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arXiv:0811.4466 · Journal of Physics B Atomic Molecular and Optical Physics

Conservation rules for entanglement transfer between qubits

Stanley Chan, M. D. Reid, Z. Ficek

Published Oct 21, 2010• 12 citations
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Abstract

We consider an entangled but non-interacting qubit pair a 1 and b 1 that is independently coupled to a set of local qubit systems, a I and b J , of 0 bit value, respectively. We derive rules for the transfer of entanglement from the pair a 1 – b 1 to an arbitrary pair a I – b J , for the case of qubit-number conserving local interactions. It is shown that the transfer rule depends strongly on the initial entangled state. If the initial entanglement is in the form of the Bell state corresponding to anti-correlated qubits, the sum of the square of the nonlocal pairwise concurrences is conserved. If the initial state is the Bell state with correlated qubits, this sum can be reduced, even to zero in some cases, to reveal a complete and abrupt loss of all nonlocal pairwise entanglement. We also identify that for the nonlocal bipartitions A – b J involving all qubits at one location, with one qubit b J at the other location, the concurrences satisfy a simple addition rule for both cases of the Bell states that the sum of the square of the nonlocal concurrences is conserved.

DOI
10.1088/0953-4075/43/21/215505

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We consider an entangled but non-interacting qubit pair a 1 and b 1 that is independently coupled to a set of local qubit systems, a I and b J , of 0 bit value, respectively. We derive rules for the transfer of entanglement from the pair a 1 – b 1 to an arbitrary pair a I – b J , for the case of qubit-number conserving local interactions. It is shown that the transfer rule depends strongly on the initial entangled state. If the initial entanglement is in the form of the Bell state corresponding to anti-correlated qubits, the sum of the square of the nonlocal pairwise concurrences is conserved. If the initial state is the Bell state with correlated qubits, this sum can be reduced, even to zero in some cases, to reveal a complete and abrupt loss of all nonlocal pairwise entanglement. We also identify that for the nonlocal bipartitions A – b J involving all qubits at one location, with one qubit b J at the other location, the concurrences satisfy a simple addition rule for both cases of the Bell states that the sum of the square of the nonlocal concurrences is conserved.
Quantum entanglement is widely renounced as one of the most fundamental concepts of quantum mechanics. Such phenomenon exhibit non-local interaction properties which cannot be explained classically. In this thesis, we address a number of problems associated with creating, transferring and engineering of entanglement between two separate parties. The work is motivated by a desire to better understand the dynamics of entanglement between systems. In particular, the research is mainly focused on the study of the dynamics of the well known maximally entangled Bell state under different influences such as decoherence and inter-qubit coupling. We show the connection between coherence and entanglement using the system sub jected to decoherence. We also confirm the transfer of entanglement between completely isolated partite using the double Jaynes-Cummings model. Based on this result, we propose a new conservation criterion proven to be general for single excitation systems. Such conservation criterion are then compared and extended to a general N qubit systems. In addition, an attempt is made to evaluate entanglement conservation rules for the EPR- like multipartite entanglement. We also describe a new technique for solving entanglement in the top-down way ignoring physical setup.
In this paper, we study the entanglement in a five qubit model. For this purpose, we consider two non-interacting systems A and B which are correlated together initially. A is a t wo atoms Tavis-Cu mmings system and B is a Jaynes-Cummings System. By choosing a convenient initial condition which shows the initially correlation, our total system (A plus B) become a system which consist of five qubits. There are ten pairwise concurrences which we must calculate. We show that initially non-entangled qubits become entangled by passing time, the process which may be interpreted as entanglement transfer. We also show that our model in special cases become two-qubit and three-qubit models and analyze entanglement in these special cases.
We consider the entanglement of two remote qubits induced by local interaction with the two-mode nonclassical state fields. We investigate the transfer of entanglement from different two-mode nonclassical state fields to a pair of separable and mixed qubits. We find that no matter what state the qubit subsystem is initially prepared in, at the specific time $gt\ensuremath{\simeq}11.07$ the entanglement of the Bell state fields can be almost completely transferred to the qubit subsystem, i.e., the Bell state fields can drive an initially separable and very mixed state of qubits into an almost pure highly entangled state suitable for quantum information processing. For the entangled coherent state fields $\ensuremath{\mid}u⟩={N}_{1}(\ensuremath{\mid}\ensuremath{\alpha},\ensuremath{-}\ensuremath{\alpha}⟩+\ensuremath{\mid}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\alpha}⟩)$ and $\ensuremath{\mid}{u}^{\ensuremath{'}}⟩={N}_{2}(\ensuremath{\mid}\ensuremath{\alpha},\ensuremath{-}\ensuremath{\alpha}⟩\ensuremath{-}\ensuremath{\mid}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\alpha}⟩)$ we find that when $\ensuremath{\mid}\ensuremath{\alpha}\ensuremath{\mid}$ is small, the dynamic behaviors of the entanglement between two qubits are quite different. For $\ensuremath{\mid}u⟩$ there is almost no entanglement between two qubits while for $\ensuremath{\mid}{u}^{\ensuremath{'}}⟩$ the dynamic behavior of the entanglement between two qubits is very similar to that of the Bell state field. For large $\ensuremath{\mid}\ensuremath{\alpha}\ensuremath{\mid}$ we find that the dynamic behaviors of entanglement between two qubits are almost the same for these two fields, and when $gt\ensuremath{\sim}\ensuremath{\mid}\ensuremath{\alpha}\ensuremath{\mid}\ensuremath{\pi}$ the entanglement between two qubits reaches its maximum and is almost independent of initial mixedness of the qubit subsystem. For the two-mode squeezed vacuum state field we find that when $r\ensuremath{\sim}0.86$ and $gt\ensuremath{\simeq}11.07$ the entanglement transfer reaches its maximum which also almost does not depend on the initial mixedness of the qubit subsystem.
The problem of a controlled transfer of an entanglement initially encoded into two two-level atoms that are successively sent through two single-mode cavities is investigated. The atoms and the cavity modes form a four-qubit system and we demonstrate the conditions under which the initial entanglement encoded into the atoms can be completely transferred to other pairs of qubits. We find that in the case of non-zero detuning between the atomic transition frequencies and the cavity mode frequencies, no complete transfer of the initial entanglement is possible to any of the other pairs of qubits. In the case of exact resonance and equal coupling strengths of the atoms to the cavity modes, an initial maximally entangled state of the atoms can be completely transferred to the cavity modes. Complete transfer of the entanglement is restricted to the cavity modes, with transfer to the other pairs being limited to 50%. We find that complete transfer of an initial entanglement to other pairs of qubits may take place if the initial state is not the maximally entangled state and the atoms couple to the cavity modes with unequal strengths. Depending on the ratio between the coupling strengths, optimal entanglement can be created between the atoms and one of the cavity modes.
The nonlocal correlations between quantum states in an entangled system are essential to many quantum communications applications. A basic quantum operation, which permits the distribution of entanglement between two initially uncorrelated systems, is entanglement swapping. Here we present a rigorous formulation of entanglement swapping of any two partially mixed two-qubit states without limiting ourselves to any specific type of state or noise. Further, for two important classes of the input states, Bell diagonal and pure states, we describe how the concurrence of the final state is related to the concurrence of the initial states. First, we consider Bell diagonal states, and find bounds on the concurrence of the final state in terms of the concurrences of the initial states. These bounds are important for communications applications because polarization mode dispersion in fibers produces Bell diagonal states up to a local unitary rotation. Second, we show that swapping pure states occasionally results in a state of higher concurrence than either of the initial states. In addition, we find that two pure states are most likely to be capable of swapping to a state of increased concurrence when the two initial states have similar concurrences. Our analysis offers a completely general framework for investigating the behavior of any pair of two-qubit states when used for entanglement swapping.