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arXiv:1305.1326 · Physical Review A

Parrondo's paradox and superactivation of classical and quantum capacity of communication channels with memory

Sergii Strelchuk

Published Sep 11, 2013• 3 citations
arXivHTMLPDF

Abstract

There exist memoryless zero-capacity quantum channels that when used jointly result in a channel with positive capacity. This phenomenon is called superactivation. Making use of Parrondo's paradox, we exhibit examples of the superactivation-like effect for the capacity of classical communication channels as well as quantum and private capacity of quantum channels with memory. There are several ingredients necessary for superactivation of quantum capacity to occur in the memoryless case. The first one is the requirement for the quantum channels which are suitable for superactivation to come from two distinct families: binding entanglement channels and erasure channels. The second one is the ability to utilize inputs which are entangled across the uses of the channels. Our construction uses a single family of erasure channels with classical memory to achieve the same superactivation-like effect for quantum capacity without any of the ingredients above.

DOI
10.1103/physreva.88.032311

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There exist memoryless zero-capacity quantum channels that when used jointly result in a channel with positive capacity. This phenomenon is called superactivation. Making use of Parrondo's paradox, we exhibit examples of the superactivation-like effect for the capacity of classical communication channels as well as quantum and private capacity of quantum channels with memory. There are several ingredients necessary for superactivation of quantum capacity to occur in the memoryless case. The first one is the requirement for the quantum channels which are suitable for superactivation to come from two distinct families: binding entanglement channels and erasure channels. The second one is the ability to utilize inputs which are entangled across the uses of the channels. Our construction uses a single family of erasure channels with classical memory to achieve the same superactivation-like effect for quantum capacity without any of the ingredients above.
Superadditivity of quantum capacity of communication channels is one of the most interesting findings of the field. Yard and Smith, finding a relation between the private capacity and the assisted quantum capacity, showed a striking example of superadditivity, i.e. two channels of zero quantum capacity could achieve a positive quantum capacity when used together [1]. The four dimensional channels they used are a 50% erasure channel (therefore zero quantum capacity, due to no-cloning theorem) and a Horodecki channel (again zero quantum capacity due to incapability of sharing free entanglement). In this work we present the more general cases of superadditivity. Directly calculating the lower bounds of joint quantum capacities without using the relation between private capacity and assisted quantum capacity, we examine scenarios considering erasure channels of arbitrary probabilities and different Horodecki channels, and discuss the roles of degradability and anti-degradability as well as the role of the private capacity in superadditivity. We also derive an upper bound for the joint quantum capacity for the superactivation case.
Superactivation of quantum capacity is the phenomenon whereby two quantum channels, each with zero quantum capacity, can exhibit a strictly positive capacity when used in tandem. In this work, we explore superactivation in the previously unexplored non-asymptotic regime of finitely many channel uses. We give a definition of finite-blocklength superactivation and propose numerical methods that can certify it. Then, focusing on the 50% erasure and positive-partial-transpose channels considered in the original work on superactivation, we show that as few as 17 uses of the joint channel already enable qubit transmission with a fidelity unattainable by any number of uses of either channel alone, demonstrating a strong finite-blocklength form of superactivation and opening the door to experimental demonstration.
In this work a new phenomenon called polaractivation is introduced. Polaractivation is based on quantum polar encoding and the result is similar to the superactivation effect — positive capacity can be achieved with zero-capacity quantum channels. However, polaractivation has many advantages over the superactivation: it is limited neither by any preliminary conditions on the quantum channel nor on the maps of other channels involved in the joint channel structure. We prove that the polaractivation works for arbitrary zero-private capacity quantum channels and we demonstrate, that the symmetric private classical capacity of arbitrary zero-private capacity quantum channels is polaractive.
Superactivation is the property that two channels with zero quantum capacity can be used together to yield a positive capacity. Here we demonstrate that this effect exists for a wide class of inequivalent channels, none of which can simulate each other. We also consider the case where one of two zero-capacity channels is applied, but the sender is ignorant of which one is applied. We find examples where the greater the entropy of mixing of the channels, the greater the lower bound for the capacity. Finally, we show that the effect of superactivation is rather generic by providing an example of superactivation using the depolarizing channel.
The zero-error classical capacity of a quantum channel is the asymptotic rate at which it can be used to send classical bits perfectly, so that they can be decoded with zero probability of error. The study of zero-error capacities dates right back to Shannon and the early days of information theory. We show that there exist pairs of quantum channels, neither of which individually have any zero-error capacity whatsoever (even if arbitrarily many uses of the channels are available), but such that access to even a single copy of both channels allows classical information to be sent perfectly reliably. In other words, we prove that the zero-error classical capacity can be superactivated. This result is the first example of superactivation of a classical capacity of a quantum channel. We further strengthen this result to show that there exist pairs of channels, neither of which have any zero-error classical capacity (as before), yet for which access to one copy of the joint channel even allows far more delicate quantum information to be transmitted perfectly. This subsumes the first result, and also implies that the quantum zero-error capacity can be superactivated. But it is strictly stronger than either of these. Indeed, this is the strongest conceivable form of superactivation, and nothing similar is possible for standard Shannon capacities of quantum channels or for zero-error capacities of classical channels.
One of the most surprising recent results in quantum Shannon theory is the superactivation of the quantum capacity of a quantum channel. This phenomenon has its roots in the extreme violation of additivity of the channel capacity and enables reliable transmission of quantum information over zero-capacity quantum channels. In this work we demonstrate a similar effect for the classical capacity of a quantum channel which previously was thought to be impossible. We show that a nonzero classical capacity can be achieved for all zero-capacity quantum channels and it only requires the assistance of an elementary photon–atom interaction process – stimulated emission.