Entanglement purification protocol for a mixture of a pure entangled state and a pure product state

Abstract

We present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state, which are orthogonal to each other. The protocol is a combination of bisection method and one-way hashing protocol. We give recursive formula for the rate of the protocol for different states, i.e., the number of maximally entangled two-qubit pairs obtained with the protocol per a single copy of the initial state. We also calculate numerically the rate for some states.Entanglement is a resource in quantum information. Maximally entangled states are a basic ingredient of fundamental quantum information protocols such as, e.g., quantum teleportation ͓1͔, dense coding ͓2͔, or Ekert's quantum cryptographic protocol ͓3͔. In these protocols the maximally entangled pair is shared by two parties and used to perform a certain task. However, in the real world the parties share noisy entangled pairs. Bennett et al. have shown that many pairs in mixed entangled states can be distilled to a smaller number of pairs in nearly maximally entangled states ͓4,5͔. In particular, they presented purification protocols that can be realized by means of local operations and classical communication. Let us suppose that the parties share n pairs of qubits, each of which is in the state . If in using a particular protocol the parties can obtain m pairs of qubits, each of which is in the maximal entangled state, from n pairs of qubits, each of which is in state , then the protocol has the rate m n for the state . Different protocols have different rates and moreover one needs different protocols for different states. The maximal rate for state , i.e., the rate of the optimal protocol for state , is called distillable entanglement. Distillable entanglement of mixed states is usually difficult to calculate, and it is only known for bound entangled states, maximally correlated states, and some other specific mixed states ͓6-11͔. For bound entangled states it is equal to zero ͓6͔ while for maximally correlated states it is equal to the relative entropy of entanglement ͓11-14͔ and can be distilled by the one-way hashing protocol ͓15,16͔. However, for many states only upper and lower bounds on distillable entanglement are known. Lower bounds are usually given by the rates of particular protocols. Upper bounds are given by entanglement measures known to be greater or equal to distillable entanglement, e.g., the relative entropy of entanglement.In this Brief Report we present an entanglement purification protocol for a mixture of a pure entangled state and a pure product state that are orthogonal to each other. The very first protocol for these states was presented in ͓5͔. Our protocol is a combination of Procrustean method of entanglement concentration ͓17͔, bisection method, and one-way hashing protocol ͓15,16͔.We assume that Alice and Bob share many copies of the stateis the maximal entangled state. State is a mixture of the maximal entangled state and a product state. The product state is orthogonal to the maximal entangled state. L...

Effect of weak measurement on entanglement distribution over noisy channels

“…The first EDP will be called a two-copy EDP, because each round of distillation only involves two copies of input states 6 . The second EDP will be referred to as a bisection EDP, because each round of distillation except the first round divides the pairs of qubits into two blocks of equal length 54 . The bisection EDP is up to now the most efficient theoretical scheme for the amplitude-damped state or 54 , although it is much more difficult than the two-copy EDP in the experiment.…”

“…In the remaining cases, the state is inseparable in terms of the partition “Alice:Bob”, that is reusable in the second round of distillation. Using the bisection method (Alice and Bob divide the pairs of qubits into two blocks of equal length) in the following rounds of distillation, the total yield of such an EDP starting from the state ρ w is given by 54 …”

Multi-photon Entanglement Concentration Protocol for Partially Entangled W States with Projection Measurement

“…If the initially maximally entangled state becomes a mixed entangled one, the two remote users can exploit entanglement purification [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61] to improve its fidelity. If the photon systems are still in a pure state with a less entanglement, i.e., a partially entangled pure state, the user can exploit entanglement concentration [62][63][64][65][66][67][68][69][70][71][72][73][74] to reconstruct a subset of systems in a maximally entangled state.…”

Entanglement concentration of partially entangled three-photon W states with weak cross-Kerr nonlinearity

“…In a sense, both entanglement concentration [7] and entanglement purification [8] aim at obtaining the desired nonlocal entanglement by local operations and classical communication. Generally, entanglement concentration [9][10][11][12][13][14][15][16][17][18] is used to increase the entanglement of pure entangled states, and entanglement purification [19][20][21][22][23][24][25][26][27] deals with mixed entangled states.…”