Parametrizations of density matrices

Brüning, Erwin; Mäkelä, Harri; Messina, A.; Petruccione, Francesco

doi:10.1080/09500340.2011.632097

Cited by 46 publications

(55 citation statements)

References 63 publications

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“…Therefore the bases {|r j } d j=1 is completely determined by U . Once d 2 − d real parameters are sufficient to specify completely an arbitrary unitary matrix U with dimensions dxd [36], it follows that d 2 − 1 independent real parameters are sufficient for a thorough description of any density matrix.…”

Section: Standard Methodsmentioning

confidence: 99%

“…One important tool for accomplishing this task is the generation and analysis of RQS [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31], which will have an analogous role to that that random numbers have in classical stochastic theories [32,33,34,35]. The parametrization of quantum states [15,36,37] is the initial step towards generating them numerically and is one of the main topics of this survey, which is organized in the following manner. In Sec.…”

Section: Introductionmentioning

confidence: 99%

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Random Sampling of Quantum States: a Survey of Methods

Maziero

2015

Braz J Phys

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The numerical generation of random quantum states (RQS) is an important procedure for investigations in quantum information science. Here we review some methods that may be used for performing that task. We start by presenting a simple procedure for generating random state vectors, for which the main tool is the random sampling of unbiased discrete probability distributions (DPD). Afterwards the creation of random density matrices is addressed. In this context we first present the standard method, which consists in using the spectral decomposition of a quantum state for getting RQS from random DPDs and random unitary matrices. In the sequence the Bloch vector parametrization method is described. This approach, despite being useful in several instances, is not in general convenient for RQS generation. In the last part of the article we regard the overparametrized method (OPM) and the related Ginibre and Bures techniques. The OPM can be used to create random positive semidefinite matrices with unit trace from randomly produced general complex matrices in a simple way that is friendly for numerical implementations. We consider a physically relevant issue related to the possible domains that may be used for the real and imaginary parts of the elements of such general complex matrices. Subsequently a too fast concentration of measure in the quantum state space that appears in this parametrization is noticed.

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Section: Standard Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Random Sampling of Quantum States: a Survey of Methods

Maziero

2015

Braz J Phys

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“…GHZ (3,4).-Let us turn to tripartite systems of dimension four ("'ququarts"') with the GHZ state defined as ρ GHZ(3,4) = 1 4 3 i,j=0 |iii jjj|. First we choose a set of indices according to ensure anti-commutativity along a partition.…”

Section: A Entanglement Detection Examplesmentioning

confidence: 99%

“…The canonical choice for a complete basis of observables is usually given by the so-called generalized Gell-Mann matrices, generators of the special unitary group [SU(d)]. Being a natural choice for higher-dimensional spin representations they have been extensively used in parametrizations of corresponding density matrices [4,13] and in entanglement detection. Other choices, such as the HeisenbergWeyl (HW) operators, and the non-Hermitian generalization of the 1/2-spin Pauli operators, have also been explored [3,[14][15][16][17][18].…”

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confidence: 99%

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Heisenberg-Weyl Observables: Bloch vectors in phase space

et al. 2016

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We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.PACS numbers: 07.10. Cm, 03.65.Ta, 03.65.Ud Introduction. The Bloch representation is a cornerstone of analyzing the characteristics of quantum systems. It was first introduced for two-level systems by Bloch [1] and has since been used in a wide variety of settings (for comprehensive reviews consult [2][3][4]). It is usually defined via a decomposition of the density matrix into a complete operator basis. Defining quantum states via the expectation values of a complete set of measurements gives a very practical account of their properties. Apart from being intuitive this approach gives convenient solutions for Hamiltonian evolutions (see, e.g., [5]) and has found many applications in entanglement theory [6][7][8][9][10][11][12].However, there is not a unique Bloch decomposition of a given quantum state; this fact may favor a particular representation over another for certain tasks. The canonical choice for a complete basis of observables is usually given by the so-called generalized Gell-Mann matrices, generators of the special unitary group [SU(d)]. Being a natural choice for higher-dimensional spin representations they have been extensively used in parametrizations of corresponding density matrices [4,13] and in entanglement detection. Other choices, such as the HeisenbergWeyl (HW) operators, and the non-Hermitian generalization of the 1/2-spin Pauli operators, have also been explored [3,[14][15][16][17][18]. While having some convenient properties, they are unitary, but not Hermitian matrices. Thus the associated Bloch vector itself has imaginary entries that do not correspond to expectation values of physical observables. This makes both the theoretical description and experimental realization more cumbersome and therefore requires more effort to identify the relevant parameters.In this article we introduce a Hermitian Bloch-basis derived from HW-operators. It conveniently combines multiple desirable properties of Bloch vector parametrizations, allowing a smooth transition to the infinite dimensional limit. We first explore properties such as (anti-) commutativity. We then proceed with the derivation of an inequality which bounds sums of anti-commuting observables with which we show in exemplary cases how one can construct powerful criteria for entanglement detection in this new basis. Finally we present a scheme for practical experimental acquisition through a Ramseytype measurement.Phase-space displacements. We start our a...

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