SIC-POVM

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.

A POVM over a ${\displaystyle d}$-dimensional Hilbert space ${\displaystyle {\mathcal {H}}}$ is a set of ${\displaystyle m}$ positive-semidefinite operators ${\displaystyle \left\{F_{i}\right\}_{i=1}^{m}}$ that sum to the identity:$${\displaystyle \sum _{i=1}^{m}F_{i}=I.}$$

If a POVM consists of at least ${\displaystyle d^{2}}$ operators which span the space of self-adjoint operators ${\displaystyle {\mathcal {L}}({\mathcal {H}})}$, it is said to be an informationally complete POVM (IC-POVM). IC-POVMs consisting of exactly ${\displaystyle d^{2}}$ elements are called minimal. A set of ${\displaystyle d^{2}}$ rank-1 projectors ${\displaystyle \left\{\Pi _{i}\right\}_{i=1}^{d^{2}}}$ which have equal pairwise Hilbert–Schmidt inner products, $${\displaystyle \mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}},}$$ defines a minimal IC-POVM with elements ${\displaystyle F_{i}={\frac {1}{d}}\Pi _{i}}$ called a SIC-POVM.

Consider an arbitrary set of rank-1 projectors ${\displaystyle (\Pi _{i})_{i=1}^{d^{2}}}$ such that ${\displaystyle F_{i}=\Pi _{i}/d}$ is a POVM, and thus ${\displaystyle {\frac {1}{d}}\sum _{i}\Pi _{i}=I}$. Asking the projectors to have equal pairwise inner products, ${\displaystyle \mathrm {Tr} (\Pi _{i}\Pi _{j})=c}$ for all ${\displaystyle i\neq j}$, fixes the value of ${\displaystyle c}$. To see this, observe that $${\displaystyle {\begin{aligned}d&=\mathrm {Tr} (I^{2})\\&={\frac {1}{d^{2}}}\sum _{i,j}\mathrm {Tr} (\Pi _{i}\Pi _{j})\\&={\frac {1}{d^{2}}}\left(d^{2}+cd^{2}(d^{2}-1)\right)\end{aligned}}}$$ implies that ${\displaystyle c={\frac {1}{d+1}}}$. Thus, $${\displaystyle \mathrm {Tr} \left(\Pi _{i}\Pi _{j}\right)={\frac {d\delta _{ij}+1}{d+1}}.}$$ This property is what makes SIC-POVMs symmetric: Any pair of elements has the same Hilbert–Schmidt inner product as any other pair.

In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map ${\displaystyle {\mathcal {L}}({\mathcal {H}})\rightarrow {\mathcal {L}}({\mathcal {H}})}$. This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map

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: