Symmetric Informationally Complete Positive Operator Valued Measures Research Articles

Construction of all general symmetric informationally complete measurements

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 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.

Systems of imprimitivity for the clifford group

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

Systems of Imprimitivity for the Clifford group

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of $k$-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

Minimum tomography of two entangled qutrits using local measurements of one-qutrit symmetric informationally complete positive operator-valued measure

An experimental demonstration of two-qutrit state tomography via one-qutrit symmetric informationally complete positive operator-valued measure (SIC-POVM) is presented. A two-qutrit state is encoded in the transversal paths of a spontaneous parametric down-converted photon pair. A spatial light modulator allows to implement the necessary 81 POVM elements to reconstruct the state. The quality of the reconstruction is evaluated by comparing independent measurements with the predicted results calculated with the reconstructed state. Entanglement in the system is calculated via negativity and generalized robustness from the two-qutrit reconstructed density operator.

利用Gauss 和与Jacobi 和构造近似MUB 和SIC-POVM

Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory. While people do not know if there exists a complete MUB for non-prime-power dimension, several versions of approximately MUB have been considered by relaxed the inner product condition. So far there are only finite number of <italic>K</italic> such that SIC- POVMs in Ck have been found. As in the MUB case, several versions of approximately SIC-POVM have been considered by relaxed the inner product condition. In this paper, we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al. For prime power q, we present simple constructions of <italic>q</italic> approximately MUB (AMUB) for dimension <italic>q</italic>-1, <italic>q</italic>+1 AMUB for dimension <italic>q</italic>-1, which shows the number of orthonormal bases of an AMUB in C^{<italic>k</italic>} can be more than <italic>K</italic>+1, and q AMUB for dimension <italic>q</italic>+1 by Gauss and Jacobi sums. We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension <italic>q</italic>-1 by Gauss sum.

Optimal quantum-state tomography with known parameters

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.

Constructing symmetric informationally complete positive-operator-valued measures in Bloch space

Here we study the construction of symmetric informationally complete positive-operator-valued measures (SIC-POVMs) in the Bloch space. In this space an SIC-POVM corresponds to a regular simplex, that is a set of real equiangular, unitary vectors. Since the Bloch space also contains vectors which do not describe quantum states, it is necessary to add an extra condition to enforce the members of the simplex to describe pure quantum states. We show that in the case of a three-dimensional quantum systems it is possible to find such a simplex in an analytical way. The solution turns out to be unitarily equivalent to a Weyl–Heisenberg covariant SIC-POVM.

A Kochen–Specker inequality from a SIC

Yu and Oh (eprint) [1] have given a state-independent proof of the Kochen–Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute a SIC (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality.

The Lie algebraic significance of symmetric informationally complete measurements

Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathrm{gl}(d,\mathbb {C})$\end{document} gl (d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of \documentclass[12pt]{minimal}\begin{document}$\mathrm{gl}(d,\mathbb {C})$\end{document} gl (d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.

Representation of Quantum States as Points in a Probability Simplex Associated to a SIC-POVM

The quantum state of a d-dimensional system can be represented by a probability distribution over the d 2 outcomes of a Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM), and then this probability distribution can be represented by a vector of \(\mathbb {R}^{d^{2}-1}\) in a (d 2−1)-dimensional simplex, we will call this set of vectors \(\mathcal{Q}\). Other way of represent a d-dimensional system is by the corresponding Bloch vector also in \(\mathbb {R}^{d^{2}-1}\), we will call this set of vectors \(\mathcal{B}\). In this paper it is proved that with the adequate scaling \(\mathcal{B}=\mathcal{Q}\). Also we indicate some features of the shape of \(\mathcal{Q}\).

Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.

The probability interpretation of Wigner function by SIC-POVM

This paper will give the probability interpretation of Wigner function. We will show that Wigner function is linear function of the probability associated with special POVM (positive operator valued measure). This special POVM is called symmetric informationally complete POVM (SIC-POVM). Therefore, Wigner function can be described by physical measurement system.

A unified approach to SIC-POVMs and MUBs

A unified approach to (symmetric informationally complete) positive operator valued measures (SIC-POVMs) and mutually unbiased bases (MUBs) is developed. The approach is based on the use of Racah unit tensors for the Wigner-Racah algebra of SU(2) ⊃ U(1). Emphasis is put on similarities and differences between SIC-POVMs and MUBs.

On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in Cn which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.