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arXiv:1406.1601 · Quantum Information and Computation

Entanglement and swap of quantum states in two qubits

Takaya Ikuto, Satoshi Ishizaka

Published Sep 1, 2015• 1 citation
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Abstract

Suppose that two distant parties Alice and Bob share an entangled state $\rho_{AB}$, and they want to exchange the subsystems of $\rho_{AB}$ by local operations and classical communication (LOCC). In general, this LOCC task (i.e. the LOCC transformation of $\rho_{AB} \to V\rho_{AB} V$ with $V$ being a swap operator) is impossible deterministically, but becomes possible probabilistically. In this paper, we study how the optimal probability is related to the amount of entanglement in the framework of positive partial transposed (PPT) operations, and numerically show a remarkable class of states whose probability is the smallest among every state in two quantum bits.

DOI
10.26421/qic15.11-12-3

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Suppose that two distant parties Alice and Bob share an entangled state ρAB\rho_{AB}ρAB​, and they want to exchange the subsystems of ρAB\rho_{AB}ρAB​ by local operations and classical communication (LOCC). In general, this LOCC task (i.e. the LOCC transformation of ρAB→VρABV\rho_{AB} \to V\rho_{AB} VρAB​→VρAB​V with VVV being a swap operator) is impossible deterministically, but becomes possible probabilistically. In this paper, we study how the optimal probability is related to the amount of entanglement in the framework of positive partial transposed (PPT) operations, and numerically show a remarkable class of states whose probability is the smallest among every state in two quantum bits.
Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a single copy, allowing finite protocols and the countable protocols specified here. Local flags are known to allow a separable part to be replaced by local preparation while an entangled block is retained. We prove that such a decomposition is necessary for two mutually convertible entangled states that are not locally unitarily equivalent. The two parts are distinguishable by local measurements. The entangled blocks have the same weight and are locally unitarily equivalent after normalization. Each entangled state has a unique smallest retained block. Two entangled states are reversibly convertible if and only if these unnormalized blocks are locally unitarily equivalent. For a state with negative partial transpose, the smallest block can be found using the negative eigenspace of its partial transpose and at most three matrix zero tests.
This thesis studies the manipulation of entanglement in three-qubit quantum systems. I consider the operational setting in which the qubits are distributed to three spatially separated parties. The parties act locally on their quantum systems and share classical communication with one another, a scenario commonly called local operations and classical communication (LOCC). In the LOCC setting, there are two different classes of entanglement in multipartite systems, called the GHZ and W classes, which are inequivalent in the sense that states from one class cannot be transformed into the other without the consumption of additional entanglement. In this thesis, I first show that the LOCC conversion of certain GHZ and W-class states becomes possible by using only one additional ebit (“entangled bit”) of shared entanglement. In many cases, this can be proven as the minimal amount of needed entanglement. I then consider the problem of one-way communication transformations from general three-qubit states into two-qubit maximally entangled states, known as EPR states. An optimal protocol in terms of success probability is provided for W-class states. The success probability of this protocol coincides with the optimal success probability if two of the parties are allowed to act jointly within the same laboratory. In other words, forcing the locality constraint on all three parties does not weaken their capabilities for obtaining bipartite entanglement when starting from a W-class state. I also present that this property holds for certain types of GHZ-class states as well.
We consider a quantum communication task between two users Alice and Bob, in which Alice and Bob exchange their respective quantum information by means of local operations and classical communication assisted by shared entanglement. Here, we assume that Alice and Bob may have quantum side information, not transferred, and classical communication is free. In this work, we derive general upper and lower bounds for the least amount of entanglement which is necessary to perfectly perform this task, called the state exchange with quantum side information. Moreover, we show that the optimal entanglement cost can be negative when Alice and Bob make use of their quantum side information. We finally provide conditions on the initial state for the state exchange with quantum side information which give the exact optimal entanglement cost.
We demonstrate the irreversibility of asymptotic entanglement manipulation under quantum operations that completely preserve the positivity of partial transpose (PPT), resolving a major open problem in quantum information theory. Our key tool is a new efficiently computable additive lower bound for the asymptotic relative entropy of entanglement with respect to PPT states, which can be used to evaluate the entanglement cost under local operations and classical communication (LOCC). We find that for any rank-two mixed state supporting on the 3⊗3 antisymmetric subspace, the amount of distillable entanglement by PPT operations is strictly smaller than one entanglement bit (ebit) while its entanglement cost under PPT operations is exactly one ebit. As a by-product, we find that for this class of states, both the Rains's bound and its regularization are strictly less than the asymptotic relative entropy of entanglement. So, in general, there is no unique entanglement measure for the manipulation of entanglement by PPT operations. We further show a computable sufficient condition for the irreversibility of entanglement distillation by LOCC (or PPT) operations.
For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the Greenberger-Horne-Zeilinger (GHZ) class and the W class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication with a nonzero probability. The optimal conversion probability, however, is only known for special cases. Here, lower and upper bounds are derived for the optimal probability of transformation from a GHZ state to other states of the GHZ class. A key idea in the derivation of the upper bounds is to consider the action of the local operations and classical communications (LOCC) protocol on a different input state, namely $1/\sqrt{2}[|000\ensuremath{\rangle}\ensuremath{-}|111\ensuremath{\rangle}]$, and to demand that the probability of an outcome remains bounded by 1. We also find an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, some of the results are generalized to the case in which each party holds a higher dimensional system. In particular, the GHZ state generalized to three qutrits; that is, $|{\mathrm{GHZ}}_{3}\ensuremath{\rangle}=1/\sqrt{3}[|000\ensuremath{\rangle}+|111\ensuremath{\rangle}+|222\ensuremath{\rangle}]$ shared among three parties can be transformed to any tripartite three-qubit pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties.