Abstract
This chapter generalizes 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 ρ R ⊗ ρ S, where ρ S is the state of the system Alice wishes to communicate and ρ 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 ρ R ⊗ ρ 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 \(\rho _S^{\prime }\) is decohered with respect to ρ S, the amount of decoherence depending on the properties of the reference token ρ R. An example of this protocol is presented in which Alice and Bob do not share a reference frame associated with the one-dimensional translation group and the fidelity between ρ S and \(\rho _S^{\prime }\) is used to quantify the success of the recovery operation.
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Notes
- 1.
This is true of the group generated by either the position or momentum operator on \(L^2(\mathbb {R})\). We note that the following construction does not rely on σ(A R) being continuous.
- 2.
More precisely, when dealing with operators with a continuous spectrum the theory is defined on a rigged Hilbert space [6]
$$\displaystyle \begin{aligned} \Phi \subset \mathcal{H}_R \subset \Phi', \end{aligned} $$where Φ is a proper subset dense in \(\mathcal {H}_R\) and Φ′ is the dual of Φ, defined through the inner product on \(\mathcal {H}_R\). In our case, Φ is the Schwarz space of smooth rapidly decreasing functions on \(\mathbb {R}\) and Φ′ is the space of tempered distributions on \(\mathbb {R}\). The eigenkets \(\left | a_R \right >\) are in Φ′.
- 3.
To the best of the author’s knowledge the question of whether such a measurement exists for any G is an open problem, as suggested by the remarks in Sec. III.4.4 of Ref. [4]. However, it is suggested in this reference that it seems plausible that such a measurement can be constructed, although there does not seem to be an easy general procedure for its construction.
- 4.
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Smith, A.R.H. (2019). Communication Without a Shared Reference Frame. In: Detectors, Reference Frames, and Time. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-11000-0_7
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