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arXiv:1004.0052 · Science

For Quantum Information, Two Wrongs Can Make a Right

Jonathan Oppenheim

Published Sep 26, 2008• 23 citations
arXivHTMLPDF

Abstract

Superactivation is the phenomenon where two quantum channels which individually have zero-capacity can have positive capacity when used together. The perspective given here provides an intuitive explanation of this discovery by Smith and Yard, and gives a protocol to activate any private channel.
DOI
10.1126/science.1164543

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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.
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.
The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity at the output. Currently, we have no theoretical background for describing all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. In this PhD Thesis I provide an algorithmic solution to the problem of superactivation and prove that superactivation effect is rooted in information geometric issues.
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.
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 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.