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arXiv:quant-ph/9907047 · Physical Review A

Distributed entanglement

Valerie Coffman, Joydip Kundu, William K. Wootters

Published Apr 10, 2000• 2,619 citations
arXivHTMLPDF

Abstract

Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the concurrence, which is related to the entanglement of formation. Specifically, we show that the squared concurrence between A and B, plus the squared concurrence between A and C, cannot be greater than the squared concurrence between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the concurrences satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of an essential three-way entanglement of the system, which is invariant under permutations of the qubits.

DOI
10.1103/physreva.61.052306

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Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the concurrence, which is related to the entanglement of formation. Specifically, we show that the squared concurrence between A and B, plus the squared concurrence between A and C, cannot be greater than the squared concurrence between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the concurrences satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of an essential three-way entanglement of the system, which is invariant under permutations of the qubits.
Quantum information and computation is one of the new fields of physics and information science. Entanglement is one of the most significant resources in quantum information processing. The most basic quantitative measure of entanglement, particularly for mixed states of a bipartite system is the entanglement of formation. This entanglement of formation is used in the concurrence, another measurement of entanglement. Here I have briefly reviewed the entanglement of formation and the concurrence of bipartite systems. I have discussed that for a three qubit pure state, if A, B and C are the three qubits, then sum of the squared concurrence between A and B and the squared concurrence between A and C never exceeds the squared concurrence between A and BC pair. The residual entanglement, obtained from the aforementioned inequality, is also discussed here. Keywords- Quantum information, entanglement, qubit.
Consider a system consisting of n d-dimensional quantum particles (qudits), and suppose that we want to optimize the entanglement between each pair. One can ask the following basic question regarding the sharing of entanglement: what is the largest possible value Emax(n,d){E}_{\mathrm{max}}(n,d)Emax​(n,d) of the minimum entanglement between any two particles in the system? (Here we take the entanglement of formation as our measure of entanglement.) For n=3n=3n=3 and d=2,d=2,d=2, that is, for a system of three qubits, the answer is known: Emax(3,2)=0.550.{E}_{\mathrm{max}}(3,2)=0.550.Emax​(3,2)=0.550. In this paper we consider first a system of d qudits and show that Emax(d,d)> 1.{E}_{\mathrm{max}}(d,d)>~1.Emax​(d,d)> 1. We then consider a system of three particles, with three different values of d. Our results for the three-particle case suggest that as the dimension d increases, the particles can share a greater fraction of their entanglement capacity.
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu, and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation cannot be freely shared and therefore should be said to be monogamous. Furthermore, the square of the entanglement of formation does obey the same relation of the squared concurrence, a fact recently noted for three particles and extended here for N particles. Therefore the entanglement of formation is as monogamous as the concurrence. We also numerically study how the entanglement is distributed in pure states of three qubits and the relation between the sum of the bipartite entanglement and the classical correlation.
We investigate the monogamy nature of entanglement in a three-qubit system. A monogamy inequality is presented to describe the exclusive relation between the $A\text{\ensuremath{-}}B$ two-qubit concurrence CAB{C}_{AB}CAB​ and the $AB\text{\ensuremath{-}}C$ three-qubit concurrence C(AB)C{C}_{(AB)C}C(AB)C​, which represents the entanglement between qubits A\mathit{A}A and B\mathit{B}B as a whole and the third qubit C\mathit{C}C. It is found that the entanglement between any two qubits in a three-qubit system is limited by the entanglement between these two qubits and another qubit. As a consequence, we present the upper bounds for the concurrence CAB{C}_{AB}CAB​, when the concurrence between qubits A\mathit{A}A and C\mathit{C}C (CAC)({C}_{AC})(CAC​) and the concurrence between qubits B\mathit{B}B and C\mathit{C}C (CBC)({C}_{BC})(CBC​) are both given or one of the two is provided.
In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger than √[(1 − C ) 2 + C 2 ] − (1 − C ), where C is the concurrence of the state. Furthermore, we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.