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Entanglement assisted communication complexity measured by distinguishability

Satyaki Manna, Ankush Pandit, Debashis Saha

Abstract

We investigate the quantum advantage that can arise in typical two-party communication scenarios, where the sender and the receiver are allowed to share prior correlations. Focusing on communication tasks constrained by the distinguishability of the sender's inputs, we demonstrate that entanglement-assisted communication, both classical and quantum, can outperform classical communication supplemented with shared randomness. We begin by developing a general framework for communication tasks with pre-shared correlations. We identify certain communication tasks that exhibit an advantage under entanglement assistance compared to classical communication. Through these results, we establish a connection between quantum communication and entanglement-assisted classical communication, and also show an equivalence between entanglement-assisted classical communication and entanglement-assisted quantum communication. We then consider the simplest scenarios in which the receiver has no input and demonstrate that entanglement-assisted strategies still offer advantages over both classical communication and quantum communication without prior entanglement. Finally, by constructing a class of communication tasks, we show that a non-maximally entangled state can, in some cases, be more useful than a maximally entangled state as a pre-shared resource.

Entanglement assisted communication complexity measured by distinguishability

Abstract

We investigate the quantum advantage that can arise in typical two-party communication scenarios, where the sender and the receiver are allowed to share prior correlations. Focusing on communication tasks constrained by the distinguishability of the sender's inputs, we demonstrate that entanglement-assisted communication, both classical and quantum, can outperform classical communication supplemented with shared randomness. We begin by developing a general framework for communication tasks with pre-shared correlations. We identify certain communication tasks that exhibit an advantage under entanglement assistance compared to classical communication. Through these results, we establish a connection between quantum communication and entanglement-assisted classical communication, and also show an equivalence between entanglement-assisted classical communication and entanglement-assisted quantum communication. We then consider the simplest scenarios in which the receiver has no input and demonstrate that entanglement-assisted strategies still offer advantages over both classical communication and quantum communication without prior entanglement. Finally, by constructing a class of communication tasks, we show that a non-maximally entangled state can, in some cases, be more useful than a maximally entangled state as a pre-shared resource.
Paper Structure (19 sections, 5 theorems, 60 equations, 2 figures)

This paper contains 19 sections, 5 theorems, 60 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General Communication Task
  3. Classical Communication with shared randomness (CC)
  4. Quantum Communication (QC)
  5. Entanglement assisted Classical Communication (EACC)
  6. Entanglement assisted Quantum Communication (EAQC)
  7. Quantifying Advantage
  8. Quantum Advantages
  9. Random Access Codes
  10. Equality Problem defined by Graphs
  11. Pair Distinguishability Task
  12. Communication task of Chaturvedi2020quantum
  13. Quantum advantage with no input on Bob's side
  14. (3,1,3) scenario
  15. (3,1,4) scenario
  16. ...and 4 more sections

Key Result

Theorem 1

In this scenario, if Alice's inputs are random (i.e., $p_x=\frac{1}{N}$) and she uses the maximally entangled state $\ket{\phi^+}=\frac{1}{\sqrt{d}}\sum_{i=1}^d\ket{ii}$ and sends the outcome as the classical message to Bob, the highest value of $\mathcal{D}^{\phi^+}_{EACC}$ for rank one projective

Figures (2)

  • Figure 1: Circuit diagrams illustrating three different communication protocols. In the circuits, single lines represent quantum systems, while double lines denote classical variables. The dashed line denotes spatial separation between the parties.
  • Figure 2: The plot illustrates the advantage of EACC over CC in the communication task derived from a Bell inequality. The independent axis represents the parameter $\theta$ of the shared entangled state, while the dependent axis shows the corresponding values of $\mathcal{S}_{EACC}(\theta)$ and $\mathcal{S}_C(\theta)$. The plot clearly demonstrates that $\mathcal{S}_{EACC}(\theta) > \mathcal{S}_C(\theta)$ for the considered range of $\theta$. Moreover, except at $\theta = \pi/4$, non-maximally entangled states provide a greater advantage than the maximally entangled state, since $\theta = \pi/4$ corresponds to maximal entanglement.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof