Monogamy of quantum correlations Sudha, A.R.Usha Devi and A.K.Rajagopal Department of Physics, Kuvempu University, Shankaraghatta Department of Physics, Bangalore University, Bangalore Inspire Institute, Alexandria, VA, USA 26th February 2012, HRI, Allahabad

Outline of the talk Introduction Measures of Quantum correlations other than entanglement: Rajagopal-Rendell Quantum Deficit (RRQD) Monogamy Relation for RRQD Majorana representation of symmetric pure states Sharing of Quantum correlations In 3-qubit symmetric pure states Polygamous nature of symmetric pure states with 2 distinct Majorana spinors Symmetric pure states with 3 distinct Majorana spinors Generalized GHZ states: Monogamous nature Generalized W states: Polygamous nature

Introduction Shareability among several parties of a composite system is one among the several features that distinguishes correlations in classical and quantum scenario. While classical correlations are known to be infinitely shareable, quantum correlations, especially quantum entanglement, have limited shareability.

V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev. A 61, 052306 (2000) If a pair of qubits A, B are in a maximally entangled state, then the system A (or B) cannot be entangled to a third system C. If A is partly entangled with B, then A (or B) can have only a limited entanglement with C V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev. A 61, 052306 (2000)

Monogamy of entanglement This unique feature of limited shareability of entanglement is qualitatively termed the Monogamy of entanglement. D. Bruss, Phys.Rev.A 60, 4344 (1999) M.Koashi, V.Buzek and N.Imoto, Phys.Rev.A 62, 050302 (2000) K.A.Dennison and W.K.Wootters, Phys.Rev.A 65, 010301R (2001) A.Higuchi and A.Sudbery, Phys.Lett.A 273, 213 (2000) B.M Terhal, quant-ph/0307120

CKW Monogamy inequality For a pure state of 3-qubits, Coffman-Kundu-Wootters have quantified the limitation on shareability of entanglement among subsystems through the following inequality

Equivalently, we have, as the CKW monogamy inequality.

For 3-qubit mixed states, the CKW monogamy inequality turns out to be where the minimum on the RHS is taken over all pure state decompositions

M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004) In fact, a system A which is maximally entangled to system B cannot even be classically correlated to system C. Also, a perfect classical correlation between A and B forbids system A (or B) from being entangled to system C M. Koashi and A. Winter, Phys. Rev. A 69, 022309 (2004)

R.Prabhu, A. Pati, A.Sen De and U.Sen quant-ph 1108.5168 Recently, enquiries regarding how non-classical correlations other than entanglement get shared among more than two parties have been raised. It is found that the quantum correlations (quantified through the measure Quantum Discord) in 3-qubit pure states do not necessarily obey monogamy inequality. R.Prabhu, A. Pati, A.Sen De and U.Sen quant-ph 1108.5168

In view of the fact that there are several measures of quantum correllation other than entanglement, our interest was to check whether the violation of monogamy is true for these measures too.

Measures of Quantum correlations other than entanglement Quantum Discord: Minimum value of the difference between two classically identical expressions for conditional entropy Optimization over the set of projective measurements is required for the evaluation of quantum discord. H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)

Measures of Quantum correlations other than entanglement Rajagopal Rendell Quantum Deficit (RRQD): A measure that quantifies non-classical correlations by finding how far a given quantum state is to its classically decohered counterpart. A.K.Rajagopal and R.W. Rendell, Phys. Rev. A. 66, 022104 (2002) S.Luo, Phys.Rev.A 77, 022301 (2008) (Measurement Induced Disturbance)

Rajagopal-Rendell Quantum Deficit It is defined as the relative entropy of the bipartite state with that of its classically decohered counterpart

No Optimization required for the evaluation of Rajagopal-Rendell Quantum Deficit, unlike that for Quantum Discord.

Monogamy relation with respect to RRQD

An illustration for the quantification of monogamous nature

Having defined the monogamy relation for the measure RRQD, we now consider 3-qubit symmetric pure states belonging to different SLOCC classes and examine whether they obey monogamy relation in general or not. To do this, we make use of the Majorana representation of symmetric states.

Ettore Majorana –Lost Genius! (1906—1938?) There are several categories of scientists in the world; those of second or third rank do their best but never get very far. Then there is the first rank, those who make important discoveries, fundamental to scientific progress. But then there are the geniuses, like Galilei and Newton. Majorana was one of these. ------- Enrico Fermi

Majorana representation of pure symmetric states Majorana had proposed an elegant geometrical representation for symmetric N qubit pure states, in terms of constituent spinors Here, corresponds to all the N! permutations of the N spinors and is the overall normalization factor. E. Majorana, Nuovo Cimento. 9, 43 (1932)

The Majorana spinors constituting the N-qubit symmetric state are given by Any N-qubit symmetric state can be represented in terms of the constituent Majorana spinors as N-points on the Bloch sphere.

Majorana representation of pure symmetric states Based on the no. of distinct spinors (degeneracy number) and their frequency of occurrence, several SLOCC classes of pure symmetric states can be identified. T. Bastin et.al., Phys.Rev.Lett., 103, 070503 (2009)

R. Usha Devi, Sudha and A. K. Rajagopal, Quantum Inf Process, DOI 10.1007/s11128-011-0280-8 (2011)

Simplification leads to Thus all symmetric states constituted by two distinct Majorana spinors, are equivalent (under local unitary transformations) to

Polygamous nature of the SLOCC Class

Polygamous nature of the SLOCC Class

Recall that and

Polygamous nature of the SLOCC Class 3-qubit states with 2 distinct Majorana spinors do not obey the monogamy inequality with respect to RRQD and thus are polygamous.

cube roots of unity

Monogamous nature of GHZ state

Polygamous nature of state

Polygamous nature of state

Generalized GHZ states

Generalized W states

Generalized W states

Generalized W states

State Nature SLOCC Class Polygamous Either Polygamous or Monogamous Generalized GHZ states Monogamous Generalized W states

Use of Rajagopal-Rendell Quantum Deficit (RRQD) as a Witness The polygamous nature of the SLOCC class of symmetric states with two distinct spinors and the strictly monogamous nature of the generalized GHZ class of states allows for the use of RRQD as a witness to identify these classes

Papers of interest. . . Monogamy properties of quantum and classical correlations Gian Luca Giorgi PRA 84, 054301 (2011) Are general quantum correlations monogamous A. Streltsov,G. Adesso, M. Piani and Dagmar Bruß, quant-ph 1112.3967 (2011) Why Entanglement of Formation is not generally monogamous? F. F. Fanchini, M. C. de Oliveira,L. K. Castelano and M. F. Cornelio quant-ph:1110.1054 (2011)

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