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arXiv:quant-ph/9604023

Sending quantum entanglement through noisy channels

Benjamin W. Schumacher

Published Apr 22, 1996• 29 citations
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Abstract

This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems $R$ and $Q$, after which $Q$ is subjected to a dynamical evolution given by the superoperator $\superop^{Q}$. Two interesting quantities can be defined for this process: the entanglement fidelity $F_{e}$ and the entropy production $S_{e}$. It turns out that neither of these quantities depends in any way on the system $R$, but only on the initial state and dynamical evolution of $Q$. $F_{e}$ and $S_{e}$ are related to various other fidelities and entropies, and are connected by an inequality reminiscent of the Fano inequality of classical information theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes.

DOI
10.48550/arxiv.quant-ph/9604023

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This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems RRR and QQQ, after which QQQ is subjected to a dynamical evolution given by the superoperator \superopQ\superop^{Q}\superopQ. Two interesting quantities can be defined for this process: the entanglement fidelity FeF_{e}Fe​ and the entropy production SeS_{e}Se​. It turns out that neither of these quantities depends in any way on the system RRR, but only on the initial state and dynamical evolution of QQQ. FeF_{e}Fe​ and SeS_{e}Se​ are related to various other fidelities and entropies, and are connected by an inequality reminiscent of the Fano inequality of classical information theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes.
This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems R and Q, after which Q is subjected to a dynamical evolution given by the superoperator EQ{\mathit{E}}^{\mathit{Q}}EQ. Two interesting quantities can be defined for this process: the entanglement fidelity Fe{\mathit{F}}_{\mathit{e}}Fe​ and the entropy exchange Se{\mathit{S}}_{\mathit{e}}Se​. It turns out that neither of these quantities depends in any way on the system R, but only on the initial state and dynamical evolution of Q. Fe{\mathit{F}}_{\mathit{e}}Fe​ and Se{\mathit{S}}_{\mathit{e}}Se​ are related to various other fidelities and entropies and are connected by an inequality reminiscent of the Fano inequality of classical information theory. Some insight can be gained from these techniques into the security of quantum cryptographic protocols and the nature of quantum error-correcting codes. \textcopyright{} 1996 The American Physical Society.
The fidelity and the entanglement fidelity are two important quantities in describing the transmission of quantum information through (possibly noisy) quantum channels. In this paper these two quantities are calculated for different kinds of state which are transmitted in noisy channels. A general equation is obtained for these fidelities. Some examples are given to illustrate the general result and possible applications of these results to quantum cryptography and teleportation via noisy channels are discussed.
In analogy with its classical counterpart, a noisy quantum channel is characterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entropy. By using block coding based on sequences of nnn entangled symbols, the average loss (defined as the overall loss of the joint nnn-symbol channel divided by nnn, when $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\infty})$ can be made lower than the loss for a single use of the channel. In this context, we examine several upper bounds on the rate at which quantum information can be transmitted reliably via a noisy channel, that is, with an asymptotically vanishing average loss while the one-symbol loss of the channel is nonzero. These bounds on the channel capacity rely on the entropic Singleton bound on quantum error-correcting codes [Phys. Rev. A 56, 1721 (1997)]. Finally, we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.
Quantum information theory studies the fundamental limits that physical laws impose on information processing tasks such as data compression and data transmission on noisy channels. This thesis presents general techniques that allow one to solve many fundamental problems of quantum information theory in a unified framework. The central theorem of this thesis proves the existence of a protocol that transmits quantum data that is partially known to the receiver through a single use of an arbitrary noisy quantum channel. In addition to the intrinsic interest of this problem, this theorem has as immediate corollaries several central theorems of quantum information theory. The following chapters use this theorem to prove the existence of new protocols for two other types of quantum channels, namely quantum broadcast channels and quantum channels with side information at the transmitter. These protocols also involve sending quantum information partially known by the receiver with a single use of the channel, and have as corollaries entanglement-assisted and unassisted asymptotic coding theorems. The entanglement-assisted asymptotic versions can, in both cases, be considered as quantum versions of the best coding theorems known for the classical versions of these problems. The last chapter deals with a purely quantum phenomenon called locking. We demonstrate that it is possible to encode a classical message into a quantum state such that, by removing a subsystem of logarithmic size with respect to its total size, no measurement can have significant correlations with the message. The message is therefore "locked" by a logarithmic-size key. This thesis presents the first locking protocol for which the success criterion is that the trace distance between the joint distribution of the message and the measurement result and the product of their marginals be sufficiently small.
Using the information-theoretic inequality proposed by Braunstein and Caves, we study the evolution of quantum correlation in the presence of different noisy channels. First, we experimentally prepare the high-fidelity two-bit entangled state and simulated the bit-flip and phase-shift noise using an all-optical experimental setup. Then, we study the quantum correlation evolution of a two-bit entangled system using the information difference of entropy under different noisy channels and noisy environments. It is found that the quantum correlation degradation depends on the type of noise and noisy channels. Our work provides a new essential tool for quantum information processing and measurement, and offers new insight into the evolution of quantum correlation in open systems.
A natural question in characterizing the information theoretic power of quantum channels is to ask at what rate entanglement is needed in order to asymptotically simulate a quantum channel in the presence of free classical communication. We call this the entanglement cost of a channel, and prove a formula describing it for all channels. We discuss two applications. Firstly, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only greatly improves the range of parameters where security could be shown previously, but allows us to prove security for storage devices for which no non-trivial statements were known before. Secondly, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.