Quantum Discord and Logarithmic Negativity in the Generalized n -qubit Werner State
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Quantum Discord and Logarithmic Negativity in the Generalized n -qubit Werner State M. S. Ramkarthik1 · Devvrat Tiwari1 · Pranay Barkataki1 Received: 6 July 2020 / Accepted: 15 November 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Quantum Discord (QD) is a measure of the total quantum non-local correlations of a quantum system. The formalism of quantum discord has been applied to various two-qubit mixed states and it has been reported that there is a non-zero quantum discord even when the states are unentangled. To this end, we have calculated the Quantum Discord for a higher than two qubit mixed state, that is, the generalized n-qubit Werner state with a bipartite split. We found that the QD saturates to a straight line with a unit slope in the thermodynamic limit. Qualitative studies of entanglement between the two subsystems using logarithmic negativity revealed that the entanglement content between them increases non-uniformly with the number of qubits leading to its saturation. We have proved the above claims both analytically and numerically. Keywords Generalized n-qubit Werner state · Quantum discord · Logarithmic negativity
1 Introduction Quantum non-local correlations have been one of the most unique and exclusive phenomena of the quantum world, which has no classical analogue. Over the years it was assumed that quantum entanglement [1] is the only source of quantum non-local correlations. However, in recent years, numerous pieces of evidence indicated in the direction that quantum entanglement is not the only non-local correlation that exists between two subsystems. Over the years it was shown that there are states that possess non-local behaviour but are not entangled [2, 3]. Zurek and Ollivier [4], Henderson and Vedral [5] independently gave a measure
M. S. Ramkarthik
[email protected] Devvrat Tiwari [email protected] Pranay Barkataki [email protected] 1
Department of Physics, Visvesvaraya National Institute of Technology, Nagpur - 440010, India
International Journal of Theoretical Physics
for the quantification of such non-local correlations known as Quantum Discord. For a bipartite composite quantum system ρAB , where A and B are the individual subsystems, the QD is defined as follows,
D(A : B) = I (A : B) − J (A|B),
(1)
where I (A : B) and J (A|B) are the total and classical correlations present between the two subsystems, respectively. Both of these terms are defined as follows, I (A : B) = S(A) + S(B) − S(A, B) and J (A|B) = S(A) − S(A|B), where S(A) and S(B) are the von Neumann entropies of the subsystem states ρA and ρB , respectively. The terms S(A, B) and S(A|B) are the joint von Neumann entropy and the quantum conditional entropy of the system, respectively. Based on these definitions the expression of the QD can be rewritten as,
D(A : B) = S(B) − S(A, B) + S(A|B).
(2)
The quantum conditional entropy S(A|B) is given as, S(A|B) = min {Πk }
HB
pk S(ρA|Πk ),
(3)
k=1
where HB is the Hilbert space dimension
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