The bilateral controlled NOT (CNOT) operation plays a key role in standard entanglement purification process, but the CNOT operation may not be the optimal joint operation in the sense that the output entanglement is maximized. In this paper, the CNOT operations in both the Schmidt-projection based entanglement concentration and the entanglement purification schemes are replaced with a general joint unitary operation, and the optimal matching relations between the entangling power of the joint unitary operation and the non-maximal entangled channel are found for optimizing the entanglement in- crement or the output entanglement. The result is somewhat counter-intuitive for entanglement concentration. The output entanglement is maximized when the entangling power of the joint unitary operation and the quantum channel satisfy certain relation. There exist a variety of joint operations with non-maximal entangling power that can induce a maximal output entanglement, which will greatly broaden the set of the potential joint operations in entanglement concentration. In addition, the entanglement increment in purification process is maximized only by the joint unitary operations (including CNOT) with maximal entangling power.
Abstract Entanglement between distant quantum systems is a critical resource for implementing quantum communication. This property is affected by external agents and can be restored by employing efficient entanglement purification protocols. In this work, we propose an entanglement purification protocol based on two entangling two-qubit operations that replace the usual controlled-NOT (CNOT) gate. These operations arise from a generalised quantum measurement and can be understood as measurement operators in a positive operator-valued measure. Furthermore, two variants of the core protocol are introduced and their performances are studied in terms of the overall success probability of reaching a Bell state and the number of purifiable states. Based on rank-two states, we can obtain analytical expressions for the success probability that we extend and refine using numerical calculations to the case of maximally entangled states. We also consider more general rank-three states to show that our procedure is in general more convenient compared to purification protocols based on Bell diagonal states. Finally, we test the protocols using initial random states. In all cases, we find a better performance and a larger amount of purifiable states using our schemes compared to the CNOT-based protocol.
How to concentrate non-maximally entangled states for quantum communication is a fundamental problem in quantum information. In this paper, we will apply generalized measurements to entanglement concentration of known non-maximally entangled pure states in arbitrary dimensional system. How to design the generalized measurements for the unambiguous discrimination of linearly independent non-orthogonal states is crucial for the concentration of the known non-maximally entangled states. The result shows that, any known non-maximally entangled pure state (for arbitrary dimensional system) can be transformed to the maximally entangled state only by introducing a qubit as ancilla and a joint unitary transformation operation on one of the entangled particles and the ancilla. In addition, because the less entangled state of each fail round will be re-concentrated too, the entanglement waste during the concentration process will be greatly reduced.
We propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a purification protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a puritlcation protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for
p>1, this formulation can be expressed in terms of a constrained maximal output Schatten
p-norm. More precisely, for a completely positive map
Ω:L(B′)→L(A), we consider the quantity
υp(Ω) defined by optimizing
∥(Ω⊗idE)(σB′E)∥p over all bipartite states
σB′E whose
B′-marginal is maximally mixed. We focus on the case
p=2. First, we compute
υ2 for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map
N:L(B′)→L(A) satisfies
N†∘N=aidA+bTr[⋅]Id for some constants
a,b≥0, where
N† denotes the Hilbert-Schmidt adjoint of
N, the quantity
υ2(N) is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map
Ω, multiplicativity of
υp(Ω) implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.
Entanglement purification protocols play an important role in the distribution of entangled systems, which is necessary for various quantum information processing applications. We consider the effects of photodetector efficiency and bandwidth, channel loss and mode mismatch on the operation of an optical entanglement purification protocol. We derive necessary detector and mode-matching requirements to facilitate practical operation of such a scheme, without having to resort to destructive coincidence-type demonstrations.