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arXiv:quant-ph/0106014 · Physical Review Letters

Aligning Reference Frames with Quantum States

E. Bagán, M. Baig, R. Muñoz-Tapia

Published Nov 1, 2001• 72 citations
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Abstract

We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary spins. We analytically obtain the optimal strategy, i.e., the best encoding state and the best measurement. For large N, we show that the average error goes to zero linearly in 1/N. Finally, we discuss the construction of finite optimal measurements.

DOI
10.1103/physrevlett.87.257903

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We analyze the problem of sending, in a single transmission, the information required to specify an orthogonal trihedron or reference frame through a quantum channel made out of N elementary spins. We analytically obtain the optimal strategy, i.e., the best encoding state and the best measurement. For large N, we show that the average error goes to zero linearly in 1/N. Finally, we discuss the construction of finite optimal measurements.
Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made up of N spins (qubits). We find that the quality of this optimal procedure, which we quantify in terms of the fidelity, depends solely on the dimension of the encoding space. We also investigate the use of spatial rotations on a quantum state, which provide a natural and less demanding encoding. In this case we prove that the fidelity is directly related to the largest zeros of the Legendre and Jacobi polynomials. We also discuss our results in terms of the information gain.
We present a procedure inspired by dense coding, which enables a highly efficient transmission of information of a continuous nature. The procedure requires the sender and the recipient to share a maximally entangled state. We deal with the concrete problem of aligning reference frames or trihedra by means of a quantum system. We find the optimal covariant measurement and compute the corresponding average error, which has a remarkably simple close form. The connection of this procedure with that of estimating unitary transformations on qubits is briefly discussed.
We consider the smallest possible directional reference frames allowed and determine the best one can ever do in preserving quantum information in various scenarios. We find that for the preservation of a single spin state, two orthogonal spins are optimal primitive reference frames; and in a product state, they do approximately 2222%22 as well as an infinite-sized classical frame. By adding a small amount of entanglement to the reference frame, this can be raised to 2(2/3)5=262(2/3){}^{5}=26%2(2/3)5=26. Under the different criterion of entanglement preservation, a very similar optimal reference frame is found; however, this time it is for spins aligned at an optimal angle of ${87}^{\ifmmode^\circ\else\textdegree\fi{}}$. In this case 2424%24 of the negativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that ${90}^{\ifmmode^\circ\else\textdegree\fi{}}$ is selected out nonmonotonically, with a peak optimal angle of ${96.5}^{\ifmmode^\circ\else\textdegree\fi{}}$ for L=3L=3L=3 spins.
We discuss the aligning of spatial reference frames from a quantum communication complexity perspective. This enables us to analyze multiple rounds of communication and give several simple examples demonstrating tradeoffs between the number of rounds and the type of communication. Using a distributed variant of a quantum computational algorithm, we give an explicit protocol for aligning spatial axes via the exchange of spin-1/2 particles which makes no use of either exchanged entangled states, or of joint measurements. This protocol achieves a worst-case fidelity for the problem of "direction finding" that is asymptotically equivalent to the optimal average case fidelity achievable via a single forward communication of entangled states.
Typical quantum communication schemes are such that to achieve perfect decoding the receiver must share a reference frame (RF) with the sender. Indeed, if the receiver only possesses a bounded-size quantum token of the sender's RF, then the decoding is imperfect, and we can describe this effect as a noisy quantum channel. We seek here to characterize the performance of such schemes, or equivalently, to determine the effective decoherence induced by having a bounded-size RF. We assume that the token is prepared in a special state that has particularly nice group-theoretic properties and that is near-optimal for transmitting information about the sender's frame. We present a decoding operation, which can be proven to be near-optimal in this case, and we demonstrate that there are two distinct ways of implementing it (corresponding to two distinct Kraus decompositions). In one, the receiver measures the orientation of the RF token and reorients the system appropriately. In the other, the receiver extracts the encoded information from the virtual subsystems that describe the relational degrees of freedom of the system and token. Finally, we provide explicit characterizations of these decoding schemes when the system is a single qubit and for three standard kinds of RF: a phase reference, a Cartesian frame (representing an orthogonal triad of spatial directions), and a reference direction (representing a single spatial direction).
The optimal NNN qubit states featuring highest sensitivity to small misalignment of Cartesian reference frames are found using the quantum Cram\'er-Rao bound. It is shown that the optimal states are supported on the symmetric subspace and hence are mathematically equivalent to a single spin J=N∕2J=N∕2J=N∕2. Majorana representation of spin states is used to reveal a beautiful connection between the states optimal for aligning reference frames and the platonic solids.