×

SIC-POVMs and compatibility among quantum states. (English) Zbl 1360.81044

Summary: An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with C. M. Caves, C: A. Fuchs and R. Schack’s paper [“Conditions for compatibility of quantum state assignments”, Preprint, arXiv:quant-ph/0206110], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.

MSC:

81P15 Quantum measurement theory, state operations, state preparations
46G10 Vector-valued measures and integration
81P05 General and philosophical questions in quantum theory
81P13 Contextuality in quantum theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] QBism: The perimeter of Quantum Bayesianismhttp://arxiv.org/abs/1003.5209
[2] DOI: 10.1103/RevModPhys.85.1693 · doi:10.1103/RevModPhys.85.1693
[3] DOI: 10.1103/PhysRevA.66.062111 · doi:10.1103/PhysRevA.66.062111
[4] DOI: 10.1038/nphys2309 · doi:10.1038/nphys2309
[5] DOI: 10.1103/PhysRevLett.112.070407 · doi:10.1103/PhysRevLett.112.070407
[7] DOI: 10.1063/1.1737053 · Zbl 1071.81015 · doi:10.1063/1.1737053
[8] DOI: 10.1063/1.3374022 · Zbl 1310.81022 · doi:10.1063/1.3374022
[9] DOI: 10.1088/1751-8113/49/8/085301 · Zbl 1342.81069 · doi:10.1088/1751-8113/49/8/085301
[10] DOI: 10.1007/s11128-013-0609-6 · Zbl 1297.81009 · doi:10.1007/s11128-013-0609-6
[11] Steiner Triple Systemshttp://designtheory.org/library/encyc/sts/g/
[12] Fuchs, Some negative remarks on operational approaches to quantum theory, Quantum Theory: Informational Foundations and Foils pp 283– (2016)
[13] DOI: 10.1073/pnas.46.4.570 · Zbl 0090.19006 · doi:10.1073/pnas.46.4.570
[14] SIC-POVMs and MUBs: Geometrical relationships in prime dimensionhttp://arxiv.org/abs/0905.1428 · Zbl 1182.81020
[15] DOI: 10.1016/j.physleta.2011.12.011 · Zbl 1255.81018 · doi:10.1016/j.physleta.2011.12.011
[17] DOI: 10.1007/s10701-010-9458-7 · Zbl 1211.81022 · doi:10.1007/s10701-010-9458-7
[18] Informational power of the Hoggar SIC-POVMhttp://arxiv.org/abs/1512.01735
[19] DOI: 10.1103/PhysRevA.86.062107 · doi:10.1103/PhysRevA.86.062107
[20] DOI: 10.1103/PhysRevA.88.012131 · doi:10.1103/PhysRevA.88.012131
[21] DOI: 10.1016/j.aop.2015.08.005 · Zbl 1343.81023 · doi:10.1016/j.aop.2015.08.005
[22] DOI: 10.1016/0003-4916(87)90176-X · doi:10.1016/0003-4916(87)90176-X
[23] Quasiprobability representations of quantum mechanics with minimal negativityhttp://arxiv.org/abs/1604.06974
[24] DOI: 10.1103/PhysRevA.70.062101 · Zbl 1227.81218 · doi:10.1103/PhysRevA.70.062101
[25] Spekkens, Quasi-quantization: Classical statistical theories with an epistemic restriction, Quantum Theory: Informational Foundations and Foils pp 83– (2016)
[26] DOI: 10.1088/1367-2630/16/1/013009 · doi:10.1088/1367-2630/16/1/013009
[27] State-independent quantum contextuality and maximum nonlocalityhttp://arxiv.org/abs/1112.5149
[28] DOI: 10.1103/PhysRevLett.114.250402 · doi:10.1103/PhysRevLett.114.250402
[29] DOI: 10.1103/RevModPhys.65.803 · doi:10.1103/RevModPhys.65.803
[30] DOI: 10.1103/PhysRevA.75.032110 · doi:10.1103/PhysRevA.75.032110
[31] DOI: 10.1007/s10701-007-9171-3 · doi:10.1007/s10701-007-9171-3
[32] DOI: 10.1103/PhysRevA.86.012103 · doi:10.1103/PhysRevA.86.012103
[33] Why protective measurement does not establish the reality of the quantum statehttp://arxiv.org/abs/1509.08893 · Zbl 1402.81026
[34] DOI: 10.1103/PhysRevA.92.012107 · doi:10.1103/PhysRevA.92.012107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.