Spatial reference frame agreement in quantum networks

Abstract

In order to communicate information in a quantum network effectively, all network nodes should share a common reference frame. Here, we propose to study how well m nodes in a quantum network can establish a common spatial reference frame from scratch, even though t of them may be arbitrarily faulty. We present a protocol that allows all correctly functioning nodes to agree on a common reference frame as long as they are fully connected and not more than < t m 3 nodes are faulty. Our protocol furthermore has the appealing property that it allows any existing two-node protocol for reference frame agreement to be lifted to a protocol for a quantum network.Keywords: quantum networks, reference frame agreement, quantum communication Quantum networks are gaining importance [1] for a variety of tasks such as quantum distributed computing [2], quantum cloud computing [3] and quantum key distribution (see e.g. [4][5][6][7]). From the current architecture of the internet one can assume that any such network will contain a large number of nodes that are distributed over widespread geographical locations on Earth or on satellites [8][9][10][11][12] and connected via quantum and classical communication channels [13]. Some of the many challenges in building a quantum network spanning long distances include Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.how to perform quantum error correction [14] and construct quantum repeaters (see e.g. [15]). Yet, before we can hope to implement even such basic building blocks effectively, we would like all nodes in the quantum network to agree on a common reference frame to enable easy quantum communication.A significant research effort has been devoted to developing protocols for agreeing on a reference frame between just two nodes [16][17][18][19][20][21][22][23]. Such protocols demand quantum communication because in the absence of a pre-shared reference frame, a node cannot meaningfully share directional information with a distant node by exchanging only classical data. Instead, a quantum system must be sent, for example, a qubit with its Bloch vector pointing in the required direction. A simple two-node protocol is thus to send many copies of the same qubit such that the receiver can approximate the direction with a certain level of accuracy.Here, our goal is to allow > m 2 number of nodes in a quantum network to agree on a common spatial reference frame, where in this first work we assume a fully connected network graph. That is, every node is connected to every other node using both classical and quantum communication channels. Why is this problem any more difficult than solving the problem for two nodes? Note that in an ideal case, where all the nodes are perfect and the channels connecting them are error-free, one node can send a reference frame to everyone else, and everyone can subsequently use th...

Asynchronous reference frame agreement in a quantum network

“…In this work we have presented the first asynchronous reference frame agreement protocol. The synchronous protocol for spatial reference frame agreement presented in [20] can tolerate up to t n 3 < faulty nodes. Whereas, the asynchronous protocol we have presented tolerates only t n 4 < faulty nodes.…”

“…So far these reference frame agreement problems are studied in a bipartite setting [13][14][15][16][17][18][19] with the exception of [20], where spatial direction are agreed on in a synchronised network of n nodes. More specifically in [20] it is assumed that the network is synchronous. That is, all the nodes of the network have a shared clock and all the link delays have known upper bound.…”

Quantum teleportation with infinite reference-frame uncertainty

“…Prior alignment of reference frames [3,28,12,13,23] may become impractical in the case of time-varying misalignment, or where the parties are far apart; prior alignment also involves communication of reference frame information, which may be cryptographically sensitive [11,2,14]. Another approach involves the use of decoherence-free subspaces [17]; because this requires larger Hilbert spaces, practical implementation can be nontrivial, although experimental solutions have been developed for optical systems [6].…”

“…• In Section 2.1, the encoding and decoding subsets for the tight scheme are the same, namely the regions R 1 and R 2 (Figure 2). In the physical (unfaithful) representation, the Pauli UEB is equivariant for the subgroup Z 8 < U (1), where a generator of Z 8 acts as the swap (12). Compatibly, the regions R 1 and R 2 are swapped under the action of a generator of Z 8 .…”

Tight quantum teleportation without a shared reference frame

“…It is now well recognized that a shared reference frame is an implicit assumption underlying the correct execution of many quantum protocols [4,18,13,34,14,31]. As quantum communication finds its way into handheld devices [35,9,10] and into space [26,38,2], it is increasingly important to develop protocols robust against reference frame error for situations where alignment is difficult [16,17,28] or undesired [3,15]. Considerable progress has already been made in this regard for quantum key distribution [8,39,36,20,29,19,30], and there is also a smaller body of work on quantum teleportation [6,21,22] without a shared reference frame, which our results extend.Main results.…”