On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States

Abstract

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.

29 pages, LaTeX

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• SIC-POVMs: A new computer study
• Tight informationally complete quantum measurements
• The SIC Question: History and State of Play
• Uncertainty relations for MUBs and SIC-POVMs in terms of generalized entropies
• Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
• Notes on general SIC-POVMs
• The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
• Quantum-Bayesian Coherence
• Doubly optimal parallel wire cutting without ancilla qubits
• On the Brukner-Zeilinger approach to information in quantum measurements
• Quantum Key Distribution Highly Sensitive to Eavesdropping
• Systems of Imprimitivity for the Clifford Group
• SU(2) nonstandard bases: the case of mutually unbiased bases
• Miscellaneous Applications of Quons
• Studies of symmetries that give special quantum states the "right to exist"
• Compressed Sensing Matrices from Fourier Matrices
• Complementarity in quantum walks
• A Note on the Set of After-measurement States in Generalized Quantum Measurement