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arXiv:quant-ph/0406219 · Quantum Information and Computation

Communicating continuous quantum variables between different Lorentz frames

Pieter Kok, Timothy C. Ralph, Gerard J Milburn

Published May 1, 2005• 8 citations
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Abstract

We show how to communicate Heisenberg-limited continuous (quantum) variables between Alice and Bob in the case where they occupy two inertial reference frames that differ by an unknown Lorentz boost. There are two effects that need to be overcome: the Doppler shift and the absence of synchronized clocks. Furthermore, we show how Alice and Bob can share Doppler-invariant entanglement, and we demonstrate that the protocol is robust under photon loss.

DOI
10.26421/qic5.3-4

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We show how to communicate Heisenberg-limited continuous (quantum) variables between Alice and Bob in the case where they occupy two inertial reference frames that differ by an unknown Lorentz boost. There are two effects that need to be overcome: the Doppler shift and the absence of synchronized clocks. Furthermore, we show how Alice and Bob can share Doppler-invariant entanglement, and we demonstrate that the protocol is robust under photon loss.
In quantum communication protocols the existence of a shared reference frame between two spatially separated parties is normally presumed. However, in many practical situations we are faced with the problem of misaligned reference frames. In this paper, we study communication between two inertial observers who have partial knowledge about the Lorentz transformation that relates their frames of reference. Since every Lorentz transformation can be decomposed into a pure boost followed by a rotation, we begin by analyzing the effects on communication when the parties have partial knowledge about the transformation relating their frames, when the transformation is either a rotation or a pure boost. This then enables us to investigate how the efficiency of communication is affected due to partially correlated inertial reference frames related by an arbitrary Lorentz transformation. Furthermore, we show how the results of previous studies where reference frames are completely uncorrelated are recovered from our results in appropriate limits.
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 present an experimental realization of a robust quantum communication scheme [Phys. Rev. Lett. 93, 220501 (2004)] using pairs of photons entangled in polarization and time. Our method overcomes errors due to collective rotation of the polarization modes (e.g., birefringence in optical fiber or misalignment), is insensitive to the phase's fluctuation of the interferometer, and does not require any shared reference frame including time reference, except the need to label different photons. The practical robustness of the scheme is further shown by implementing a variation of the Bennett-Brassard 1984 quantum key distribution protocol over 1 km optical fiber.
Synchronizing clocks using quantum entanglement works on the principle that two clocks Alice and Bob share between them a singlet state which is a stationary state that is immune to evolution under bare atomic Hamiltonian. A major obstacle to its realization is the hidden assumption of a common phase reference between the clocks. Without hidden assumption, a clock state of Alice or Bob is not a uniquely defined quantum state because the phase of the state is arbitrary. This results in an unknown relative phase in a two-particle entangled state defined by the clocks. We show that using entanglement purification, an entanglement-based clock synchronization is achieved despite earlier results showing the contrary. This closes the loophole for entanglement based quantum clock synchronization protocols, which is a non-local approach to synchronize two clocks independent of the properties of the intervening medium.
Quantum teleportation, a key building block for quantum technologies, enables distant parties, Alice and Bob, to transfer an unknown quantum state using entanglement, local operations, and classical communication. In this work, we propose a quantum teleportation protocol for single-photon time-frequency continuous variables, where time is the time of arrival at a detector. We detail the operational requirements of the protocol, notably the nonlinear interactions needed for the photon-photon time-frequency entangling gate. By explicitly treating the detector position as the reference frame for time of arrival, we show that, even with ideal classical and quantum channels, the input-output fidelity is not unity unless the receiving party can correct the detector-position mismatch. This results in a trade-off between teleportation fidelity and the temporal bandwidth of the input photon wavepacket. The protocol itself can be used for remote detector synchronization. In particular, we show that time-frequency cat and Gottesman-Kitaev-Preskill states provide an internal temporal ruler through fidelity revivals. We further derive the classical and best-copy fidelity thresholds for single-photon Gaussian wavepackets, finding that the relevant vacuum-equivalent shot noise is input-dependent, in contrast with quadrature-based continuous-variable quantum teleportation. Our proposal derives from a quantum theory of space-time events, whereafter we establish space as a reference for the time degree of freedom, thereby grounding our work in ongoing research on reference frames and superselection rules in quantum optics and quantum information. Beyond protocols harnessing time-frequency continuous variables of single photons, this work points toward remote synchronization in quantum communication architectures.
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.