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arXiv:2012.01672 · ACM Transactions on Quantum Computing

Rigidity of Superdense Coding

Ashwin Nayak, Henry Yuen

Published May 2, 2023• 7 citations

Abstract

The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender’s encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid . In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of d 2 possible messages by sending a d -dimensional quantum state, for general dimensions d . Unlike the d =2 case (i.e., sending a single qubit), there can be inequivalent superdense coding protocols for higher d . We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest.

DOI: 10.1145/3593593

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The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender’s encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid . In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of d 2 possible messages by sending a d -dimensional quantum state, for general dimensions d . Unlike the d =2 case (i.e., sending a single qubit), there can be inequivalent superdense coding protocols for higher d . We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest.

In quantum superdense coding, two parties previously sharing entanglement can communicate a two bit message by sending a single qubit.We study this feature in the broader framework of general probabilistic theories.We consider a particular class of theories in which the local state space of the communicating parties corresponds to Euclidian hyperballs of dimension n (the case n = 3 corresponds to the Bloch ball of quantum theory).We show that a single n-ball can encode at most one bit of information, independently of n.We introduce a bipartite extension of such theories for which there exist dense coding protocols such that n log 1 2 + ( ) bits are communicated if entanglement is previously shared by the communicating parties.For n 3, > these protocols are more powerful than the quantum one, because more than two bits are communicated by transmission of a system that locally encodes at most one bit.We call this phenomenon hyperdense coding (HDC).Our HDC protocols imply superadditive classical capacities: two entangled systems can encode n log 1 2 2 + > ( ) bits, even though each system individually encodes at most one bit.In our examples, HDC and superadditivity of classical capacities come at the expense of violating tomographic locality or dynamical continuous reversibility.

Quantum dense coding is a foundational protocol in quantum communication, allowing two classical bits to be transmitted by sending a single qubit when a maximally entangled pair is shared. In this work, we consider Embedded Dense Coding (EDC)-a generalization of deterministic dense coding that embeds one subsystem into a higher-dimensional Hilbert space. To assess the operational advantage of EDC compared to standard dense coding, we consider the probability of transmission error when fixing the rate of entanglement consumed per classical message sent. We first demonstrate that EDC enables a smaller one-shot transmission error compared to standard dense coding when using quantum channels with nonzero rates of dephasing and loss. We then demonstrate that even with noiseless communication channels, EDC leads to smaller overall errors when the sender and receiver have noisy local processors. This advantage is shown through concrete implementations of EDC on IBM's Heron processor.

This chapter demonstrates the power of both coherent communication from Chapter 7 and the particular protocol for entanglement-assisted classical coding from the previous chapter. Recall that coherent dense coding is a version of the dense coding protocol in which the sender and receiver perform all of its steps coherently. Since our protocol for entanglement-assisted classical coding from the previous chapter is really just a glorified dense coding protocol, the sender and receiver can perform each of its steps coherently, generating a protocol for entanglement-assisted coherent coding. Then, by exploiting the fact that two coherent bits are equivalent to a qubit and an ebit, we obtain a protocol for entanglement-assisted quantum coding that consumes far less entanglement than a naive strategy would in order to accomplish this task. We next combine this entanglement-assisted quantum coding protocol with entanglement distribution (Section 6.2.1) and obtain a protocol for which the channel's coherent information (Section 12.5) is an achievable rate for quantum communication. This sequence of steps demonstrates an alternate proof of the direct part of the quantum channel coding theorem stated in Chapter 23. Entanglement-assisted classical communication is one generalization of superdense coding, in which the noiseless qubit channel becomes an arbitrary noisy quantum channel while the noiseless ebits remain noiseless. Another generalization of super-dense coding is a protocol named noisy super-dense coding , in which the shared entanglement becomes a shared noisy state ρ AB and the noiseless qubit channels remain noiseless.

The complementarity of two observables is often captured in uncertainty relations, which quantify an inevitable trade-off in knowledge. Here we study complementarity in the context of an information-processing task: we link the complementarity of two observables to their usefulness for superdense coding (SDC). In SDC, Alice sends two classical dits of information to Bob by sending a single qudit. However, we show that encoding with commuting unitaries prevents Alice from sending more than one dit per qudit, implying that complementarity is necessary for SDC to be advantageous over a classical strategy for information transmission. When Alice encodes with products of Pauli operators for the

X

and

Z

bases, we quantify the complementarity of these encodings in terms of the overlap of the

X

and

Z

basis elements. Our main result explicitly solves for the SDC capacity as a function of the complementarity, showing that the entropy of the overlap matrix gives the capacity, when the preshared state is maximally entangled. We generalize this equation to resources with symmetric noise such as a preshared Werner state. In the most general case of arbitrary noisy resources, we obtain an analogous lower bound on the SDC capacity. Our results shed light on the role of complementarity in determining the quantum advantage in SDC and also seem fundamentally interesting since they bear a striking resemblance to uncertainty relations.