Reference : Multiqubit symmetric states with maximally mixed one-qubit reductions
Scientific congresses and symposiums : Poster
Physical, chemical, mathematical & earth Sciences : Physics
Multiqubit symmetric states with maximally mixed one-qubit reductions
Baguette, Dorian mailto [Université de Liège > Département de physique > Optique quantique >]
Bastin, Thierry mailto [Université de Liège > Département de physique > Spectroscopie atomique et Physique des atomes froids >]
Martin, John mailto [Université de Liège > Département de physique > Optique quantique >]
5th International Colloquium on Quantum Information, Foundations & Applications
from 17-11-2014 to 19-11-2014
Ecole Normale Supérieure de Lyon
[en] Anticoherence ; Quantum Information ; Entanglement
[en] We present a comprehensive study on the remarquable properties shared by maximally entangled symmetric states of arbitrary numbers of qubits in the sense of the maximal mixedness of the one-qubit reduced density operator. Such states are of great interest in quantum information as they maximize several measures of entanglement, such as Meyer-Wallach entropy [1] and any entanglement monotone based on linear homogenous positive functions of pure state within their SLOCC classes of states [2, 3]. When they exist, they are unique up to local unitaries within their SLOCC classes [3, 4]. They play a specific role in the determination of the local unitary equivalence of multiqubit states [5]. Moreover, they are maximally fragile (in the sense that they are the states which are the most sensitive to noise) and have therefore been proposed as ideal candidates for ultrasensitive sensors [6]. They appear in the litterature under various names : maximally entangled states [6], 1-uniform states [7], normal forms [3, 4] and nongeneric states [5].
We present a general criterion to easily identify whether given symmetric states are maximally entangled or not [9]. We show that these maximally entangled symmetric (MES) states are the only symmetric states for which the expectation value of the associated collective spin S of the system vanishes, which coincides with the definition of anticoherence to order one of spin states. This definition also coincides with the cancellation of the dipole moment of the Husimi function of the state. We then generalize these properties and show that a state is anticoherent to order t, <(S.n)^k> is independent of n for k = 1, . . . , t, where n is a unit vector, iff it has maximally mixed t-qubit reductions or iff all moments up to order 2t of its Husimi function vanish. We also establish the equivalence between anticoherent states to order t and unpolarized light states to order t [8], thereby encompassing various state characterizations under the same banner [9, 10].
We provide a nonexistence criterion allowing us to know immediately whether SLOCC classes of symmetric states can contain MES states or not. We show in particular that the symmetric Dicke state SLOCC classes never contain such MES states, with the only exception of the balanced Dicke state class for even numbers of qubits. We analyze the 4-qubit system exhaustively and identify and characterize all MES states of this system as well as the only 4-qubit state anticoherent to order 2. Finally, we analyze the entanglement content of MES states with respect to the geometric [11] and barycentric [12] measures of entanglement.
[1] D. A. Meyer and N. R. Wallach, J. Math. Phys. 43, 4273 (2002).
[2] Classes of states equivalent through stochastic local operations with classical communication.
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[4] G. Gour and N. Wallach, N. J. Phys. 13, 073013 (2011).
[5] B. Kraus, Phys. Rev. Lett. 104, 020504 (2010).
[6] N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A 246, 1 (1998).
[7] A. J. Scott, Phys. Rev. A 69, 052330 (2004).
[8] L. L. Sánchez-Soto, A. B. Klimov, P. de la Hoz, and G. Leuchs J. Phys. B : At. Mol. Opt. Phys. 46, 104011 (2013).
[9] D. Baguette, T. Bastin, and J. Martin, Phys. Rev. A 90, 032314 (2014).
[10] O. Giraud, D. Braun, D. Baguette, T. Bastin, and J. Martin, arXiv :1409.1106.
[11] T.-C. Wei and P. M. Goldbart, Phys. Rev. A 68, 042307 (2003).
[12] W. Ganczarek, M. Kus, and K. Zyczkowski, Phys. Rev. A 85, 032314 (2012).

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