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arXiv:1301.2123 · Physical Review A

Optimal quantum tomography of permutationally invariant qubits

A. B. Klimov, Gunnar G. Bjork, Luis Lorenzo Sánchez-Soto

Published Jan 10, 2013• 25 citations
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Abstract

Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exists, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.

DOI
10.1103/physreva.87.012109

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Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exists, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.
Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is presented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to minimize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems.
We study the determination of an unknown mixed state of a ddd-dimensional quantum system by means of unambiguous state discrimination. We show that optimal and nonoptimal unambiguous state discrimination can be used to reconstruct unknown states of a qubit. This result is extended to the case of a qudit by a sequence of reconstructions in two-dimensional subspaces. The total number of projections scales approximately as 2d22{d}^{2}2d2 for ddd large; this is twice as much as in the case of tomography based on mutually unbiased bases or symmetric informationally complete positive-operator-valued measures, and less than the d3{d}^{3}d3 projections required by standard tomography.
d+1d+1d+1 Measurement Bases are Sufficient for Determining ddd-Dimensional Quantum States: Theory and Experiment
A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only d+1d+1d+1 projective measurement bases to fully reconstruct an arbitrary ddd-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for d=6d=6d=6, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.
We study the physical implementation of an optimal tomographic reconstruction scheme for the case of determining the state of a multiqubit system, where trapped ions are used for defining qubits. The protocol is based on the use of mutually unbiased measurements and on the physical information described in H. H\"affner et al. [Nature (London) 438, 643 (2005)]. We introduce the concept of physical complexity for different types of unbiased measurements and analyze their generation in terms of one and two qubit gates for trapped ions.