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arXiv:2608.03184

Characterizing pairwise swapping capabilities of dense coding channels

Abhishek Muhuri, Nirman Ganguly, Aditi Sen De, Indranil Chakrabarty

quant-phSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations

Abstract

We introduce a novel multipartite entanglement-assisted classical communication task, referred to as dense coding swapping, in which legitimate parties collaboratively swap the dense codeability from one communication channel to another through suitable joint unitary operations. Due to the dense coding (DC) exclusion principle, the scheme enhances the dense codeability of a target pair while simultaneously reducing it for a non-target branch in the network. This swapping capability has broader implications, as it may be viewed as a form of process swapping, distinct from resource swapping, while also providing a prevention measure when one of the receivers is compromised. We derive necessary and sufficient conditions, expressed in terms of the Schmidt coefficients, for three-qubit pure states to support DC swapping, while we obtain a sufficient criterion for mixed states using their Bloch correlation parameters. Furthermore, we identify the optimal two-qubit unitary operators capable of realizing the swapping of dense codeability between communication channels. We further examine the tolerance of these eligible states against both colored and white noise, demonstrating the resilience of the proposed task under environmental perturbations. We also show that multipartite states supporting DC swapping require only a small amount of genuine multipartite entanglement and that this requirement decreases with increasing system size.

Comments: 17 pages, 8 figures

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Quantum dense coding could be used to transmit two classical bits with one qubit when a maximally entangled state is shared. In realistic channels, entanglement degradation reduces the channel capacity, while bit-flip noise increases decoding errors. To address these issues, we propose a novel probabilistic controlled dense coding protocol that employs the three-qubit repetition code for error correction and an auxiliary qubit for probabilistic decoding. Moreover, this proposed scheme includes a third party to supervise the communication based on a three-qubit entanglement state. The implementation steps of our protocol are presented in detail, and numerical simulations show that it achieves higher average information than dense coding without error correction. The scheme provides a robust solution for quantum communication under noisy conditions and non-maximally entangled state.

In order to understand the resourcefulness of a natural quantum system in quantum communication tasks, we study the dense coding capacity (DCC) and teleportation fidelity of Haar uniformly generated random multipartite states of various ranks. We prove that when a rank-2 two-qubit state, a Werner state, and a pure state possess the same amount of entanglement, the DCC of a rank-2 state belongs to the envelope made by pure and Werner states. In a similar way, we obtain an upper bound via the generalized Greenberger-Horne-Zeilinger state for rank-2 three-qubit states when the dense coding with two senders and a single receiver is performed and entanglement is measured in the senders:receiver bipartition. The normalized frequency distributions of DCC for randomly generated two-, three-, and four-qubit density matrices with global as well as local decodings at the receiver's end are reported. The estimation of mean DCC for two-qubit states is found to be in good agreement with the numerical simulations. Universally, we observe that the performance of random states for dense coding as well as teleportation decreases with the increase of the rank of states, which we have shown to be surmounted by the local preprocessing operations performed on the shared states before starting the protocols, irrespective of the rank of the states. The local preprocessing employed here is based on positive operator valued measurements along with classical communication and we show that unlike dense coding with two-qubit random states, the senders' operations are always helpful to probabilistically enhance the capabilities of implementing dense coding as well as teleportation.

We present an analytical study of the standard two-party deterministic dense-coding protocol, under which communication of perfectly distinguishable messages takes place via a qudit from a pair of nonmaximally entangled qudits in a pure state $|\ensuremath{\psi}⟩$. Our results include the following: (i) We prove that it is possible for a state $|\ensuremath{\psi}⟩$ with lower entanglement entropy to support the sending of a greater number of perfectly distinguishable messages than one with higher entanglement entropy, confirming a result suggested via numerical analysis in Mozes et al. [Phys. Rev. A 71, 012311 (2005)]. (ii) By explicit construction of families of local unitary operators, we verify, for dimensions

d=3

and

d=4

, a conjecture of Mozes et al. about the minimum entanglement entropy that supports the sending of

d+j

messages, $2\ensuremath{\le}j\ensuremath{\le}d\ensuremath{-}1$; moreover, we show that the

j=2

and $j=d\ensuremath{-}1$ cases of the conjecture are valid in all dimensions. (iii) Given that $|\ensuremath{\psi}⟩$ allows the sending of

K

messages and has $\sqrt{{\ensuremath{\lambda}}_{0}}$ as its largest Schmidt coefficient, we show that the inequality ${\ensuremath{\lambda}}_{0}\ensuremath{\le}d/K$, established by Wu et al. [Phys. Rev. A 73, 042311 (2006)], must actually take the form ${\ensuremath{\lambda}}_{0}<d/K$ if

K=d+1

, while our constructions of local unitaries show that equality can be realized if

K=d+2

or $K=2d\ensuremath{-}1$.