The Bloch vector for N-level systems

Kimura, Gen

doi:10.1016/s0375-9601(03)00941-1

Cited by 437 publications

(558 citation statements)

References 18 publications

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“…To obtain a tighter upper bound by using the lowest order relaxation to (18), we found it most convenient to express (18) in terms of the real optimization variables, r A , r B are simply the coherence vectors that have been studied in the literature [53].…”

Section: Appendix C: Lowest Order Relaxation With Observables Of Fixementioning

confidence: 99%

Bounds on quantum correlations in Bell-inequality experiments

Liang

Doherty

2007

Phys. Rev. A

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Bell-inequality violation is one of the most widely known manifestations of entanglement in quantum mechanics; indicating that experiments on physically separated quantum mechanical systems cannot be given a local realistic description. However, despite the importance of Bell inequalities, it is not known in general how to determine whether a given entangled state will violate a Bell inequality. This is because one can choose to make many different measurements on a quantum system to test any given Bell inequality and the optimization over measurements is a high-dimensional variational problem. In order to better understand this problem we present algorithms that provide, for a given quantum state and Bell inequality, both a lower bound and an upper bound on the maximal violation of the inequality. In many cases these bounds determine measurements that would demonstrate violation of the Bell inequality or provide a bound that rules out the possibility of a violation. Both bounds apply techniques from convex optimization and the methodology for creating upper bounds allows them to be systematically improved. Examples are given to illustrate how these algorithms can be used to conclude definitively if some quantum states violate a given Bell inequality.

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Section: Appendix C: Lowest Order Relaxation With Observables Of Fixementioning

confidence: 99%

Bounds on quantum correlations in Bell-inequality experiments

Liang

Doherty

2007

Phys. Rev. A

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“…[23] using the parametric form of an arbitrary d-dimensional density matrix, written in terms of the generators, G i , of S U(d) [28,[30][31][32][33], as…”

Section: Reciprocity Between Quantum Coherence and Mixednessmentioning

confidence: 99%

Quantum coherence: Reciprocity and distribution

Kumar

2017

Physics Letters A

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Quantum coherence is the outcome of the superposition principle. Recently, it has been theorized as a quantum resource, and is the premise of quantum correlations in multipartite systems. It is therefore interesting to study the coherence content and its distribution in a multipartite quantum system. In this work, we show analytically as well as numerically the reciprocity between coherence and mixedness of a quantum state. We find that this trade-off is a general feature in the sense that it is true for large spectra of measures of coherence and of mixedness. We also study the distribution of coherence in multipartite systems by looking at monogamytype relation-which we refer to as additivity relation-between coherences of different parts of the system. We show that for the Dicke states, while the normalized measures of coherence violate the additivity relation, the unnormalized ones satisfy the same.

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“…However, in dimensionality N = 3 or higher, it has been be shown that the physical constraint of positivity on the density matrix restricts the set of valid states to an irregular convex region that is a proper subset of the enclosing hypersphere. This physical region touches the surface of the enclosing hypersphere only in some places (where the fully polarized states lie) [25,26]. That is, many states within the hypersphere and on its surface are unphysical since they would create density matrices that are not positive.…”

Section: F Sskf Puritymentioning

confidence: 99%

“…Classes of cross section of the eight dimensional space in which the generalized Bloch vectors live, based on a figure by Kimura [26]. In each diagram, the shaded region represents the allowable states, while the outer circle is a cross section of the enclosing hypersphere.…”

Section: Figmentioning

confidence: 99%

Measures of quantum state purity and classical degree of polarization

Gamel

James

2012

Phys. Rev. A

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There is a well known mathematical similarity between two dimensional classical polarization optics and two level quantum systems, where the Poincare and Bloch spheres are identical mathematical structures. This analogy implies classical degree of polarization and quantum purity are in fact the same quantity. We make extensive use of this analogy to analyze various measures of polarization for higher dimensions proposed in the literature, and in particular, the N = 3 case, illustrating interesting relationships that emerge as well the advantages of each measure. We also propose a possible new class of measures of entanglement based on purity of subsystems.

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The Bloch vector for N-level systems

Abstract: We determine the set of the Bloch vectors for N -level systems, generalizing the familiar Bloch ball in 2-level systems. An origin of the structural difference from the Bloch ball in 2-level systems is clarified.

Cited by 437 publications

References 18 publications

Bounds on quantum correlations in Bell-inequality experiments

Bounds on quantum correlations in Bell-inequality experiments

Quantum coherence: Reciprocity and distribution

Measures of quantum state purity and classical degree of polarization

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