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arXiv:1209.3494 · Physical Review A

Optimal fidelity for a quantum channel may be attained by nonmaximally entangled states

Somshubhro Bandyopadhyay, Anindita Ghosh

Published Aug 24, 2012• 20 citations
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Abstract

We consider the problem of establishing a two-qubit entangled state of optimal fidelity across a noisy quantum channel when only a single use of the channel and local postprocessing by trace-preserving operations are allowed. We show that the optimal fidelity is obtained only when part of an appropriate nonmaximally entangled state is transmitted through the channel. The entanglement of this state can be vanishingly small when the channel becomes very noisy. Moreover, in the optimal case no further local processing is required to enhance the fidelity. We further show that local postprocessing can enhance fidelity if and only if the amount of noise is larger than a critical value and entanglement of the transmitted state is bounded from below. A notable consequence of these results is that the ordering of states under an entanglement monotone can be reversed even when the states undergo the same local interaction via a trace-preserving, completely positive map.

DOI
10.1103/physreva.86.020304

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We consider the problem of establishing a two-qubit entangled state of optimal fidelity across a noisy quantum channel when only a single use of the channel and local postprocessing by trace-preserving operations are allowed. We show that the optimal fidelity is obtained only when part of an appropriate nonmaximally entangled state is transmitted through the channel. The entanglement of this state can be vanishingly small when the channel becomes very noisy. Moreover, in the optimal case no further local processing is required to enhance the fidelity. We further show that local postprocessing can enhance fidelity if and only if the amount of noise is larger than a critical value and entanglement of the transmitted state is bounded from below. A notable consequence of these results is that the ordering of states under an entanglement monotone can be reversed even when the states undergo the same local interaction via a trace-preserving, completely positive map.
Maximally entangled states---a resource for quantum information processing---can only be shared through noiseless quantum channels, whereas in practice channels are noisy. Here we ask: Given a noisy quantum channel, what is the maximum attainable purity (measured by singlet fraction) of shared entanglement for single channel use and local trace preserving operations? We find an exact formula of the maximum singlet fraction attainable for a qubit channel and give an explicit protocol to achieve the optimal value. The protocol distinguishes between unital and nonunital channels and requires no local postprocessing. In particular, the optimal singlet fraction is achieved by transmitting part of an appropriate pure entangled state, which is maximally entangled if and only if the channel is unital. A linear function of the optimal singlet fraction is also shown to be an upper bound on the distillable entanglement of the mixed state dual to the channel.
We consider the problem of establishing entangled states of optimal singlet fraction and negativity between two remote parties for every use of a noisy quantum channel and trace-preserving local operations and classical communication (LOCC) under the assumption that the parties do not share prior correlations. We show that for a family of quantum channels in every finite dimension $d\ensuremath{\ge}3$, one-shot optimal singlet fraction and entanglement negativity are attained only with appropriate nonmaximally entangled states. A consequence of our results is that the ordering of entangled states in all finite dimensions may not be preserved under trace-preserving LOCC.
We consider a single copy of a mixed state of two qubits and derive the optimal trace-preserving local operations assisted by classical communication such as to maximize the fidelity of teleportation that can be achieved with this state. These optimal local operations turn out to be implementable by one-way communication and always yield a teleportation fidelity larger than 2/3 if the original state is entangled. This maximal achievable fidelity is an entanglement measure and turns out quantifying the minimal amount of mixing required to destroy the entanglement in a quantum state.
We consider a single copy of a mixed state of two qubits and prove tight lower and upper bounds for its fidelity in terms of its concurrence or negativity. Next we derive the optimal filtering operations and the optimal trace-preserving local operations assisted by classical communication (LOCC) such as to maximize the fidelity of a given copy of a mixed state of two qubits. The optimal LOCC operations turn out to be very simple and implementable by 1-LOCC protocols, and always yield a fidelity larger then 1/2 if the original state was entangled.
We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.
Entanglement subject to noise cannot be shielded against decaying. But, in case of many noisy channels, the degradation can be partially prevented by using local unitary operations. We consider the effect of local noise on shared quantum states and evaluate the amount of entanglement that can be preserved from deterioration. The amount of saved entanglement not only depends on the strength of the channel but also on the type of the channel, and in particular, it always vanishes for the depolarizing channel. The main motive of this work is to analyze the reason behind this dependency of saved entanglement by inspecting properties of the corresponding channels. In this context, we quantify and explore the biasnesses of channels towards the different states on which they act. We postulate that all biasness measures must vanish for depolarizing and unitary channels, and subsequently introduce a few measures of biasness. We also consider the entanglement capacities of channels. We observe that the joint behavior of the biasness quantifiers and the entanglement capacity explains the nature of saved entanglement. Furthermore, we find a pair of upper bounds on saved entanglement which are noticed to imitate the graphical nature of the latter.