Deterministic and Unambiguous Dense Coding

Abstract

Optimal dense coding using a partially-entangled pure state of Schmidt rank $\bar{D}$ and a noiseless quantum channel of dimension $D$ is studied both in the deterministic case where at most $L_d$ messages can be transmitted with perfect fidelity, and in the unambiguous case where when the protocol succeeds (probability ${\tau }_x$) Bob knows for sure that Alice sent message $x$, and when it fails (probability $1-{\tau }_x$) he knows it has failed. Alice is allowed any single-shot (one use) encoding procedure, and Bob any single-shot measurement. For $\bar{D}\le D$ a bound is obtained for $L_d$ in terms of the largest Schmidt coefficient of the entangled state, and is compared with published results by Mozes et al. For $\bar{D}>D$ it is shown that $L_d$ is strictly less than $D^2$ unless $\bar{D}$ is an integer multiple of $D$, in which case uniform (maximal) entanglement is not needed to achieve the optimal protocol. The unambiguous case is studied for $\bar{D}\le D$, assuming ${\tau }_x>0$ for a set of $\bar{D}D$ messages, and a bound is obtained for the average $\lgl1/τ\rgl$. A bound on the average $\lglτ\rgl$ requires an additional assumption of encoding by isometries (unitaries when $\bar{D}=D$) that are orthogonal for different messages. Both bounds are saturated when ${\tau }_x$ is a constant independent of $x$, by a protocol based on one-shot entanglement concentration. For $\bar{D}>D$ it is shown that (at least) $D^2$ messages can be sent unambiguously. Whether unitary (isometric) encoding suffices for optimal protocols remains a major unanswered question, both for our work and for previous studies of dense coding using partially-entangled states, including noisy (mixed) states.

Short new section VII added. Latex 23 pages, 1 PSTricks figure in text

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• The state space for two qutrits has a phase space structure in its core
• The states of W-class as shared resources for perfect teleportation and superdense coding
• Experimental Multi-Party Sequential State Discrimination
• Atemporal diagrams for quantum circuits
• Additive Classical Capacity of Quantum Channels Assisted by Noisy Entanglement
• Maximum-confidence discrimination among symmetric qudit states
• Deterministic dense coding and entanglement entropy
• Experimental scheme for unambiguous discrimination of linearly independent symmetric states
• Optimal dense coding with arbitrary pure entangled states
• Improving the applicability of the Pauli kinetic energy density based semilocal functional for solids
• Boundary effect of deterministic dense coding
• Deterministic Quantum Dense Coding Networks
• One-shot conclusive multiport quantum dense coding capacities
• Optimal probabilistic dense coding schemes
• Secure $N$-dimensional Simultaneous Dense Coding and Applications
• Non-optimality of unitary operations for dense coding
• Deterministic distributed dense coding with stabilizer states
• Phase boundaries in deterministic dense coding
• Superiority in dense coding through non-Markovian stochasticity
• Overcoming a limitation of deterministic dense coding with a non-maximally entangled initial state
• The augmented message-matrix approach to deterministic dense coding theory
• Initial-Phase Spectroscopy as a Control of Entangled Systems
• Signaling versus distinguishing different superpositions of same pure quantum state
• Entanglement-invertible channels