Coherence Scaling in Quantum Communication Protocols

TL;DR

The paper develops a coherence-based framework to analyze how quantum coherence scales and is redistributed in the circuit implementations of LOCC quantum communication protocols. Using the relative entropy of coherence, it shows that generalized superdense coding can achieve scalable communication with logarithmic or constant coherence growth, whereas quantum teleportation incurs a universal linear coherence cost of $2$ bits per teleported qubit at the maximal-coherence stage, independent of the input. It further derives a general LOCC coherence accounting framework, including a branching-entropy identity and an upper bound, showing that large intermediate coherence can arise purely from protocol structure while remaining consistent with the Holevo bound. These results provide operational insights for quantum networks and the quantum internet, offering an internal diagnostic of coherence overheads in entanglement swapping and repeater architectures and guiding design choices for shared-resource protocols.

Abstract

We investigate how quantum coherence scales and is redistributed in quantum communication protocols, using superdense coding and quantum teleportation as paradigmatic case studies. Employing the relative entropy of coherence as a circuit-level resource measure, we show that multipartite resource states relevant to generalized superdense coding can enable scalable communication while exhibiting only logarithmic or even constant coherence growth, depending on their entanglement structure. In sharp contrast, quantum teleportation displays an unavoidable, protocol-induced coherence cost that grows linearly with the number of teleported qubits and is independent of the input state. Through a stage-resolved analysis of the teleportation circuit, we separate protocol-generated coherence from message-dependent contributions and identify a universal two-bit coherence offset per teleported qubit at the maximal-coherence stage. We further demonstrate explicitly that this extensive intermediate coherence generation is fully consistent with information-theoretic bounds, including the Holevo limit, and does not correspond to an increase in accessible classical information.

Figures (5)

• Figure 1: Relative entropy of coherence for two paradigmatic multipartite resource states relevant to generalized superdense coding. The $W_n$ state exhibits logarithmic scaling, $C_{r}=\log_2 n$, while the $GHZ_n$ state has constant coherence $C_{r}=1$ independent of $n$. This illustrates that coherence scaling need not track communication capability.
• Figure 2: Quantum circuit representing the protocol for quantum teleportation based on NielsenChuang2011, the gray vertical lines separate the three stages of the process, first creating the EPR pair, second the teleportation process and the third and last phase the message extraction.
• Figure 3: Total relative entropy of coherence across teleportation stages. Coherence is generated by entangling gates, peaks immediately before measurement, and is suppressed by non-selective measurement.
• Figure 4: Single-qubit reduced-state coherences during teleportation of an incoherent input state $\ket{0}$, illustrating redistribution of coherence among subsystems.
• Figure 5: Message-dependent coherence contribution for teleportation of a single-qubit state $\ket{\psi(\phi)}=\cos\phi\ket{0}+\sin\phi\ket{1}$.

Theorems & Definitions (7)

• Lemma 1: Additivity of the relative entropy of coherence
• Proposition 1: Stage-resolved coherence decomposition
• Lemma 2: Coherence is not an information monotone
• Lemma 3: Coherence of a branch superposition
• proof
• Proposition 2: Coherence overhead bound for LOCC-assisted protocols
• proof : Proof sketch