[Paper Review] Length of separable states and symmetrical informationally complete (SIC) POVM

Lin Chen|arXiv (Cornell University)|Feb 4, 2013

Quantum Mechanics and Non-Hermitian Physics11 references3 citations

TL;DR

This paper establishes a deep connection between the existence of symmetric informationally complete positive operator-valued measures (SIC-POVMs) and the minimal length of separable states. It proves that the conjecture $ L( ho_2) = d^2 $ — that the length of the state $ ho_2 = rac{2}{d^2 + d} S_d $ equals $ d^2 $ — is equivalent to the existence of a SIC-POVM in dimension $ d $, thereby reducing the SIC-POVM existence problem to a separability length conjecture.

ABSTRACT

This short note reviews the notion and fundamental properties of SIC-POVM and its connection with the length of separable states. We also review the t-design.

Motivation & Objective

• To clarify the relationship between the length of separable states and the existence of symmetric informationally complete positive operator-valued measures (SIC-POVMs).
• To show that the conjecture $ L( ho_2) = d^2 $, where $ ho_2 = \frac{2}{d^2 + d} S_d $, is equivalent to the existence of a SIC-POVM in dimension $ d $.
• To frame the SIC-POVM existence problem as a separability length problem, offering a new pathway to resolve this long-standing open question.
• To connect SIC-POVMs to t-designs, particularly tight 2-designs, and show that their existence is equivalent to the length conjecture for $ \rho_2 $.

Proposed method

• Uses the definition of a SIC-POVM as $ d^2 $ subnormalized projectors $ \Pi_j = \frac{1}{d} |\psi_j\rangle\langle\psi_j| $ satisfying $ |\langle\psi_j|\psi_k\rangle|^2 = \frac{1 + d\delta_{jk}}{d+1} $.
• Applies the 2-design property of SIC-POVMs, shown via the identity $ \sum_{i=1}^{d^2} |\psi_i\rangle\!\langle\psi_i|^{\otimes 2} = \frac{2d}{d+1} S_d $, where $ S_d $ is the symmetrizer on $ \mathbb{C}^d \otimes \mathbb{C}^d $.
• Analyzes the state $ \rho_2 = \frac{2}{d^2 + d} S_d $, showing its partial transpose is $ \rho_2^\Gamma = \frac{1}{d^2 + d}(I + |\Psi_d\rangle\!\langle\Psi_d|) $, with $ |\Psi_d\rangle = \sum_{i=1}^d |ii\rangle $.
• Establishes that $ L(\rho_2) \geq \max\{ r(\rho_2), r(\rho_2^\Gamma) \} = d^2 $, and that equality holds if and only if a SIC-POVM exists in dimension $ d $.
• Relates the problem to t-designs, showing that tight 2-designs exist if and only if $ L(\rho_2) = d^2 $, and that this is equivalent to SIC-POVM existence.
• Uses known results on t-designs to show that tight 2-designs exist for $ d = 2, \dots, 16, 19, 24, 28, 31, 35, 37, 43, 48 $, consistent with known SIC-POVM constructions.

Experimental results

Research questions

• RQ1 Is the conjecture $ L(\rho_2) = d^2 $ true for all dimensions $ d \geq 2 $, where $ \rho_2 = \frac{2}{d^2 + d} S_d $?
• RQ2 Does the existence of a SIC-POVM in dimension $ d $ imply that the length of $ \rho_2 $ is exactly $ d^2 $?
• RQ3 Is the existence of a tight 2-design in dimension $ d $ equivalent to the existence of a SIC-POVM in that dimension?
• RQ4 Can the SIC-POVM existence problem be reduced to a separability length conjecture for the state $ \rho_2 $?
• RQ5 Are there dimensions $ d $ for which $ L(\rho_2) > d^2 $, implying the non-existence of SIC-POVMs in those dimensions?

Key findings

• The state $ \rho_2 = \frac{2}{d^2 + d} S_d $ is separable and has birank $ \left(\frac{d^2 + d}{2}, d^2\right) $, implying $ L(\rho_2) \geq d^2 $.
• For $ d = 2 $, the length $ L(\rho_2) = d^2 $ holds, and this is consistent with the known analytical SIC-POVM in dimension 2.
• The equality $ L(\rho_2) = d^2 $ holds for all $ d \in \{2, \dots, 16, 19, 24, 28, 31, 35, 37, 43, 48\} $, where SIC-POVMs are known to exist.
• The conjecture $ L(\rho_2) = d^2 $ is equivalent to the existence of a SIC-POVM in dimension $ d $, so a positive answer would resolve the SIC-POVM existence problem.
• The existence of a tight 2-design in dimension $ d $ is equivalent to $ L(\rho_2) = d^2 $, and thus also equivalent to SIC-POVM existence.
• The paper shows that either $ L(\rho_2) = d^2 $ for all $ d \geq 2 $, implying SIC-POVMs exist universally, or $ L(\rho_2) > d^2 $ for some $ d $, implying SIC-POVMs do not exist in that dimension.