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In the Bloch sphere representation of a qubit, the states of a SIC-POVM form a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography[1] and quantum cryptography,[2] and a possible connection has been discovered with Hilbert's twelfth problem.[3]

Contents

DefinitionEdit

  Unsolved problem in mathematics:
Do SIC-POVMs exist in all dimensions?
(more unsolved problems in mathematics)

Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used throughout this article to represent elements in a Hilbert space.

A POVM over a  -dimensional Hilbert space   is a set of   positive-semidefinite operators   on the Hilbert space that sum to the identity:

 
If a POVM consists of at least   operators which span  , it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly   elements are called minimal. A set of   rank-1 projectors   which have equal pairwise Hilbert–Schmidt inner products,
 
defines a minimal IC-POVM   called a SIC-POVM.

PropertiesEdit

SymmetryEdit

The condition that the projectors   defined above have equal pairwise inner products actually fixes the value of this constant. Recall that   and set  . Then

 
implies that  . Thus,
 
This property is what makes SIC-POVMs symmetric; with respect to the Hilbert–Schmidt inner product, any pair of elements is equivalent to any other pair.

SuperoperatorEdit

In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map  . This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map

 

This operator acts on a SIC-POVM element in a way very similar to identity, in that

 

But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:

  where  

From here, the left inverse can be calculated[4] to be  , and so with the knowledge that

 ,

an expression for a state   can be created in terms of a quasi-probability distribution, as follows:

 

where   is the Dirac notation for the density operator viewed in the Hilbert space  . This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state   is given by

 

Finding SIC setsEdit

Group covarianceEdit

General group covarianceEdit

A SIC-POVM   is said to be group covariant if there exists a group   with a  -dimensional unitary representation such that

  •  
  •  

The search for SIC-POVMs can be greatly simplified by exploiting the property of group covariance. Indeed, the problem is reduced to finding a normalized fiducial vector   such that

 .

The SIC-POVM is then the set generated by the group action of   on  .

The case of Zd × ZdEdit

So far, most SIC-POVM's have been found by considering group covariance under  .[5] To construct the unitary representation, we map   to  , the group of unitary operators on d-dimensions. Several operators must first be introduced. Let   be a basis for  , then the phase operator is

  where   is a root of unity

and the shift operator as

 

Combining these two operators yields the Weyl operator   which generates the Heisenberg-Weyl group. This is a unitary operator since

 

It can be checked that the mapping   is a projective unitary representation. It also satisfies all of the properties for group covariance,[6] and is useful for numerical calculation of SIC sets.

Zauner's conjectureEdit

Given some of the useful properties of SIC-POVMs, it would be useful if it was positively known whether such sets could be constructed in a Hilbert space of arbitrary dimension. Originally proposed in the dissertation of Zauner,[7] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.

More specifically,

For every dimension   there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator   under the Weyl–Heisenberg group  . What is more,   commutes with an element T of the Jacobi group  . The action of T on   modulo the center has order three.

Utilizing the notion of group covariance on  , this can be restated as [8]

For any dimension  , let   be an orthonormal basis for  , and define

 

Then   such that the set   is a SIC-POVM

Partial resultsEdit

Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is  .[7][8][9][10][11][12][13] Furthermore, using the Heisenberg group covariance on  , numerical solutions have been found for all integers up through  .[5][8][10][14][15][16]

The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,[6] but is an ongoing field of research in the quantum mechanics community.

Relation to spherical t-designsEdit

A spherical t-design is a set of vectors   on the d-dimensional generalized hypersphere, such that the average value of any  -order polynomial   over   is equal to the average of   over all normalized vectors  . Defining   as the t-fold tensor product of the Hilbert spaces, and

 

as the t-fold tensor product frame operator, it can be shown that[8] a set of normalized vectors   with   forms a spherical t-design if and only if

 

It then immediately follows that every SIC-POVM is a 2-design, since

 

which is precisely the necessary value that satisfies the above theorem.

Relation to MUBsEdit

In a d-dimensional Hilbert space, two distinct bases  are said to be mutually unbiased if

 

This seems similar in nature to the symmetric property of SIC-POVMs. Wootters points out that a complete set of   unbiased bases yields a geometric structure known as a finite projective plane, while a SIC-POVM (in any dimension that is a prime power) yields a finite affine plane, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged. In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.[17]

In dimension  , the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.[18] The 9 vectors of the SIC-POVM, together with the 12 vectors of the mutually unbiased bases, form a set that can be used in a Kochen–Specker proof.[19] However, in 6-dimensional Hilbert space, a SIC-POVM is known, but no complete set of mutually unbiased bases has yet been discovered, and it is widely believed that no such set exists.[20][21]

ReferencesEdit

  1. ^ C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown Quantum States: The Quantum de Finetti Representation”, J. Math. Phys. 43, 4537–4559 (2002).
  2. ^ C. A. Fuchs and M. Sasaki, “Squeezing Quantum Information through a Classical Channel: Measuring the ‘Quantumness’ of a Set of Quantum States”, Quant. Info. Comp. 3, 377–404 (2003).
  3. ^ Appleby, Marcus; Flammia, Steven; McConnell, Gary; Yard, Jon (2017-04-24). "SICs and Algebraic Number Theory". Foundations of Physics. 47 (8): 1042–1059. arXiv:1701.05200. Bibcode:2017FoPh..tmp...34A. doi:10.1007/s10701-017-0090-7. ISSN 0015-9018.
  4. ^ C.M. Caves (1999); http://info.phys.unm.edu/~caves/reports/infopovm.pdf
  5. ^ a b Robin Blume-Kohout, Joseph M. Renes, Andrew J. Scott, Carlton M. Caves, http://info.phys.unm.edu/papers/reports/sicpovm.html
  6. ^ a b Appleby, D. M. (2005). "SIC-POVMs and the Extended Clifford Group". Journal of Mathematical Physics. 46 (5): 052107. arXiv:quant-ph/0412001. Bibcode:2005JMP....46e2107A. doi:10.1063/1.1896384.
  7. ^ a b G. Zauner, Quantendesigns – Grundzüge einer nichtkommutativen Designtheorie. Dissertation, Universität Wien, 1999.
  8. ^ a b c d Renes, Joseph M.; Blume-Kohout, Robin; Scott, A. J.; Caves, Carlton M. (2004). "Symmetric Informationally Complete Quantum Measurements". Journal of Mathematical Physics. 45 (6): 2171. arXiv:quant-ph/0310075. Bibcode:2004JMP....45.2171R. doi:10.1063/1.1737053.
  9. ^ A. Koldobsky and H. König, “Aspects of the Isometric Theory of Banach Spaces,” in Handbook of the Geometry of Banach Spaces, Vol. 1, edited by W. B. Johnson and J. Lindenstrauss, (North Holland, Dordrecht, 2001), pp. 899–939.
  10. ^ a b Scott, A. J.; Grassl, M. (2010). "SIC-POVMs: A new computer study". Journal of Mathematical Physics. 51 (4): 042203. arXiv:0910.5784. Bibcode:2010JMP....51d2203S. doi:10.1063/1.3374022.
  11. ^ TY Chien. ``Equiangular lines, projective symmetries and nice error frames. PhD thesis University of Auckland (2015); https://www.math.auckland.ac.nz/~waldron/Tuan/Thesis.pdf
  12. ^ "Exact SIC fiducial vectors". University of Sydney. Retrieved 2018-03-07.
  13. ^ Appleby, Marcus; Chien, Tuan-Yow; Flammia, Steven; Waldron, Shayne (2018). "Constructing exact symmetric informationally complete measurements from numerical solutions". Journal of Physics A: Mathematical and Theoretical. 51 (16): 165302. arXiv:1703.05981. doi:10.1088/1751-8121/aab4cd.
  14. ^ Fuchs, Christopher A.; Stacey, Blake C. (2016-12-21). "QBism: Quantum Theory as a Hero's Handbook". arXiv:1612.07308 [quant-ph].
  15. ^ Scott, A. J. (2017-03-11). "SICs: Extending the list of solutions". arXiv:1703.03993 [quant-ph].
  16. ^ Fuchs, Christopher A.; Hoang, Michael C.; Stacey, Blake C. (2017-03-22). "The SIC Question: History and State of Play". Axioms. 6 (4): 21. arXiv:1703.07901. doi:10.3390/axioms6030021.
  17. ^ Wootters, William K. (2004). "Quantum measurements and finite geometry". arXiv:quant-ph/0406032.
  18. ^ Stacey, Blake C. (2016). "SIC-POVMs and Compatibility among Quantum States". Mathematics. 4 (2): 36. arXiv:1404.3774. doi:10.3390/math4020036.
  19. ^ Bengtsson, Ingemar; Blanchfield, Kate; Cabello, Adán (2012). "A Kochen–Specker inequality from a SIC". Physics Letters A. 376 (4): 374–376. arXiv:1109.6514. Bibcode:2012PhLA..376..374B. doi:10.1016/j.physleta.2011.12.011.
  20. ^ Grassl, Markus (2004). "On SIC-POVMs and MUBs in Dimension 6". arXiv:quant-ph/0406175.
  21. ^ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of quantum states : an introduction to quantum entanglement (Second ed.). Cambridge, United Kingdom: Cambridge University Press. pp. 313–354. ISBN 9781107026254. OCLC 967938939.

See alsoEdit