In the context of quantum mechanics and quantum information theory, symmetric, informationally complete, positive operator-valued measures (SIC-POVMs) are a particular type of generalized measurement (POVM). SIC-POVMs are particularly notable thanks to their defining features of (1) being informationally complete; (2) having the minimal number of outcomes compatible with informational completeness, and (3) being highly symmetric. In this context, informational completeness is the property of a POVM of allowing to fully reconstruct input states from measurement data. The properties of SIC-POVMs make them an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. SIC-POVMs have several applications in the context of quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem.
Definition
A POVM over a d {\displaystyle d} -dimensional Hilbert space H {\displaystyle {\mathcal {H}}} is a set of m {\displaystyle m} positive-semidefinite operators { F i } i = 1 m {\displaystyle \left\{F_{i}\right\}_{i=1}^{m}} that sum to the identity: ∑ i = 1 m F i = I . {\displaystyle \sum _{i=1}^{m}F_{i}=I.}
If a POVM consists of at least d 2 {\displaystyle d^{2}} operators which span the space of self-adjoint operators L ( H ) {\displaystyle {\mathcal {L}}({\mathcal {H}})} , it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly d 2 {\displaystyle d^{2}} elements are called minimal. A set of d 2 {\displaystyle d^{2}} rank-1 projectors { Π i } i = 1 d 2 {\displaystyle \left\{\Pi _{i}\right\}_{i=1}^{d^{2}}} which have equal pairwise Hilbert–Schmidt inner products,
T r ( Π i Π j ) = d δ i j + 1 d + 1 , {\displaystyle \mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}},}
defines a minimal IC-POVM with elements F i = 1 d Π i {\displaystyle F_{i}={\frac {1}{d}}\Pi _{i}} called a SIC-POVM.
Properties
Symmetry Consider an arbitrary set of rank-1 projectors ( Π i ) i = 1 d 2 {\displaystyle (\Pi _{i})_{i=1}^{d^{2}}} such that F i = Π i / d {\displaystyle F_{i}=\Pi _{i}/d} is a POVM, and thus 1 d ∑ i Π i = I {\displaystyle {\frac {1}{d}}\sum _{i}\Pi _{i}=I} . Asking the projectors to have equal pairwise inner products, T r ( Π i Π j ) = c {\displaystyle \mathrm {Tr} (\Pi _{i}\Pi _{j})=c} for all i ≠ j {\displaystyle i\neq j} , fixes the value of c {\displaystyle c} . To see this, observe that
d = T r ( I 2 ) = 1 d 2 ∑ i , j T r ( Π i Π j ) = 1 d 2 ( d 2 + c d 2 ( d 2 − 1 ) ) {\displaystyle {\begin{aligned}d&=\mathrm {Tr} (I^{2})\\&={\frac {1}{d^{2}}}\sum _{i,j}\mathrm {Tr} (\Pi _{i}\Pi _{j})\\&={\frac {1}{d^{2}}}\left(d^{2}+cd^{2}(d^{2}-1)\right)\end{aligned}}}
implies that c = 1 d + 1 {\displaystyle c={\frac {1}{d+1}}} . Thus,
T r ( Π i Π j ) = d δ i j + 1 d + 1 . {\displaystyle \mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}}.}
This property is what makes SIC-POVMs symmetric: Any pair of elements has the same Hilbert–Schmidt inner product as any other pair.
Superoperator In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map L ( H ) → L ( H ) {\displaystyle {\mathcal {L}}({\mathcal {H}})\rightarrow {\mathcal {L}}({\mathcal {H}})} . This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map
G : L ( H ) → L ( H ) A ↦ ∑ α | ψ α ⟩ ⟨ ψ α | A | ψ α ⟩ ⟨ ψ α | {\displaystyle {\begin{aligned}{\mathcal {G}}:{\mathcal {L}}({\mathcal {H}})&\rightarrow {\mathcal {L}}({\mathcal {H}})\\A&\mapsto \displaystyle \sum _{\alpha }|\psi _{\alpha }\rangle \langle \psi _{\alpha }|A|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\end{aligned}}}
This operator acts on a SIC-POVM element in a way very similar to identity, in that
G ( Π β ) = ∑ α Π α | ⟨ ψ α | ψ β ⟩ | 2 = Π β + 1 d + 1 ∑ α ≠ β Π α = d d + 1 Π β + 1 d + 1 Π β + 1 d + 1 ∑ α ≠ β Π α = d d + 1 Π β + d d + 1 ∑ α 1 d Π α = d d + 1 ( Π β + I ) {\displaystyle {\begin{aligned}{\mathcal {G}}(\Pi _{\beta })&=\displaystyle \sum _{\alpha }\Pi _{\alpha }\left|\langle \psi _{\alpha }|\psi _{\beta }\rangle \right|^{2}\\&=\displaystyle \Pi _{\beta }+{\frac {1}{d+1}}\sum _{\alpha \neq \beta }\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\Pi _{\beta }+{\frac {1}{d+1}}\Pi _{\beta }+{\frac {1}{d+1}}\sum _{\alpha \neq \beta }\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\Pi _{\beta }+{\frac {d}{d+1}}\sum _{\alpha }{\frac {1}{d}}\Pi _{\alpha }\\&=\displaystyle {\frac {d}{d+1}}\left(\Pi _{\beta }+I\right)\end{aligned}}}
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 = d d + 1 ( I + I ) {\displaystyle G={\frac {d}{d+1}}\left({\mathcal {I}}+I\right)} where I ( A ) = A and I ( A ) = T r ( A ) I {\displaystyle I(A)=A{\text{ and }}{\mathcal {I}}(A)=\mathrm {Tr} (A)I}
From here, the left inverse can be calculated to be G − 1 = 1 d [ ( d + 1 ) I − I ] {\displaystyle G^{-1}={\frac {1}{d}}\left[\left(d+1\right)I-{\mathcal {I}}\right]} , and so with the knowledge that
I = G − 1 G = 1 d ∑ α [ ( d + 1 ) Π α ⊙ Π α − I ⊙ Π α ] {\displaystyle I=G^{-1}G={\frac {1}{d}}\sum _{\alpha }\left[(d+1)\Pi _{\alpha }\odot \Pi _{\alpha }-I\odot \Pi _{\alpha }\right]} , an expression for a state ρ {\displaystyle \rho } can be created in terms of a quasi-probability distribution, as follows:
ρ = I | ρ ) = ∑ α [ ( d + 1 ) Π α − I ] ( Π α | ρ ) d = ∑ α [ ( d + 1 ) Π α − I ] T r ( Π α ρ ) d = ∑ α p α [ ( d + 1 ) Π α − I ] where p α = T r ( Π α ρ ) / d = − I + ( d + 1 ) ∑ α p α | ψ α ⟩ ⟨ ψ α | = ∑ α [ ( d + 1 ) p α − 1 d ] | ψ α ⟩ ⟨ ψ α | {\displaystyle {\begin{aligned}\rho =I|\rho )&=\displaystyle \sum _{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]{\frac {(\Pi _{\alpha }|\rho )}{d}}\\&=\displaystyle \sum _{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]{\frac {\mathrm {Tr} (\Pi _{\alpha }\rho )}{d}}\\&=\displaystyle \sum _{\alpha }p_{\alpha }\left[(d+1)\Pi _{\alpha }-I\right]\quad {\text{ where }}p_{\alpha }=\mathrm {Tr} (\Pi _{\alpha }\rho )/d\\&=\displaystyle -I+(d+1)\sum _{\alpha }p_{\alpha }|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\\&=\displaystyle \sum _{\alpha }\left[(d+1)p_{\alpha }-{\frac {1}{d}}\right]|\psi _{\alpha }\rangle \langle \psi _{\alpha }|\end{aligned}}}
where | ρ ) {\displaystyle |\rho )} is the Dirac notation for the density operator viewed in the Hilbert space L ( H ) {\displaystyle {\mathcal {L}}({\mathcal {H}})} . This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state ρ {\displaystyle \rho } is given by
( d + 1 ) p α − 1 d {\displaystyle (d+1)p_{\alpha }-{\frac {1}{d}}}
Finding SIC sets
Simplest example For d = 2 {\displaystyle d=2} the equations that define the SIC-POVM can be solved by hand, yielding the vectors
| ψ 1 ⟩ = | 0 ⟩ | ψ 2 ⟩ = 1 3 | 0 ⟩ + 2 3 | 1 ⟩ | ψ 3 ⟩ = 1 3 | 0 ⟩ + 2 3 e i 2 π 3 | 1 ⟩ | ψ 4 ⟩ = 1 3 | 0 ⟩ + 2 3 e i 4 π 3 | 1 ⟩ , {\displaystyle {\begin{aligned}|\psi _{1}\rangle &=|0\rangle \\|\psi _{2}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}|1\rangle \\|\psi _{3}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}e^{i{\frac {2\pi }{3}}}|1\rangle \\|\psi _{4}\rangle &={\frac {1}{\sqrt {3}}}|0\rangle +{\sqrt {\frac {2}{3}}}e^{i{\frac {4\pi }{3}}}|1\rangle ,\end{aligned}}}
which form the vertices of a regular tetrahedron in the Bloch sphere. The projectors that define the SIC-POVM are given by Π i = | ψ i ⟩ ⟨ ψ i | {\displaystyle \Pi _{i}=|\psi _{i}\rangle \langle \psi _{i}|} , and the elements of the SIC-POVM are thus F i = Π i / 2 = | ψ i ⟩ ⟨ ψ i | / 2 {\displaystyle F_{i}=\Pi _{i}/2=|\psi _{i}\rangle \!\langle \psi _{i}|/2} . For higher dimensions this is not feasible, necessitating the use of a more sophisticated approach.
Group covariance
General group covariance A SIC-POVM P {\displaystyle P} is said to be group covariant if there exists a group G {\displaystyle G} with a d 2 {\displaystyle d^{2}} -dimensional unitary representation such that
∀ | ψ ⟩ ⟨ ψ | ∈ P , ∀ U g ∈ G , U g | ψ ⟩ ∈ P {\displaystyle \forall |\psi \rangle \langle \psi |\in P,\quad \forall U_{g}\in G,\quad U_{g}|\psi \rangle \in P}
∀ | ψ ⟩ ⟨ ψ | , | ϕ ⟩ ⟨ ϕ | ∈ P , ∃ U g ∈ G , U g | ϕ ⟩ = | ψ ⟩ {\displaystyle \forall |\psi \rangle \langle \psi |,|\phi \rangle \langle \phi |\in P,\quad \exists U_{g}\in G,\quad U_{g}|\phi \rangle =|\psi \rangle }
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 ⟩ {\displaystyle |\phi \rangle } such that
| ⟨ ϕ | U g | ϕ ⟩ | 2 = 1 d + 1 ∀ g ≠ i d {\displaystyle |\langle \phi |U_{g}|\phi \rangle |^{2}={\frac {1}{d+1}}\ \forall g\neq id} . The SIC-POVM is then the set generated by the group action of U g {\displaystyle U_{g}} on | ϕ ⟩ {\displaystyle |\phi \rangle } .
The case of Zd × Zd So far, most SIC-POVM's have been found by considering group covariance under Z d × Z d {\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}} . To construct the unitary representation, we map Z d × Z d {\displaystyle \mathbb {Z} _{d}\times \mathbb {Z} _{d}} to U ( d ) {\displaystyle U(d)} , the group of unitary operators on d-dimensions. Several operators must first be introduced. Let | e j ⟩ {\displaystyle |e_{j}\rangle } be a basis for H {\displaystyle {\mathcal {H}}} , then the phase operator is
T | e j ⟩ = ω j | e j ⟩ {\displaystyle T|e_{j}\rangle =\omega ^{j}|e_{j}\rangle } where ω = e 2 π i d {\displaystyle \omega =e^{\frac {2\pi i}{d}}} is a root of unity and the shift operator as
S | e j ⟩ = | e j + 1 ( mod d ) ⟩ {\displaystyle S|e_{j}\rangle =|e_{j+1{\pmod {d}}}\rangle }
Combining these two operators yields the unitary Weyl operator W ( p , q ) = S p