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Qudits offer no advantages over dits for sending random messages

Ronit Shah

TL;DR

The paper analyzes a one-shot messenger–receiver task where Alice sends one quantum system of dimension d to Bob, seeking to maximize the probability Bob correctly identifies the message drawn from a known prior. It proves that, in the absence of preshared entanglement, a qudit offers no advantage over a classical d-ary symbol: the maximum success probability is bounded by the sum of the top-d priors, and pure encodings suffice. It further derives a mixed-state spectral bound, P_succ^{qu} ≤ ∑_{k=1}^{d} λ'_{(k)}, where λ'_{ik} = p_i λ_{ik} are the weighted eigenvalues, and proves sharpness for both bounds. The results strengthen the standard dimension-based ceiling and connect the classical and quantum limits through a unified, spectrum-aware framework, with implications for understanding information transmission without entanglement and for minimum-error discrimination strategies.

Abstract

We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit to Bob, without preshared entanglement, there is never any advantage over sending him a classical dit. This result was previously known only for a uniform distribution. We also prove a mixed-state generalization of this result in the form of an upper bound on the success probability of discriminating between mixed quantum states with a single measurement. This bound is based solely on the dimension, probability distribution, and eigenvalues of the states and is sharp among such bounds.

Qudits offer no advantages over dits for sending random messages

TL;DR

The paper analyzes a one-shot messenger–receiver task where Alice sends one quantum system of dimension d to Bob, seeking to maximize the probability Bob correctly identifies the message drawn from a known prior. It proves that, in the absence of preshared entanglement, a qudit offers no advantage over a classical d-ary symbol: the maximum success probability is bounded by the sum of the top-d priors, and pure encodings suffice. It further derives a mixed-state spectral bound, P_succ^{qu} ≤ ∑_{k=1}^{d} λ'_{(k)}, where λ'_{ik} = p_i λ_{ik} are the weighted eigenvalues, and proves sharpness for both bounds. The results strengthen the standard dimension-based ceiling and connect the classical and quantum limits through a unified, spectrum-aware framework, with implications for understanding information transmission without entanglement and for minimum-error discrimination strategies.

Abstract

We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit to Bob, without preshared entanglement, there is never any advantage over sending him a classical dit. This result was previously known only for a uniform distribution. We also prove a mixed-state generalization of this result in the form of an upper bound on the success probability of discriminating between mixed quantum states with a single measurement. This bound is based solely on the dimension, probability distribution, and eigenvalues of the states and is sharp among such bounds.

Paper Structure

This paper contains 22 sections, 3 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical case.
  3. Quantum case.
  4. Structure.
  5. Related work
  6. Dimension-only quantum ceilings.
  7. Overlap-/geometry–sensitive bounds.
  8. Measurement model
  9. Compressing to the joint support.
  10. Measurement and post-processing.
  11. Success probability.
  12. Pure-state bound
  13. Bounded by $1$.
  14. Total budget at most $d$.
  15. Linear-programming viewpoint.
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let Alice’s messages have priors $(p_i)_{i=1}^m$, and let these priors be sorted as $p_{(1)}\ge\cdots\ge p_{(m)}$. Suppose that for each $i$ Alice encodes message $i$ into a pure state $\ket{\psi_i}$, and that the span of the states $\{\ket{\psi_i}\}$ has dimension $d$. Then for any one-shot measure This bound is tight: if the $d$ most likely states are mutually orthogonal, Bob can measure in a ba

Figures (1)

  • Figure :

Theorems & Definitions (5)

  • Theorem 1.1: Qudits vs. dits for pure encodings
  • Theorem 1.2: Mixed-state spectral bound
  • Proposition 5.1: Sharpness for mixed encodings
  • proof
  • Remark 5.2