[Paper Review] Construction of all general symmetric informationally complete measurements

Amir Kalev, Gilad Gour|arXiv (Cornell University)|May 28, 2013

Numerical methods in inverse problems1 references18 citations

TL;DR

This paper constructs the complete family of all general symmetric informationally complete positive operator-valued measures (SIC POVMs), showing that every orthonormal basis of a real vector space of dimension $d^2 - 1$ corresponds to a unique general SIC POVM. The key contribution is proving that general SIC POVMs exist in all finite dimensions, with a continuous parameter $a = \mathrm{Tr}(P_\alpha^2)$ controlling their rank, ranging from full-rank ($a \to 1/d^3$) to nearly rank-1 ($a \to 1/d^2$), including weak SIC POVMs that minimally disturb quantum states.

ABSTRACT

We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.

Motivation & Objective

• To construct the complete set of all general (non-rank-1) symmetric informationally complete POVMs in any finite dimension $d$.
• To resolve the open question of whether general SIC POVMs exist in all finite dimensions.
• To characterize the full family of SIC POVMs via a continuous parameter $a = \mathrm{Tr}(P_\alpha^2)$, which determines the rank and disturbance of the POVM.
• To show that weak SIC POVMs—where elements are close to $\frac{1}{d^2}I$—exist for all $d$, minimizing state disturbance.

Proposed method

• Parametrize all general SIC POVMs using a real orthogonal matrix $A$ acting on an orthonormal basis $\{F_\alpha\}$ of the traceless Hermitian space $\mathrm{T}_d$ of dimension $d^2 - 1$.
• Construct SIC POVM elements as $P_\alpha = \frac{1}{d^3}I + t \sum_{\beta=1}^{d^2-1} A_{\alpha\beta} R_\beta$, where $R_\beta$ are traceless Hermitian operators forming an orthonormal basis of $\mathrm{T}_d$.
• Use the Hilbert-Schmidt inner product to enforce symmetry: $\mathrm{Tr}(P_\alpha P_\beta) = b$ for $\alpha \neq \beta$, and $\mathrm{Tr}(P_\alpha^2) = a$ for all $\alpha$, with $a$ and $b$ related by $b = \frac{1 - da}{d(d^2 - 1)}$.
• Prove linear independence of the $P_\alpha$ operators via trace constraints and orthogonality of the basis $\{F_\alpha\}$, ensuring they form a valid POVM.
• Demonstrate that any orthonormal basis of $\mathbb{R}^{d^2 - 1}$ corresponds to a unique general SIC POVM, establishing a one-to-one correspondence.
• Show that the parameter $a$ ranges continuously from $1/d^3$ (full rank) to $1/d^2$ (nearly rank-1), with $a = 1/d^2$ corresponding to rank-1 SIC POVMs if they exist.

Experimental results

Research questions

• RQ1 Does a complete family of general SIC POVMs exist in all finite dimensions $d$?
• RQ2 Can all general SIC POVMs be parameterized by orthonormal bases of a real vector space of dimension $d^2 - 1$?
• RQ3 What is the role of the parameter $a = \mathrm{Tr}(P_\alpha^2)$ in determining the rank and disturbance of a SIC POVM?
• RQ4 Do weak SIC POVMs—where elements are close to $\frac{1}{d^2}I$—exist for all $d$, and can they be constructed explicitly?
• RQ5 Is the existence of rank-1 SIC POVMs in all dimensions equivalent to whether $a = 1/d^2$ is achievable within the constructed family?

Key findings

• All general SIC POVMs in dimension $d$ are in one-to-one correspondence with orthonormal bases of a real vector space of dimension $d^2 - 1$, establishing a complete classification.
• The parameter $a = \mathrm{Tr}(P_\alpha^2)$ fully characterizes the type of a general SIC POVM, with $a \in (1/d^3, 1/d^2]$, where $a = 1/d^2$ corresponds to rank-1 SIC POVMs.
• Weak SIC POVMs—where all elements are arbitrarily close to $\frac{1}{d^2}I$—exist for all $d$, minimizing disturbance to the measured state.
• The construction proves that general SIC POVMs exist in all finite dimensions, resolving an open question about their existence.
• The family of general SIC POVMs is continuous in $a$, so if any $a_0 > 1/d^3$ is achievable, then all $a \in (1/d^3, a_0]$ are also achievable.
• The maximal value $a_{\max} = 1/d^2$ is achievable if and only if a rank-1 SIC POVM exists in dimension $d$, which remains an open problem.