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arXiv:1901.01503 · Physical Review A

Entanglement-assisted communication in the absence of shared reference frame

Ali Beheshti, Sadegh Raeisi, Vahid Karimipour

Published Apr 22, 2019• 5 citations
arXivHTMLPDF

Abstract

Alice wants to convey the value of a parameter to Bob with whom she does not share a reference frame. What physical object can she use for this task? Shall she encode this value into the angle between two physical vectors such as the angle between two spins? Can she benefit from using entanglement? We investigate these questions here and show that an entangled state of two qubits has three parameters that are invariant under changes of the reference frame. We also calculate for specific examples the average information gain for different circumstances, where one of these parameters is used for communication. We compare our result with the special case of separable states and find that entanglement enhances the information gain.

DOI
10.1103/physreva.99.042330

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Alice wants to convey the value of a parameter to Bob with whom she does not share a reference frame. What physical object can she use for this task? Shall she encode this value into the angle between two physical vectors such as the angle between two spins? Can she benefit from using entanglement? We investigate these questions here and show that an entangled state of two qubits has three parameters that are invariant under changes of the reference frame. We also calculate for specific examples the average information gain for different circumstances, where one of these parameters is used for communication. We compare our result with the special case of separable states and find that entanglement enhances the information gain.
We generalize a quantum communication protocol introduced by Bartlett et al. [New J. Phys. 11, 063013 (2009)], in which two parties communicating do not share a classical reference frame, to the case where changes of their reference frames form a one-dimensional noncompact Lie group. Alice sends to Bob the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$, where ${\ensuremath{\rho}}_{S}$ is the state of the system Alice wishes to communicate and ${\ensuremath{\rho}}_{R}$ is the state of an ancillary system serving as a token of her reference frame. Because Bob is ignorant of the relationship between his reference frame and Alice's, he will describe the state ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$ as an average over all possible reference frames. Bob measures the reference token and applies a correction to the system Alice wished to communicate conditioned on the outcome of the measurement. The recovered state ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ is decohered with respect to ${\ensuremath{\rho}}_{S}$, the amount of decoherence depending on the properties of the reference token ${\ensuremath{\rho}}_{R}$. We present an example of this protocol when Alice and Bob do not share a reference frame associated with the one-dimensional translation group and use the fidelity between ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ to quantify the success of the recovery operation.
Abstract This chapter discusses some of the ways in which quantum entanglement can be used for quantum communication. Suppose Alice and Bob share an entangled state: they can use it for dense coding, in which Alice sends Bob just one qubit of an entangled pair, which delivers two bits of classical information. Alice and Bob can also use an entangled state for teleportation, in which a qubit in a unknown quantum state is teleported from Alice to Bob when Alice sends Bob two classical bits. Entanglement swapping is an example of a tripartite communication protocol. If Alice and Bob share entanglement, and Bob and Charlie share entanglement, entanglement swapping can be used to entangle Alice and Charlie. Even if Alice and Bob are entangled, Alice cannot communicate with Bob without sending him classical bits or qubits, which prevents Alice and Bob from breaking the laws of relativity. This chapter also discusses Pauli matrices and the extension of Hilbert spaces.
Quantum dense coding is considered to be one of the most remarkable demonstrations of the existence of quantum entanglement. In this paper, we analyze the role of correlations (both classical and quantum) in quantum communication protocols. We study how the capacity of a single-qubit quantum channel depends on correlations between Alice and Bob. In particular, we consider a scenario when Alice and Bob share a pair of qubits in a state such that Alice's qubit is in the state described by the density operator $\ensuremath{\varrho}=\frac{1}{2}1+\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\cdot}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}.$ We evaluate a capacity of the noiseless channel for two cases: (1) when Bob performs measurement just on the particle received from Alice, and (2) in the case when he utilizes correlations that have been established prior to the communication. We present a simple classical scenario which might serve as a prototype of quantum dense coding. We generalize our results also for qudits. In addition, we address some aspects of the issue of security in correlation-assisted communication protocols.
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are kkk senders, Alice1_11​, …\ldots…, Alicek_kk​, and a single receiver, Bob. Each Alicei_ii​ has a data vector Wi∈FmiW_i \in \mathbb{F}^{m_i}Wi​∈Fmi​, where F\mathbb{F}F is a finite field. Bob wants to compute the linear combination Y=V1W1+V2W2+⋯+VkWk∈FmY = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^mY=V1​W1​+V2​W2​+⋯+Vk​Wk​∈Fm, where ViV_iVi​ is an m×mim \times m_im×mi​ matrix over F\mathbb{F}F. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information WiW_iWi​, i=1,…,ki=1,\ldots,ki=1,…,k, into their local subsystems and transmit them to Bob so that he can recover YYY through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.