Papyrus 91

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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. Furthermore, it has been shown that applications exist in quantum state tomography[1] and quantum cryptography.[2]

Definition

Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used to represent elements associated with the Hilbert space.

In general, a POVM over a d-dimensional Hilbert space is defined as a set M of positive semidefinite operators {Fi} on the Hilbert space that sum to the identity:

i=1MFi=I.

A SIC-POVM is more restrictive in that the operators must be subnormalized projectors related to one another such that they have the properties of symmetry and informational completeness.

In this context informational completeness means that the probabilities of observing the various outcomes entirely determines any quantum state being measured by the scheme. This requires d2 linearly independent operators. Symmetry means that the inner product of all pairs of subnormalized projectors Fi,Fj is a constant:

Tr(FiFj)=Tr(ΠiΠj)d2=|ψi|ψj|2d2=1d2(d+1)ij.

The combination of symmetry and informational completeness means M is composed entirely of operators of the form

Fi=1dΠi,

where Πi is a rank-one projector.

Properties

Symmetry

As defined above, the distinct pairwise inner product of the pure states must be a constant. Remembering that 1dαΠα=I and setting Tr(ΠαΠβ)=μ2, its value can be thus demonstrated:

d=Tr(I2)=1d2α,βTr(ΠαΠβ)=1d2(d2+μ2d2(d21))

From which it follows in general that

Tr(ΠiΠj)=|ψi|ψj|2=1d+1ij

Superoperator

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

𝒢:()()Aα|ψαψα|A|ψαψα|

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

𝒢(Πβ)=αΠα|ψα|ψβ|2=Πβ+1d+1αβΠα=dd+1Πβ+1d+1Πβ+1d+1αβΠα=dd+1Πβ+dd+1α1dΠα=dd+1(Πβ+I)

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:

G=dd+1(+I) where (A)=A and I(A)=Tr(A)I

From here, the left inverse can be calculated[3] to be G1=1d[(d+1)I], and so with the knowledge that

I=G1G=1dα[(d+1)ΠαΠαIΠα],

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

ρ=I|ρ)=α[(d+1)ΠαI](Πα|ρ)d=α[(d+1)ΠαI]Tr(Παρ)d=αpα[(d+1)ΠαI] where pα=Tr(Παρ)/d=I+(d+1)αpα|ψαψα|=α[(d+1)pα1d]|ψαψα|

where |ρ) 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 ρ is given by

(d+1)pα1d

Finding SIC sets

Group covariance

General group covariance

A SIC-POVM P is said to be group covariant if there exists a group G with a d2-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 |ϕ such that

|ϕ|Ug|ϕ|2=1d+1gid.

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

The case of Zd × Zd

So far, most SIC-POVM's have been found by considering group covariance under d×d.[4] To construct the unitary representation, we map d×d to U(d), the group of unitary operators on d-dimensions. Several operators must first be introduced. Let |ei be a basis for , then the phase operator is

T|ei=ωi|ei where ω=e2πid is a root of unity

and the shift operator as

S|ei=|ei+1modd

Combining these two operators yields the Weyl operator W(p,q)=SpTq which generates the Heisenberg-Weyl group. This is a unitary operator since

W(p,q)W(p,q)=SpTqTqSp=Id

It can be checked that the mapping (p,q)d×dW(p,q) is a projective unitary representation. It also satisfies all of the properties for group covariance,[5] and is useful for numerical calculation of SIC sets.

Zauner's conjecture

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,[6] a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.

More specifically,

For every dimension d2 there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator E0 under the Heisenberg group Hd. What is more, E0 commutes with an element T of the Jacobi group Jd=HdSL(2,d). The action of T on Hd modulo the center has order three.

Utilizing the notion of group covariance on d×d, this can be restated as [7]

For any dimension d, let {k}k=0d1 be an orthonormal basis for d, and define

ω=e2πid,Dj,k=ωjk2m=0d1ωjm|k+mmoddm|

Then |ϕd such that the set {Dj,k|ϕ}j,k=1d is a SIC-POVM

Partial results

Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is d=1,,15,19,24,35,48.[6][7][8][9] Furthermore, using the Heisenberg group covariance on d×d, numerical solutions have been found for all integers less than d=67.[4][7][9]

The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,[5] but is an ongoing field of research in the quantum mechanics community.

Relation to spherical t-designs

A spherical t-design is a set of vectors S={|ϕk:|ϕk𝕊d} on the d-dimensional generalized hypersphere, such that the average value of any tth-order polynomial ft(ψ) over S is equal to the average of ft(ψ) over all normalized vectors |ψ. Defining t=i=1t as the t-fold tensor product of the Hilbert spaces, and

St=k=1n|ΦktΦkt|,|Φkt=|ϕkt

as the t-fold tensor product frame operator, it can be shown that[7] a set of normalized vectors {|ϕk𝕊d}k=1n with n(t+d1d1) forms a spherical t-design if and only if

Tr[St2]=j,k|ϕj|ϕk|2t=n2t!(d1)!(t+d1)!

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

Tr(S22)=j,k|ϕj|ϕk|4=2d3d+1

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

Relation to MUBs

In a d-dimensional Hilbert space, two distinct bases {|ψi},{|ϕj}are said to be mutually unbiased if

|ψi|ϕj|2=1d,i,j

This seems similar in nature to the symmetric property of SIC-POVMs. In fact, the problem of finding a SIC-POVM is precisely the problem of finding equiangular lines in d; whereas mutually unbiased bases are analogous to affine spaces. In fact it can be shown that the geometric analogy of finding a "complete set of N+1 mutually unbiased bases is identical to the geometric structure analogous to a SIC-POVM[10] ". It is important to note that the equivalence of these problems is in the strict sense of an abstract geometry, and since the space on which each of these geometric analogues differs, there's no guarantee that a solution on one space will directly correlate with the other.

An example of where this analogous relation has yet to necessarily produce results is the case of 6-dimensional Hilbert space, in which a SIC-POVM has been analytically computed using mathematical software, but no complete mutually unbiased bases has yet been discovered.[11]

References

  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. C.M. Caves (1999); http://info.phys.unm.edu/~caves/reports/infopovm.pdf
  4. 4.0 4.1 Robin Blume-Kohout, Joseph M. Renes, Andrew J. Scott, Carlton M. Caves, http://info.phys.unm.edu/papers/reports/sicpovm.html
  5. 5.0 5.1 D. M. Appleby, SIC-POVMs and the Extended Clifford Group, http://arxiv.org/abs/quant-ph/0412001 (2004).
  6. 6.0 6.1 G. Zauner, Quantendesigns – Grundz¨uge einer nichtkommutativen Designtheorie. Dissertation, Universitat Wien, 1999.
  7. 7.0 7.1 7.2 7.3 J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric Informationally Complete Quantum Measurements, Journal of Mathematical Physics, 45 (2004), pp. 2171–2180.http://arxiv.org/abs/quant-ph/0310075.
  8. A. Koldobsky and H. K¨onig, “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.
  9. 9.0 9.1 A.J. Scott, M. Grassl, "SIC-POVMs: A new computer study", Journal Mathematical Physics, Volume 51, 042203 (2010); http://arxiv.org/abs/0910.5784
  10. W. K. Wootters, Quantum measurements and finite geometry. http://arxiv.org/abs/quant-ph/0406032v2, 2004.
  11. M. Grassl, On SIC-POVMs and MUBs in Dimension 6. http://arxiv.org/abs/quant-ph/0406175v1, 2004

See also