbitrary \(\Pi\) satisfying \(\Pi(\mathfrak{S})=\overline{S}\) we have by (iia)
and taking \(A=\mathfrak{S}\) in (iib) we obtain (9) whence (i) follows. \(\Box\)
Appendix. Let \(F\) be a convex functional on a convex subset \(\mathfrak{S}\) of a linear subspace \(L\) and \(G\) be a convex map of \(\mathfrak{S}\) into partially ordered Banach space \(L_{1}.\)
Consider the optimization problem
The following duality theorem holds (see e.g. [6], pp. 217, 224):
Theorem. Assume that the positive cone of \(L_{1}\) contains an inner point, and there exists \(x_{1}\in S\) such that \(\left\langle\lambda,G(x_{1})\right\rangle<0\) for all \(\lambda\in L_{1}^{\ast},\lambda>0.\) Then if the quantity (11) is finite,
where
is the dual functional.
Let \(\lambda_{0}\) be a solution of the dual problem in the right-hand side of (12). If the infimum in the left-hand side of (12) is attained on \(x_{0},\) then \(x_{0}\) is a solution of the problem \(\min\left\{F(x)+\left\langle\lambda_{0},G(x)\right\rangle;x\in\mathfrak{S}\right\}\) and \(\left\langle\lambda_{0},G(x_{0})\right\rangle=0.\)
REFERENCES
A. S. Holevo, Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd ed. (De Gruyter, Berlin, 2019).
A. S. Holevo and M. E. Shirokov, ‘‘Continuous ensembles and the capacity of infinite-dimensional quantum channels,’’ Theory Probab. Appl. 50, 86–98 (2005).
A. S. Holevo, ‘‘Statistical decision theory for quantum systems,’’ J. Multivariate Anal. 3, 337–394 (1973).
A. S. Holevo, ‘‘On a vector-valued integral in the noncommutative statistical decision theory,’’ J. Multivariate Anal. 5, 462–465 (1975).
A. S. Holevo, ‘‘Studies in general theory of statistical decisions,’’ Proc. Steklov Math. Inst. 124 (1978).
D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
M. E. Shirokov, ‘‘On properties of the space of quantum states and their application to construction of entanglement monotones,’’ Izv.: Math. 74, 849–882 (2010).
Funding
This work is supported by Russian Science Foundation under the grant no. 19-11-00086, https://rscf.ru/project/19-11-00086/. The author is grateful to M. E. Shirokov for useful remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Holevo, A.S. On Optimization Problem for Positive Operator-Valued Measures. Lobachevskii J Math 43, 1646–1650 (2022). https://doi.org/10.1134/S1995080222100158
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080222100158