Symmetric informationally complete quantum measurements

By Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, Carlton M. Caves. 2003

1,076 citations recorded by the source corpus.

We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 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 d2 equiangular lines in Cd. 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.

Subfields: Applied Mathematics, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d06ebdbcab48d34b1a486e71d2ea285faf7154c7

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Citations in the corpus

On SIC-POVMs and MUBs in Dimension 6 (2004, background): “A weaker form of his conjecture—the existence of SIC-POVMs that are the orbit under the Heisenberg group—has been numerically verified up to dimension d = 45 [6] (as of May 25, 2009, Andrew Scott has found numerical solutions up to d = 67 [7]).”
Quasi-probability representations of quantum theory with applications to quantum information science (2010, background): “In Renes et al (2004), the authors conjecture13 that the set {U(p,q)φ} for some fiducial φ ∈ H forms a symmetric informationally complete positive operator valued measure (SIC-POVM).”
Geometry of Quantum States: Index (2007, methodology): “30 The acronym is short for Symmetric Informationally Complete Positive Operator Valued Measure [772], and is rarely spelled out.”
Barren plateaus in quantum neural network training landscapes (2018, background): “This motivates the concept of unitary t -designs [42–44], which satisfy the above properties for restricted classes of f ( U ), often requiring only modest polynomial resources.”
The road to equal-norm Parseval frames (2010, background): “This type of equal-norm Parseval frames is particularly important because of their applications to signal processing [33, 31, 6, 34, 27] and to quantum information theory [35, 30, 17, 5].”
On Weyl-Heisenberg orbits of equiangular lines (2008, background): “It is widely believed that for every d there exists a fiducial vector in C (for example see [2, 7, 9, 11, 13, 17]).”
Equiangular lines with a fixed angle (2019, background): “Complex equiangular lines, also known under the name SIC-POVM, play an important role in quantum information theory [20].”
Unknown Quantum States and Operations, a Bayesian View (2004, neutral): “We give here a simple way to produce one, though there are surely more elegant ways with greater symmetry [69,70].”
Optimal simplices and codes in projective spaces (2013, background): “SIC-POVMs are d(2) –point simplices in CP 1 , which play an important role in quantum information theory (see Renes, Blume-Kohout, Scott and Caves [67]).”
Steiner equiangular tight frames (2010, background): “In the complex setting, much attention has focused on the maximal case of M(2) vectors in HM [2, 15, 18, 21, 23].”
Emergent Quantum State Designs from Individual Many-Body Wave Functions (2021, methodology): “To quantify the degree of randomness of an ensemble, we use a well-established notion from quantum information theory, namely quantum state k-designs [26,27].”
The randomized measurement toolbox (2022, background): “Single-qubit symmetric informationally complete measurements 157,158 seem an ideal candidate, because they may alleviate the need for choosing measurement settings at random while still allowing for fast state reconstruction and fast estimation of arbitrary polynomial functionals of the quantum…”
Disentangling Scrambling and Decoherence via Quantum Teleportation (2018, background): “[45] Equation (3) holds since the Pauli operators form a unitary 1-design [46,47].”
Quantum-Bayesian Coherence (2009, background): “Do SICs exist for every finite dimension d? Despite much effort [60, 61, 62, 63, 64, 65, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98], no one presently knows.”