Exchange symmetry and multipartite entanglement

Ichikawa, Tsuneki; Sasaki, Toshihiko; Tsutsui, Izumi; Yonezawa, Nobuhiro

doi:10.1103/physreva.78.052105

Cited by 33 publications

(39 citation statements)

References 38 publications

(17 reference statements)

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“…By virtue of the results of Refs. [11,13], the latter can hold only if |φ i = |e ⊗|S| for any i. For completeness, let us recall the proof of this fact.…”

Section: Theorem 2 Let Us Consider An N -Qubit Ppt Symmetric State ρmentioning

confidence: 99%

“…This straightforwardly implies that entangled symmetric pure and thus mixed states have genuine multipartite entanglement (see also Ref. [13]). Indeed, if a symmetric ρ can be written as a convex combination of density matrices, each separable across some, in general different, bipartition, then ρ has pure separable vectors in its range.…”

Section: A Separabilitymentioning

confidence: 99%

“…[11,13]). This straightforwardly implies that entangled symmetric pure and thus mixed states have genuine multipartite entanglement (see also Ref.…”

Section: A Separabilitymentioning

confidence: 99%

“…The so-called symmetric states have recently been attracting much attention [11][12][13][14][15][16][17][18]. In particular, the underlying symmetry allowed for the use of the Majorana representation [19] for an identification of SLOCC classes of multipartite symmetric states [14] (see also Refs.…”

Section: Introductionmentioning

confidence: 99%

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Entangled symmetric states ofNqubits with all positive partial transpositions

et al. 2012

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From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with positive partial transposition and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)] With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd number of qubits we find a single configuration of ranks for which there are extremal states.

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“…By virtue of the results of Refs. [11,13], the latter can hold only if |φ i = |e ⊗|S| for any i. For completeness, let us recall the proof of this fact.…”

Section: Theorem 2 Let Us Consider An N -Qubit Ppt Symmetric State ρmentioning

confidence: 99%

Section: A Separabilitymentioning

confidence: 99%

“…[11,13]). This straightforwardly implies that entangled symmetric pure and thus mixed states have genuine multipartite entanglement (see also Ref.…”

Section: A Separabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Entangled symmetric states ofNqubits with all positive partial transpositions

et al. 2012

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“…(5), φ(x) is a single-particle function, which determines the spatial properties of the system. As we prove in the Supplementary Materials, rephrasing the arguments of [35,36], the general separable state of N identical bosons is a mixture of different states (4),…”

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confidence: 99%

Cauchy-Schwarz inequality and particle entanglement

et al. 2014

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The Glauber-Sudarshan P-representation is used in quantum optics to distinguish between semi-classical and genuinely quantum electromagnetic fields. We employ the analog of the Prepresentation to systems of identical bosons and show that the violation of the Cauchy-Schwarz inequality for the second-order correlation function is a proof of particle entanglement. The present derivation applies to any quantum system of identical bosons, with either fixed or fluctuating number of particles, provided that there is no coherence between different number states. In the light of recent experimental advances in single-particle detection, the violation of the Cauchy-Schwarz inequality may become an easily accessible entanglement probe in correlated many-body systems.Although the foundations of quantum and classical physics are much different, it is often difficult to construct a simple criterion of "quantumness" of a particular system. A good example of a non-classical behavior is, according to the Schrödinger equation, the ability of particles to exist in superpositions of quantum states. The most prominent manifestation of such superposition is the Young double-slit experiment for massive particles, which confirms their wave character. For optical waves it is the opposite -the non-classical electromagnetic field is that consisting of individual photons. The challenging question whether, and in what sense, the pulse of light is quantum, was among the key issues triggering the development of quantum optics.The problem was formalized by Glauber and Sudarshan in their studies on coherence in the context of correlation functions [1,2]. They employed the coherent states |Φ defined by the relationÊ (+) (x) |Φ = Φ(x) |Φ , whereÊ (+) (x) is the positive-frequency part of the electromagnetic fieldÊ(x), and expressed the density matrix using the so-called P-representationThe symbol DΦ denotes the integration measure over the set of complex fields Φ. The state of light is classical if the outcome of the measurement can be explained in terms of classical electromagnetic fields, which happens when the P-representation can be interpreted as a probability distribution, which means that it is normalized andfor any volume V. When the P-representation does not satisfy condition (2), the field is said to be quantum. Once the electromagnetic field is quantized, photons can be treated on a more equal foot with other particles. It is then reasonable to ask the question about correlations between individual particles and in this context the concept of particle entanglement emerges [3,4]. The possibility for particles to be entangled, which is a purely quantum phenomenon, has rather dramatic consequences. The quantumness of entanglement is underlined by the word "paradox" often used to describe some highly counter-intuitive phenomena such as the Einstein-Podolsky-Rosen (EPR) paradox [5] and the related Schrödinger's cat problem. Apart from fundamental aspects, systems of entangled particles have applications in quantum information [6], teleporta...

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