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4 ZONOIDS AND SPARSIFICATION OF QUANTUM MEASUREMENTS

Guillaume Aubrun, Cécilia Lancien

Published Aug 16, 2016• 12 citations
OpenAlex

Abstract

In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory. This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on $$\mathbf {C}^d$$ , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only $$O(d^2)$$ outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on $$\mathbf {C}^d$$ . By considering an appropriate notion of tensor product for zonoids, we extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on $$\mathbf {C}^{d_1}\otimes \cdots \otimes \mathbf {C}^{d_k}$$ can be approximated by a discrete POVM which is local and has $$O(d_1^2 \times \cdots \times d_k^2)$$ outcomes.

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In this paper, we establish a connection between zonoids (a concept from classical convex geometry) and the distinguishability norms associated to quantum measurements or POVMs (Positive Operator-Valued Measures), recently introduced in quantum information theory. This correspondence allows us to state and prove the POVM version of classical results from the local theory of Banach spaces about the approximation of zonoids by zonotopes. We show that on Cd\mathbf {C}^dCd , the uniform POVM (the most symmetric POVM) can be sparsified, i.e. approximated by a discrete POVM having only O(d2)O(d^2)O(d2) outcomes. We also show that similar (but weaker) approximation results actually hold for any POVM on Cd\mathbf {C}^dCd . By considering an appropriate notion of tensor product for zonoids, we extend our results to the multipartite setting: we show, roughly speaking, that local POVMs may be sparsified locally. In particular, the local uniform POVM on Cd1⊗⋯⊗Cdk\mathbf {C}^{d_1}\otimes \cdots \otimes \mathbf {C}^{d_k}Cd1​⊗⋯⊗Cdk​ can be approximated by a discrete POVM which is local and has O(d12×⋯×dk2)O(d_1^2 \times \cdots \times d_k^2)O(d12​×⋯×dk2​) outcomes.
We study the distinguishability associated to either a single POVM or a family of POVMs. These norms quantify how quantum measurements perform in the task of discriminating two quantum states. We address the following questions regarding their behaviour in the high-dimensional regime: (i) How many POVMs are needed to approximate the family of all POVMs on C^d? We show that the answer is exp(O(d^2)). (ii) How many outcomes are required for a POVM to emulate the uniform POVM on C^d? Improving on previous results, we show that O(d^2) outcomes are necessary and sufficient. Moreover, using the concept of zonoid and approximation results from Banach space theory, we show that this property extends (in a weaker form) to an arbitrary POVM. (iii) On a bipartite space, what are the relative strengths of standard classes of locally restricted measurements? We show that the class of PPT measurements typically performs almost as well as the class of all measurements whereas restricting to separable measurements implies a substantial loss.
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.
Similarly to quantum states, also quantum measurements can be 'mixed', corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are indecomposable , i.e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVMs form a convex set, and in this language the indecomposable apparatuses are represented by extremal points—the analogous of 'pure states' in the convex set of states. Differently from the case of states, however, indecomposable POVMs are not necessarily rank-one, e.g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVMs, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, 'informationally complete' measurements are analysed in this respect. The convex set of POVMs is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVMs.
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 subject to affine constraints. These examples include sets of quantum states with given expectation values of some observables of interest, 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 characterisation 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 are used to retrieve classic results such as Choi's characterisation of the extreme quantum channels, Parhasarathy's characterisation 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 normalisation, and we characterise the extremal POVMs.
In quantum mechanics the statistics of the outcomes of a measuring apparatus is described by a positive operator valued measure (POVM). A quantum channel transforms POVMs into POVMs, generally irreversibly, thus losing some of the information retrieved from the measurement. This poses the problem of which POVMs are “undisturbed,” i.e., they are not irreversibly connected to another POVM. We will call such POVMs clean. In a sense, the clean POVMs would be “perfect,” since they would not have any additional “extrinsical” noise. Quite unexpectedly, it turns out that such a “cleanness” property is largely unrelated to the convex structure of POVMs, and there are clean POVMs that are not extremal and vice versa. In this article we solve the cleannes classification problem for number n of outcomes n⩽d (d dimension of the Hilbert space), and we provide a set of either necessary or sufficient conditions for n>d, along with an iff condition for the case of informationally complete POVMs for n=d2.
We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.