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Symmetric Informationally Complete Quantum Measurements
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We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
Forward citations
Cited by 5 Pith papers
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Tight designs minimize p-frame energies over all probability measures for p between consecutive even integers, and the 600-cell does so on S3 for p in [8,10].
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A general estimation framework for continuous-variable systems
In continuous-variable quantum estimation, informational completeness is not enough: stable reconstruction needs a positive lower frame bound in a σ-regularized geometry, and quasiprobability singularities (the P func...
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Uncountably many inequivalent maximally entangled measurements for two qutrits
There are uncountably many locally inequivalent maximally entangled measurement bases for two qutrits, constructed from qutrit SICs, including the first wild error bases in dimension 3.
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Quantum randomness beyond projective measurements
Unbiased extremal rank-one measurements generate characterized randomness in dimension 2, with tetrahedral SIC having the least, and SICs achieve maximal 2 log d randomness device-dependently in dimensions where they exist.
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Energy on spheres and discreteness of minimizing measures
For non-even p, every minimizer of the p-frame energy on the sphere has support with empty interior, and for potentials with finitely many positive Gegenbauer coefficients a discrete minimizer always exists.
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