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arXiv:quant-ph/0302064

On Symmetric Sets of Projectors for Reconstruction of a Density Matrix

Alexander Yu. Vlasov

Published Feb 10, 2003• 4 citations
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Abstract

In this work are presented sets of projectors for reconstruction of a density matrix for an arbitrary mixed state of a quantum system with the finite-dimensional Hilbert space. It was discussed earlier [quant-ph/0104126] a construction with (2n-1)n projectors for the dimension n. For n=2 it is a set with six projectors associated with eigenvectors of three Pauli matrices, but for n>2 the construction produces not such a `regular' set. In this paper are revisited some results of previous work [quant-ph/0104126] and discussed another, more symmetric construction with the Weyl matrix pair (as the generalization of Pauli matrices). In the particular case of prime n it is the mutually unbiased set with (n+1)n projectors. In appendix is shown an example of application of complete sets for discussions about separability and random robustness.

DOI
10.48550/arxiv.quant-ph/0302064

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In this work are presented sets of projectors for reconstruction of a density matrix for an arbitrary mixed state of a quantum system with the finite-dimensional Hilbert space. It was discussed earlier [quant-ph/0104126] a construction with (2n-1)n projectors for the dimension n. For n=2 it is a set with six projectors associated with eigenvectors of three Pauli matrices, but for n>2 the construction produces not such a `regular' set. In this paper are revisited some results of previous work [quant-ph/0104126] and discussed another, more symmetric construction with the Weyl matrix pair (as the generalization of Pauli matrices). In the particular case of prime n it is the mutually unbiased set with (n+1)n projectors. In appendix is shown an example of application of complete sets for discussions about separability and random robustness.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius sqrt[3/20] located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design-a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.
Recently there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: (1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the space of density operators as possible; and (2) in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant SIC-sets are intimately related: The latter represent minimum uncertainty states for the former in the sense of Wootters and Sussman. Finally, we contribute to the question of existence by conjecturing a quadratic redundancy in the equations for Weyl-Heisenberg SIC-sets.
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $(6\ifmmode\times\else\texttimes\fi{}6)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.
We study a quantum process reconstruction based on the use of mutually unbiased projectors (MUB projectors) as input states for a DDD-dimensional quantum system, with DDD being a power of a prime number. This approach connects the results of quantum-state tomography using mutually unbiased bases with the coefficients of a quantum process, expanded in terms of MUB projectors. We also study the performance of the reconstruction scheme against random errors when measuring probabilities at the MUB projectors.
A complete set of d + 1 mutually unbiased bases exists in a Hilbert space of dimension d , whenever d is a power of a prime. We discuss a simple construction of d + 1 disjoint classes (each one having d − 1 commuting operators) such that the corresponding eigenstates form sets of unbiased bases. Such a construction works properly for prime dimension. We investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements. One of these classes is diagonal, and can be mapped to cyclic operators by means of the finite Fourier transform, which allows one to understand complementarity in a similar way as for the position–momentum pair in standard quantum mechanics. The relevant examples of two and three qubits and two qutrits are discussed in detail.