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Quantum Information Processing

Examples of conditional SIC-POVMs

Hiromichi Ohno, Dėnes Petz

Published Aug 3, 2015• 2 citations
Publisher

Abstract

The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.

DOI
10.1007/s11128-015-1085-y

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The state of a quantum system is a density matrix with several parameters. The concern herein is how to recover the parameters. Several possibilities exist for the optimal recovery method, and we consider some special cases. We assume that a few parameters are known and that the others are to be recovered. The optimal positive-operator-valued measure (POVM) for recovering unknown parameters with an additional condition is called a conditional symmetric informationally complete POVM (SIC-POVM). In this paper, we study the existence or nonexistence of conditional SIC-POVMs. We provide a necessary condition for existence and some examples.
This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in Cn\mathbb{C}^{n}Cn whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such n2n^{2}n2 vectors in Cn\mathbb{C}^{n}Cn for any nnn .But the existence is only proved when n≤15n\leq 15n≤15 and n=19n=19n=19 , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in Cn\mathbb{C}^{n}Cn .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 ρ∈Mn(C)\rho\in M_{n}(\mathbb{C})ρ∈Mn​(C) is called a density matrix (or state) if ρ≥0\rho\geq 0ρ≥0 and ′b(ρ)=1.\prime b(\rho)=1.′b(ρ)=1.For any density matrix ρ\rhoρ , we can define a state ρ^\hat{\rho}ρ^​ by ρ^(X)=h(ρX)\hat{\rho}(X)=h(\rho X)ρ^​(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t (≻(\succ(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators {Pi}i=1k⊂Mn(C)\{P_{i}\}_{i=1}^{k}\subset M_{n}(\mathbb{C}){Pi​}i=1k​⊂Mn​(C) is called a positive operator valued measure (POVM) if Pi≥0(1≤i≤k)P_{i}\geq 0(1\leq i\leq k)Pi​≥0(1≤i≤k) and ∑i=1kP=I.\sum_{i=1}^{k}P=I.∑i=1k​P=I.If P1P_{1}P1​ is a projection, then {Pi}i=1k\{P_{i}\}_{i=1}^{k}{Pi​}i=1k​ is called a projection valued measure (PVM)(PVM)(PVM) .
It is a well-known fact that the optimal positive operator valued measure (POVM) for quantum-state tomography is the symmetric, informationally complete POVM (SIC-POVM). We investigate the same problem only in the case when there is some a priori information about the state, specifically when some parameters are known. In this paper, we mainly focus on solving a three-dimensional optimization problem, which gives us a non-trivial example for the so-called conditional SIC-POVMs, a straightforward generalization of the concept of SIC-POVMs. We also present other special cases to show further applications of the proposed numerical methods and to illustrate the complexity of this topic.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a few known classes of them, and we do not even know whether there exists an infinite class of them, thus constructing approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) has its own meaning. In this paper, we use character sums over finite fields to present two constructions of ASIC-POVMs. We show that there are some classes of infinite families of ASIC-POVMs by using some special functions over finite fields.