An unavoidable task in quantum information processing is how to obtain data about the state of an individual system by suitable measurements. Informationally complete measurements are relevant in quantum state tomography, quantum cryptography, quantum cloning, and other questions. Symmetric informationally complete measurements (SIC-POVMs) form an especially important class of such measurements. We formulate some novel properties and relations for general SIC-POVMs in a finite-dimensional Hilbert space. For a given density matrix and any general SIC-POVM, the so-called index of coincidence of generated probability distribution is exactly calculated. Using this result, we obtain state-dependent entropic bounds for a single general SIC-POVM. Lower entropic bounds are derived in terms of the Rényi α -entropies for and the Tsallis α -entropies for . A lower bound on the min-entropy of an SIC-POVM is separately examined. For a pair of general SIC-POVMs, entropic uncertainty relations of the Maassen–Uffink type are considered.
We construct inequalities between R\'enyi $\ensuremath{\alpha}$-entropy and the indexes of coincidence of probability distributions, based on which we obtain improved state-dependent entropic uncertainty relations for general symmetric informationally complete positive operator-valued measures (SIC-POVMs) and mutually unbiased measurements (MUMs). We show that our uncertainty relations for general SIC-POVMs and MUMs can be tight for sufficiently mixed states, and, moreover, comparisons to the numerically optimal results are made via information diagrams.
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 whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such
n2 vectors in
Cn for any
n .But the existence is only proved when
n≤15 and
n=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 .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) is called a density matrix (or state) if
ρ≥0 and
′b(ρ)=1.For any density matrix
ρ , we can define a state
ρ^ by
ρ^(X)=h(ρX) .Conversely, any state is written by the above form.Therefore, there exists a one-t
(≻ one correspondence between density matrices and states.Definition 2.2 A set of positive operators
{Pi}i=1k⊂Mn(C) is called a positive operator valued measure (POVM) if
Pi≥0(1≤i≤k) and
∑i=1kP=I.If
P1 is a projection, then
{Pi}i=1k is called a projection valued measure
(PVM) .
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.
Abstract A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important role in quantum tomography, quantum cryptography and foundations of quantum mechanics. The most important and difficult problem about SIC-POVM is to show its existence in every dimension n, which is suggested by many numerical evidences. In this paper, this existence problem of SIC-POVMs is investigated by considering their overlap phases. Specifically, the symmetric requirements of SIC-POVMs are encoded into the overlap phases between the fiducial project and the n 2 displacement operators. In this way, an equivalent condition of the existence of Weyl-Heisenberg group covariant SIC-POVMs is obtained in terms of these phases, so that a set of complex equations is established, which enables one to study the existence problem in a more explicit and direct fashion. In particular, we show that all the numbers appearing in the overlap phases have to be algebraic numbers. The significance of Zauner’s conjecture in our equivalent condition is also discussed, and therefore the SIC-POVMs in dimension four is constructed explicitly. Actually, our system of equations always admits a solution, the real difficulty lies in how to ensure these complex number solutions are of norm one, and therefore indeed phases. Our result shows that constructing SIC-POVMs from the overlap phases is possible, and symmetries can be applied directly to reduce the number of equations, so that it provide new insights into the difficult problem of SIC-POVMs, especially about the algebraic properties of these overlap phases.