Coherent Communication of Classical Messages

Abstract

We define coherent communication in terms of a simple primitive, show it is equivalent to the ability to send a classical message with a unitary or isometric operation, and use it to relate other resources in quantum information theory. Using coherent communication, we are able to generalize super-dense coding to prepare arbitrary quantum states instead of only classical messages. We also derive single-letter formulae for the classical and quantum capacities of a bipartite unitary gate assisted by an arbitrary fixed amount of entanglement per use. Beyond qubits and cbitsThe basic units of communication in quantum information theory are usually taken to be qubits and cbits, meaning one use of a noiseless quantum or classical channel respectively. Qubits and cbits can be converted to each other and to and from other quantum communication resources, but often only irreversibly. For example, one qubit can yield at most one cbit, but to obtain a qubit from cbits requires the additional resource of entanglement. By introducing a new resource, intermediate in power between a qubit and a cbit, we will show that many resource transformations can be modified to be reversible and thus more efficient.If {|x } x=0,1 is a basis for C 2 , then a qubit channel can be described as the isometry |x A → |x B and a cbit can be written as |x A → |x B |x E . Here A is the sender Alice, B is the receiver Bob and E denotes an inaccessible environment, sometimes personified as a malicious Eve. Tracing out Eve yields the traditional definition of a cbit channel with the basis {|x } considered the computational basis.Define a coherent bit (or "cobit") of communication as the ability to perform the map |x A → |x A |x B . Since Alice is free to copy or destroy her channel input, 1 qubit 1 cobit 1 cbit, where X Y means that the resource X can be used to simulate the resource Y . We will also write X Y (c) when X + Z Y + Z for some resource Z used as a catalyst and X Y (a) when the conversion is asymptotic, by which we mean that X ⊗n Y ⊗≈n for n sufficiently large.In Prop. 1, we will see that coherent bits come from any method for sending bits using a coherent procedure (a unitary or isometry on the joint Alice-Bob Hilbert space); hence their name. From their definitions, we see that cobits can be thought of as cbits where Alice controls the environment, making them like classical channels with quantum feedback. Further connections between cobits and cbits will be seen later in this Letter, where we show that irreversible resource transformations are often equivalent to performing "1 cobit 1 cbit," and in [1] where several communication protocols are "made coherent" with the effect of replacing cbits with cobits.In the first half of this paper we will describe how to obtain coherent bits and then how to use them, allowing us to exactly describe the power of cobits in terms of conventional resources. The purpose of this paper is thus not to define a new incomparable quantum resource, but rather to introduce a technique for relating and composin...

Triangle of Dualities between Quantum Communication Protocols

“…The state ψ RAB is defined by (6), noting that entropic coefficients are independent of the choice of U : A ′ → AB for which N = Tr A • U. The first observation is that there is a protocol implementing (7) that merely requires the relative resource N A ′ →B : ρ A ′ instead of the full…”

“…It is shown in [9] that the protocol can be made "coherent" [2,6], yielding the fully quantum reverse Shannon…”

“…However, a dynamic version of this decomposition does hold. The construction of coherent versions of teleportation and super-dense coding leads to the "coherent communication" resource equality [6] 2…”

Coherent communication with continuous quantum variables

“…Insights into quantum protocols occur by replacing classical bits with cobits-replacing a measurement and feedforward classical communication with a coherent channel and replacing a conditional unitary with a controlled unitary. Coherent teleportation and superdense coding for DVs are dual under resource reversal [1,4]. Two protocols are dual under resource reversal if one protocol generates the same resources that the other protocol consumes and vice versa.…”

“…Finally, we show that the quality of squeezing diminishes through successive compositions of coherent teleportation and superdense coding. The coherent bit (cobit) channel is a useful resource for quantum communication with discrete variables (DV) [1]. The cobit channel ∆ σZ copies σ Z eigenstates coherently from Alice to Bob: |i A → |i A |i B .…”