# Optimal probabilistic dense coding schemes

## TL;DR

This paper introduces optimal probabilistic decoding schemes for dense coding with non-maximally entangled states, balancing success probability and confidence, and enhancing information transfer in quantum communication.

## Contribution

It proposes novel optimal probabilistic decoding schemes, including quantum-state separation and multistage decoding, improving upon existing methods for dense coding with non-maximally entangled states.

## Key findings

- Quantum-state separation increases message distinguishability with optimal success probability.
- Multistage decoding enhances mutual information for qudits.
- The schemes interpolate between error-prone and error-free decoding approaches.

## Abstract

Dense coding with non-maximally entangled states has been investigated in many different scenarios. We revisit this problem for protocols adopting the standard encoding scheme. In this case, the set of possible classical messages cannot be perfectly distinguished due to the non-orthogonality of the quantum states carrying them. So far, the decoding process has been approached in two ways: (i) The message is always inferred, but with an associated (minimum) error; (ii) the message is inferred without error, but only sometimes; in case of failure, nothing else is done. Here, we generalize on these approaches and propose novel optimal probabilistic decoding schemes. The first uses quantum-state separation to increase the distinguishability of the messages with an optimal success probability. This scheme is shown to include (i) and (ii) as special cases and continuously interpolate between them, which enables the decoder to trade-off between the level of confidence desired to identify the received messages and the success probability for doing so. The second scheme, called multistage decoding, applies only for qudits ($d$-level quantum systems with $d>2$) and consists of further attempts in the state identification process in case of failure in the first one. We show that this scheme is advantageous over (ii) as it increases the mutual information between the sender and receiver.