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arXiv:2010.14868 · Quantum

Bipartite quantum measurements with optimal single-sided distinguishability

Jakub Czartowski, Karol Życzkowski

Published Apr 26, 2021• 11 citations
arXivHTMLPDF

Abstract

We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN . Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.

DOI
10.22331/q-2021-04-26-442

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We analyse orthogonal bases in a composite N×N Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the N2 reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case N=2 of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for N=3 and provide a general construction of N2 states forming such an optimal basis in HN⊗HN . Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.
We introduce random-matrix theory to study the tomographic efficiency of a wide class of measurements constructed out of weighted 2-designs, including symmetric informationally complete (SIC) probability operator measurements (POMs). In particular, we derive analytic formulas for the mean Hilbert-Schmidt distance and the mean trace distance between the estimator and the true state, which clearly show the difference between the scaling behaviors of the two error measures with the dimension of the Hilbert space. We then prove that the product SIC POMs, the multipartite analog of the SIC POMs, are optimal among all product measurements in the same sense as the SIC POMs are optimal among all joint measurements. We further show that, for bipartite systems, there is only a marginal efficiency advantage of the joint SIC POMs over the product SIC POMs. In marked contrast, for multipartite systems, the efficiency advantage of the joint SIC POMs increases exponentially with the number of parties.
We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits an approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The basis of the first measurement and the four bases of the second measurements are pairwise unbiased---a hint at a possibly profound link between SIC POMs and mutually unbiased bases.
We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, ‘as close as possible’ to orthonormal bases for the space of quantum states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows ‘painless’ quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter being the unique minimal rank-one members. Recast as ensembles of pure quantum states, the rank-one members are in fact equivalent to weighted 2-designs in complex projective space. These measures are shown to be optimal for quantum cloning and linear quantum state tomography.