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arXiv:2203.17116 · Physical Review A

Optimal supplier of single-error-type entanglement via coherent-state transmission

Koji Azuma, Nobuyuki Imoto, Masato Koashi

Published Jun 21, 2022• 2 citations
arXivHTMLPDF

Abstract

Compared with entanglement with multiple types of noise, entanglement including only one type of error is a favorable fundamental resource not only for quantum communication but also for distributed quantum computation. We consider protocols that present single-error-type entanglement for distant qubits via coherent-state transmission over a lossy channel. Such a protocol is regarded as a subroutine to serve entanglement for larger protocol to yield a final output, such as ebits or pbits. In this paper, we provide a subroutine protocol which achieves the global optimal for typical jointly convex yield functions monotonically nondecreasing with respect to the singlet fraction, such as an arbitrary convex function of a singlet fraction, two-way distillable entanglement, and two-way distillable key. Entanglement generation based on remote nondestructive parity measurement protocol [K. Azuma et al., Phys. Rev. A 85, 062309 (2012)] is identified as such an optimal subroutine.

DOI
10.1103/physreva.105.062432

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Compared with entanglement with multiple types of noise, entanglement including only one type of error is a favorable fundamental resource not only for quantum communication but also for distributed quantum computation. We consider protocols that present single-error-type entanglement for distant qubits via coherent-state transmission over a lossy channel. Such a protocol is regarded as a subroutine to serve entanglement for larger protocol to yield a final output, such as ebits or pbits. In this paper, we provide a subroutine protocol which achieves the global optimal for typical jointly convex yield functions monotonically nondecreasing with respect to the singlet fraction, such as an arbitrary convex function of a singlet fraction, two-way distillable entanglement, and two-way distillable key. Entanglement generation based on remote nondestructive parity measurement protocol [K. Azuma et al., Phys. Rev. A 85, 062309 (2012)] is identified as such an optimal subroutine.
Entanglement is a fundamental resource not only for quantum communication but also for distributed quantum computation. Especially, entanglement including only one type of error is favorable, compared with one including multiple types of noise. In this talk, we consider protocol that presents single-error-type entanglement for distant qubits via coherent-state transmission over lossy channels. This protocol is considered to be a subroutine to serve entanglement for larger protocol to yield a final output, such as ebits or pbits. A protocol based on remote non-destructive parity measurement (RNPM) [K. Azuma, H. Takeda, M. Koashi, and N. Imoto, Phys. Rev. A 85, 062309 (2012)] is identified as a subroutine which achieves the global optimal for typical yield functions monotonically non-decreasing with respect to the singlet fraction, such as an arbitrary convex function of a singlet fraction and two-way distillable entanglement/key.
The first stage of the hybrid quantum repeaters is entanglement generation based on transmission of pulses in coherent states over a lossy channel. Protocols to make entanglement with only one type of error are favorable for rendering subsequent entanglement distillation efficient. Here we provide the tight upper bound on performances of these protocols that is determined only by the channel loss. In addition, we show that this bound is achievable by utilizing a proposed protocol [K. Azuma, N. Sota, R. Namiki, \ifmmode \mbox{\c{S}}\else \c{S}\fi{}. K. \"Ozdemir, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. A 80, 060303(R) (2009)] composed of a simple combination of linear optical elements and photon-number-resolving detectors.
Recent work has shown that use of quantum feedback can significantly enhance both the speed and success rate of measurement-based remote entanglement generation, but it is generally unknown what feedback protocols are optimal for these tasks. Here we consider two common measurements that are capable of projecting into pairwise entangled states, namely half- and full-parity measurements of two qubits, and determine in each case a globally optimal protocol for generation of entanglement. For the half-parity measurement, we rederive a previously described protocol using more general methods and prove that it is globally optimal for several figures of merit, including maximal concurrence or fidelity, and minimal time to reach a specified concurrence or fidelity. For the full-parity measurement, we derive a protocol for rapid entanglement generation related to that of (C. Hill, J. Ralph, Phys. Rev. A 77, 014305 (2008)), and then map the dynamics of the concurrence of the state to the Bloch vector length of an effective qubit. This mapping allows us to prove several optimality results for feedback protocols with full-parity measurements. We further show that our full-parity protocol transfers entanglement optimally from one qubit to the other amongst all measurement-based schemes. The methods developed here will be useful for deriving feedback protocols and determining their optimality properties in many other quantum systems subject to measurement and unitary operations.
Abstract Recent work has shown that the use of quantum feedback can significantly enhance both the speed and success rate of measurement-based remote entanglement generation, but it is generally unknown what feedback protocols are optimal for these tasks. Here we consider two common measurements that are capable of projecting into pairwise entangled states, namely half- and full-parity measurements of two qubits, and determine in each case a globally optimal protocol for generation of entanglement. For the half-parity measurement, we rederive a previously described protocol using more general methods and prove that it is globally optimal for several figures of merit, including maximal concurrence or fidelity and minimal time to reach a specified concurrence or fidelity. For the full-parity measurement, we derive a protocol for rapid entanglement generation related to that of (Hill, Ralph, Phys. Rev. A 77 , 014305), and then map the dynamics of the concurrence of the state to the Bloch vector length of an effective qubit. This mapping allows us to prove several optimality results for feedback protocols with full-parity measurements. We further show that our full-parity protocol transfers entanglement optimally from one qubit to the other amongst all measurement-based schemes. The methods developed here will be useful for deriving feedback protocols and determining their optimality properties in many other quantum systems subject to measurement and unitary operations.
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 propose a scheme to measure the parity of two distant qubits, while ensuring that losses on the quantum channel between them does not destroy coherences within the parity subspaces. This capability enables deterministic preparation of highly entangled qubit states whose fidelity is not limited by the transmission loss. The key observation is that, for a probe electromagnetic field in a particular quantum state; namely, a superposition of two coherent states of opposite phases, the transmission loss stochastically applies a near-unitary backaction on the probe state. This leads to a parity measurement protocol where the main effect of the transmission losses is a decrease in the measurement strength. By repeating the nondestructive (weak) parity measurement, one achieves a high-fidelity entanglement in spite of a significant transmission loss.