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arXiv:1809.07366

Joint measurability meets Birkhoff-von Neumann's theorem

Leonardo Guerini, Alexandre Baraviera

Published Sep 19, 2018• 0 citations
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Abstract

Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.

DOI
10.48550/arxiv.1809.07366

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Quantum measurements can be interpreted as a generalisation of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalisation of doubly stochastic matrices that we call doubly normalised tensors (DNTs), and formulate a corresponding version of Birkhoff-von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. Conversely, we also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
Quantum measurements can be interpreted as a generalization of probability vectors, in which non-negative real numbers are replaced by positive semi-definite operators. We extrapolate this analogy to define a generalization of doubly stochastic matrices that we call doubly normalized tensors (DNTs), and investigate a corresponding version of Birkhoff–von Neumann's theorem, which states that permutations are the extremal points of the set of doubly stochastic matrices. We prove that joint measurability appears naturally as a mathematical feature of DNTs in this context and that this feature is necessary and sufficient for a characterization similar to Birkhoff–von Neumann's. Conversely, we also show that DNTs arise from a particular instance of a joint measurability problem, remarking the relevance of this quantum-theoretical property in general operator theory.
Quantum magic squares are generalizations of doubly stochastic matrices to arbitrary, possibly non-commutative algebras, more specifically C∗-algebras. Birkhoff-von Neumann Theorem states that every doubly stochastic matrix is a convex combination of permutation matrices. One inevitable question is whether a quantum magic square is a convex combinations of quantum permutation matrices. It has been proven by Cuevas, Drescher, and Netzer that the generalization of the Birkhoff-von Neumann Theorem in the quantum setting is not true in general. A quantum magic square is a convex combination of Arveson extreme points but not necessarily a convex combination of quantum permutation matrices. Additionally, the relationships between different subsets of quantum magic squares, including their inclusion properties, have been studied in this context.
Birkhoff's Theorem states that doubly stochastic matrices are convex combinations of permutation matrices. Quantum mechanically these matrices are doubly stochastic channels, i.e. they are completely positive maps preserving both the trace and the identity. We expect these channels to be convex combinations of unitary channels and yet it is known that some channels cannot be written that way. Recent work has suggested that nnn copies of a single channel might approximate a mixture (convex combination) of unitaries. In this paper we show that n(n+1)/2n(n+1)/2n(n+1)/2 copies of a symmetric unital quantum channel may be arbitrarily-well approximated by a mixture (convex combination) of unitarily implemented channels. In addition, we prove that any extremal properties of a channel are preserved over nnn (and thus n(n+1)/2n(n+1)/2n(n+1)/2) copies. The result has the potential to be completely generalized to include non-symmetric channels.
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage MS\mathcal{M}_SMS​ consisting of a finite number of POVMs and then prove that the joint measurability of MS\mathcal{M}_SMS​ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage MS\mathcal{M}_SMS​ is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS\textbf{M}_SMS​ composed of MS\mathcal{M}_SMS​ can be expressed as an “operator convex combination" of {0,IS}\{0,I_S\}{0,IS​}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0 ,n1)\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})M(n0 ​,n1​) induced by two three-dimensional real unit vectors ni(i=0,1)\textbf{n}_{i}(i=0,1)ni​(i=0,1) is jointly measurable if and only if n0=±n1\textbf{n}_{0}=\pm\textbf{n}_{1}n0​=±n1​. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with M(n0,n1)\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})M(n0​,n1​).
We introduce a framework for simulating quantum measurements based on classical processing of a set of accessible measurements. Well-known concepts such as joint measurability and projective simulability naturally emerge as particular cases of our framework, but our study also leads to novel results and questions. First, a generalisation of joint measurability is derived, which yields a hierarchy for the incompatibility of sets of measurements. A similar hierarchy is defined based on the number of outcomes necessary to perform a simulation of a given measurement. This general approach also allows us to identify connections between different kinds of simulability and, in particular, we characterise the qubit measurements that are projective-simulable in terms of joint measurability. Finally, we discuss how our framework can be interpreted in the context of resource theories.
Decoherence, when viewed in the Heisenberg picture, can turn incompatible measurements into jointly measurable ones. This provides an observable-level description of emergent classicality in terms of the incompatibility destroyed by a noise channel. The corresponding loss of incompatibility can be captured by the channel's \emph{compatibility region}---the observables in a probe class that become jointly measurable with every noisy observable. The geometry and volume of this region provide a refined operational way to compare noise channels. We apply this framework to decoherence channels, each of which acts by Schur multiplication with extreme points of the convex set of unit-diagonal positive semidefinite matrices. Restricting to extremals eliminates convex mixing and reveals subtle phase-dependent decoherence effects thar are not captured by damping rates or conventional coherence quantifiers. Such channels have a rigid dilation structure described by a rank-one operator frame, which reduces joint measurability to a positivity test, making the compatibility region analytically tractable, and its volume computable from the frame's Gram matrix. If the frame can be normalised to a minimal informationally complete (MIC) POVM, the compatibility region acquires a geometric interpretation in the associated probability representation of the quantum state space, familiar with Qbism. Among maximal-rank extremals, those associated with symmetric informationally complete (SIC) POVMs maximise the compatibility volume and hence destroy the greatest amount of incompatibility, providing an operational characterisation of the special role of SICs in terms of joint measurability. This leads to a new formulation of the well-known SIC existence problem as a joint measurability question for noisy mutually unbiased bases.