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| A '''symmetric, informationally complete, positive operator valued measure''' (SIC-[[POVM]]) is a special case of a generalized [[Measurement in quantum mechanics|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]]<ref>C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown Quantum States: The Quantum de Finetti Representation,” J. Math. Phys. 43, 4537–4559 (2002).</ref> and [[quantum cryptography]].<ref>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).</ref>
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| ==Definition==
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| Due to the use of SIC-POVMs primarily in quantum mechanics, [[Dirac notation]] will be used to represent elements associated with the [[Hilbert space]].
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| In general, a POVM over a <math>d</math>-dimensional Hilbert space <math> \mathcal{H} </math> is defined as a set <math>M</math> of [[positive definite kernel|positive semidefinite operators]] <math>\{F_i\} </math> on the Hilbert space that sum to the [[Identity matrix|identity]]:
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| :<math>\sum_{i=1}^M F_i = I.</math>
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| A SIC-POVM is more restrictive in that the operators must be subnormalized [[Projection (linear algebra)|projectors]] related to one another such that they have the properties of symmetry and informational completeness.
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| 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 <math>d^2</math> linearly independent operators. Symmetry means that the [[Trace_(linear_algebra)#Inner_product|inner product]] of all pairs of subnormalized projectors <math>F_i,F_j</math> is a constant:
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| :<math> \mathrm{Tr}\left( F_i F_j \right) = \frac{\mathrm{Tr}\left( \Pi_i \Pi_j \right)}{d^2} = \frac{\left| \langle \psi_i | \psi_j \rangle \right|^2}{d^2} = \frac{1}{d^2(d+1)} \quad i \ne j.</math>
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| The combination of symmetry and informational completeness means <math>M</math> is composed entirely of operators of the form
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| :<math> F_i=\frac{1}{d} \Pi_i,</math> | |
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| where <math>\Pi_i</math> is a rank-one projector.
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| ==Properties==
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| ===Symmetry===
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| As defined [[SICPOVM#Definition|above]], the distinct pairwise inner product of the pure states must be a constant. Remembering that <math> \frac{1}{d} \sum_\alpha \Pi_\alpha = I</math> and setting <math> \mathrm{Tr}(\Pi_\alpha \Pi_\beta ) = \mu^2 \;</math>, its value can be thus demonstrated:
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| :<math> \begin{align} d &= \mathrm{Tr}(I^2) \\
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| &= \displaystyle \frac{1}{d^2} \sum_{\alpha,\beta} \mathrm{Tr}(\Pi_\alpha \Pi_\beta) \\
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| &= \displaystyle \frac{1}{d^2} \left( d^2 + \mu^2 d^2 (d^2-1) \right) \end{align} </math>
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| From which it follows in general that
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| : <math> \mathrm{Tr}\left( \Pi_i \Pi_j \right) = \left| \langle \psi_i | \psi_j \rangle \right|^2 = \frac{1}{d+1} \quad i \ne j</math>
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| ===Superoperator===
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| In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map <math> \mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H}) </math>. This operator is most useful in considering the [[SICPOVM#Relation to spherical t-designs|relation of SIC-POVMs with spherical t-designs]]. Consider the map
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| :<math> \begin{align} \mathcal{G}: \mathcal{B}(\mathcal{H}) &\rightarrow \mathcal{B}(\mathcal{H})\\
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| A &\mapsto \displaystyle \sum_\alpha |\psi_\alpha \rangle \langle \psi_\alpha | A |\psi_\alpha \rangle \langle \psi_\alpha | \end{align}</math>
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| This operator acts on a SIC-POVM element in a way very similar to identity, in that
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| :<math> \begin{align} \mathcal{G}(\Pi_\beta) &= \displaystyle \sum_\alpha \Pi_\alpha \left| \langle \psi_\alpha | \psi_\beta \rangle \right|^2 \\
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| &= \displaystyle \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
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| &= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{1}{d+1} \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
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| &= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{d}{d+1}\sum_\alpha \frac{1}{d}\Pi_\alpha \\
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| &= \displaystyle \frac{d}{d+1} \left( \Pi_\beta + I \right) \end{align}</math>
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| 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:
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| :<math> G = \frac{d}{d+1} \left( \mathcal{I} + I \right) </math> where <math> \mathcal{I}(A) = A \text{ and } I(A)=\mathrm{Tr}(A)I </math>
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| From here, the [[inverse element|left inverse]] can be calculated<ref>C.M. Caves (1999); http://info.phys.unm.edu/~caves/reports/infopovm.pdf</ref> to be <math> G^{-1} = \frac1d \left[ \left(d+1\right)I - \mathcal{I} \right]</math>, and so with the knowledge that
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| :<math> I=G^{-1}G = \frac1d \sum_\alpha \left[ (d+1)\Pi_\alpha \odot \Pi_\alpha - I\odot \Pi_\alpha \right]</math>,
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| an expression for a state <math> \rho </math> can be created in terms of a [[quasi-probability distribution]], as follows:
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| :<math> \begin{align} \rho = I | \rho ) &= \displaystyle \sum_\alpha \left[ (d+1)\Pi_\alpha - I \right] \frac{ (\Pi_\alpha|\rho)}{d} \\
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| &= \displaystyle \sum_\alpha \left[ (d+1)\Pi_\alpha - I \right] \frac{ \mathrm{Tr}(\Pi_\alpha\rho)}{d} \\
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| &= \displaystyle \sum_\alpha p_\alpha \left[ (d+1)\Pi_\alpha - I \right] \quad \text{ where } p_\alpha = \mathrm{Tr}(\Pi_\alpha\rho)/d\\
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| &= \displaystyle -I + (d+1) \sum_\alpha p_\alpha |\psi_\alpha \rangle \langle \psi_\alpha | \\
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| &= \displaystyle \sum_\alpha \left[ (d+1)p_\alpha - \frac1d \right] |\psi_\alpha \rangle \langle \psi_\alpha |
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| \end{align} </math>
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| where <math> | \rho ) </math> is the Dirac notation for the density operator viewed in the Hilbert space <math> \mathcal{B} (\mathcal{H}) </math>. This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state <math> \rho </math> is given by
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| :<math>(d+1)p_\alpha - \frac1d</math>
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| ==Finding SIC sets==
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| ===Group covariance ===
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| ====General group covariance====
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| A SIC-POVM <math>P</math> is said to be ''group covariant'' if there exists a group <math> G </math> with a <math>d^2</math>-dimensional [[Unitary representation|unitary]] [[group representation|representation]] such that
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| * <math> \forall |\psi\rangle\langle \psi | \in P, \quad \forall U_g \in G,\quad U_g|\psi\rangle \in P </math>
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| * <math> \forall |\psi\rangle\langle \psi |, |\phi \rangle\langle \phi | \in P, \quad \exists U_g \in G, \quad U_g |\phi \rangle = | \psi \rangle </math>
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| 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'' <math> | \phi \rangle </math> such that
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| :<math> | \langle \phi | U_g | \phi \rangle |^2 = \frac{1}{d+1} \ \forall g \neq id </math>.
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| The SIC-POVM is then the set [[Generating set of a group|generated]] by the [[group action]] of <math> U_g </math> on <math> |\phi \rangle </math>.
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| ====The case of '''Z'''<sub>''d''</sub> × '''Z'''<sub>''d''</sub> ====
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| So far, most SIC-POVM's have been found by considering group covariance under <math> \mathbb{Z}_d \times \mathbb{Z}_d </math>.<ref name="BlumeKohoutNumerical">Robin Blume-Kohout, Joseph M. Renes, Andrew J. Scott, Carlton M. Caves, http://info.phys.unm.edu/papers/reports/sicpovm.html</ref> To construct the unitary representation, we map <math> \mathbb{Z}_d \times \mathbb{Z}_d</math> to <math> U(d) </math>, the group of unitary operators on d-dimensions. Several operators must first be introduced. Let <math> |e_i \rangle </math> be a basis for <math> \mathcal{H}</math>, then the ''phase operator'' is
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| :<math> T|e_i \rangle = \omega^i |e_i \rangle </math> where <math> \omega = e^{\frac{2\pi i}{d}}</math> is a root of unity
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| and the ''shift operator'' as
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| :<math> S|e_i \rangle = |e_{i+1 \mod{d}} \rangle </math>
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| Combining these two operators yields the ''Weyl operator'' <math> W(p,q) = S^p T^q </math> which generates the Heisenberg-Weyl group. This is a unitary operator since
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| :<math> \begin{align} W(p,q) W^\dagger (p,q) &= S^p T^q T^{-q} S^{-p} \\
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| &= Id \end{align} </math>
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| It can be checked that the mapping <math> (p,q) \in \mathbb{Z}_d \times \mathbb{Z}_d \rightarrow W(p,q) </math> is a projective unitary representation. It also satisfies all of the properties for group covariance,<ref name="Appleby2004"/> and is useful for numerical calculation of SIC sets.
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| ===Zauner's conjecture===
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| 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,<ref name="Zauner1999">G. Zauner, Quantendesigns – Grundz¨uge einer nichtkommutativen Designtheorie. Dissertation, Universitat Wien, 1999.</ref> a conjecture about the existence of a fiducial vector for arbitrary dimensions was hypothesized.
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| More specifically,
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| <blockquote> | |
| For every dimension <math>d\geq 2</math> there exists a SIC-POVM whose elements are the orbit of a positive rank-one operator <math>E_0</math> under the [[Heisenberg group]] <math> H_d </math>. What is more, <math>E_0</math> commutes with an element T of the Jacobi group <math>J_d=H_d \rtimes SL(2,\mathbb{Z}_d)</math>. The action of T on <math>H_d</math> modulo the center has order three.
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| </blockquote> | |
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| Utilizing the notion of group covariance on <math> \mathbb{Z}_d \times \mathbb{Z}_d </math>, this can be restated as <ref name="Renes2004"/>
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| <blockquote>
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| For any dimension <math> d \in \mathbb{N} </math>, let <math> \left\{ k \right\}_{k=0}^{d-1} </math> be an orthonormal basis for <math> \mathbb{C}^d </math>, and define
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| : <math> \displaystyle \omega = e^{\frac{2\pi i}{d}}, \quad \quad D_{j,k} = \omega^{\frac{jk}{2}} \sum_{m=0}^{d-1}\omega^{jm} | k+m\mod{d} \rangle \langle m |</math>
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| Then <math> \exists |\phi \rangle \in \mathbb{C}^d </math> such that the set <math> \left\{ D_{j,k} |\phi \rangle \right\}_{j,k=1}^d </math> is a SIC-POVM
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| </blockquote>
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| ===Partial results===
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| Algebraic and analytical results for finding SIC sets have been shown in the limiting case where the dimension of the Hilbert space is <math> d=1,\dots,15,19,24,35,48 </math>.<ref name="Zauner1999"/><ref name="Renes2004"/><ref>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.</ref><ref name="ScottGrassl">A.J. Scott, M. Grassl, "SIC-POVMs: A new computer study", ''Journal Mathematical Physics,'' Volume 51, 042203 (2010); http://arxiv.org/abs/0910.5784</ref> Furthermore, using the Heisenberg group covariance on <math> \mathbb{Z}_d\times \mathbb{Z}_d </math>, numerical solutions have been found for all integers less than <math> d=67 </math>.<ref name="BlumeKohoutNumerical"/><ref name="Renes2004"/><ref name="ScottGrassl"/>
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| The proof for the existence of SIC-POVMs for arbitrary dimensions remains an open question,<ref name="Appleby2004">D. M. Appleby, SIC-POVMs and the Extended Clifford Group, http://arxiv.org/abs/quant-ph/0412001 (2004).</ref> but is an ongoing field of research in the quantum mechanics community.
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| ==Relation to spherical t-designs== | |
| A ''[[spherical t-design]]'' is a set of vectors <math> S=\left\{ | \phi_k \rangle : |\phi_k \rangle \in \mathbb{S}^d \right\} </math> on the d-dimensional generalized [[hypersphere]], such that the average value of any <math> t^{th}</math>-order polynomial <math> f_t(\psi) </math> over <math> S </math> is equal to the average of <math> f_t(\psi) </math> over all normalized vectors <math> | \psi \rangle </math>. Defining <math> \mathcal{H}_t = \displaystyle \bigotimes_{i=1}^t \mathcal{H} </math> as the t-fold [[tensor product]] of the Hilbert spaces, and
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| : <math>S_t = \displaystyle \sum_{k=1}^n | \Phi_k^t \rangle \langle \Phi_k^t | , \quad |\Phi_k^t\rangle = |\phi_k\rangle^{\otimes t} </math>
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| as the t-fold tensor product [[Frame of a vector space|frame]] operator, it can be shown that<ref name="Renes2004">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.</ref> a set of normalized vectors <math> \left\{ | \phi_k \rangle \in \mathbb{S}^d \right\}_{k=1}^n </math> with <math> n \geq {t+d-1 \choose d-1} </math> forms a spherical t-design if and only if
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| ::<math> \displaystyle \mathrm{Tr}\left[ S_t^2 \right] = \sum_{j,k} \left| \langle \phi_j | \phi_k \rangle \right|^{2t} = \frac{n^2 t! (d-1)!}{(t+d-1)!} </math>
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| It then immediately follows that every SIC-POVM is a 2-design, since
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| :<math> \mathrm{Tr}(S^2_2) = \displaystyle \sum_{j,k} |\langle \phi_j |\phi_k \rangle |^4 = \frac{2d^3}{d+1} </math>
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| which is precisely the necessary value that satisfies the above theorem.
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| ==Relation to MUBs ==
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| In a ''d''-dimensional Hilbert space, two ''distinct'' bases <math> \left\{|\psi_i\rangle \right\}, \left\{ |\phi_j \rangle \right\} </math>are said to be [[mutually unbiased bases|mutually unbiased]] if
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| : <math>\displaystyle |\langle \psi_i | \phi_j \rangle|^2 = \frac{1}{d}, \quad \forall i,j </math>
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| 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 <math> \mathbb{C}^d </math>; whereas mutually unbiased bases are analogous to [[affine space]]s. In fact it can be shown that the geometric analogy of finding a "complete set of <math>N+1</math> mutually unbiased bases is identical to the geometric structure analogous to a SIC-POVM<ref>W. K. Wootters, Quantum measurements and finite geometry. http://arxiv.org/abs/quant-ph/0406032v2, 2004.</ref> ". 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.
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| 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.<ref>M. Grassl, On SIC-POVMs and MUBs in Dimension 6. http://arxiv.org/abs/quant-ph/0406175v1, 2004</ref>
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| ==References ==
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| <references/>
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| ==See also==
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| * [[Measurement in quantum mechanics]]
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| * [[Mutually unbiased bases]]
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| * [[POVM]]
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| * [[Quantum Bayesianism]]
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| [[Category:Quantum measurement]]
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| [[Category:Quantum mechanics]]
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