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arXiv:1106.6170 · Journal of Physics A Mathematical and Theoretical

Information–disturbance tradeoff in sending direction information via antiparallel quantum spin

ShengLi Zhang, Xu‐Bo Zou, ChuanFeng Li, Chenhui Jin, Guang‐Can Guo

Published May 13, 2010• 5 citations
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Abstract

For sending unknown direction information, antiparallel spins contains more direction information than parallel spins(Gisin and Popescu, 1999, \textit{Phys. Rev. Lett.} 83, 432).In this paper, the optimal information-disturbance tradeoff bound for antiparallel spins is derived. The quantum measurements which attain the optimal tradeoff bound are obtained. This result can be of practical relevance for posing some general limits on Eve's eavesdropping process. Finally, we also present a comparison between the bound for antiparallel spins and the bound for parallel spins.

DOI
10.1088/1751-8113/43/23/235301

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For sending unknown direction information, antiparallel spins contains more direction information than parallel spins(Gisin and Popescu, 1999, \textit{Phys. Rev. Lett.} 83, 432).In this paper, the optimal information-disturbance tradeoff bound for antiparallel spins is derived. The quantum measurements which attain the optimal tradeoff bound are obtained. This result can be of practical relevance for posing some general limits on Eve's eavesdropping process. Finally, we also present a comparison between the bound for antiparallel spins and the bound for parallel spins.
We determine the optimal information--disturbance tradeoff for estimating an unknown spatial direction encoded in two antiparallel spins. Rotational covariance reduces the optimization over all instruments to a finite-dimensional Choi problem: a positive seed operator obeys one trace constraint for each irreducible sector of the input representation, while both the directional score and the operation fidelity are linear functionals of this seed. For two antiparallel spin-1/21/21/2 particles, whose physical representation decomposes as 0⊕10\oplus10⊕1, we derive the two-multiplier dual problem and characterize the optimal instrument from the kernel vectors of the dual slack operator. The optimal operation is a covariant filter with scalar--vector coherence and is generally not a convex interpolation between the identity channel and a measure-and-reprepare strategy. At maximum information we recover the Gisin--Popescu score, but the least disturbing output state is optimized independently, giving a smaller disturbance than both the parallel-spin benchmark and antiparallel measure-and-reprepare. We also formulate the parallel benchmark and, as a central extension of the method, treat antiparallel spin-coherent states of arbitrary spin jjj. In this case the signal coherently occupies all sectors ℓ=0,…,2j\ell=0,\ldots,2jℓ=0,…,2j of j⊗jj\otimes jj⊗j, the endpoint information is governed by nearest-neighbor sector coherences, and the endpoint disturbance is obtained from an explicit finite block-diagonal eigenvalue problem.
We revisit the issue of extracting information from parallel and antiparallel spins, as initiated by Gisin and Popescu [Phys. Rev. Lett. 83, 432 (1999)], from the viewpoint of parameter estimation. By comparing two fundamental figures of merit based on fidelity and quantum Fisher information for assessing information quality, we demonstrate that these criteria yield different strategies for optimally extracting spin information. A surprising observation is that while for a single spin, as well as for parallel spins, quantum Fisher information cannot be fully extracted by any uniform measurement, this is not the case for antiparallel spins. A simple uniform measurement fully extracting quantum Fisher information in antiparallel spins is identified. This reveals a significant feature of antiparallel spins from the perspective of Fisher information and provides an alternative illustration of the idea that antiparallel spins carry more information than parallel spins.
We consider a pair of antiparallel spins polarized in a random direction to encode quantum information. We wish to extract as much information as possible on the polarization direction attainable by an unentangled measurement, i.e., by a measurement whose outcomes are associated with product states. We develop analytically the upper bound 0.7935bits0.7935\phantom{\rule{0.3em}{0ex}}\text{bits}0.7935bits to the Shannon mutual information obtainable by an unentangled measurement, which is definitely less than the value 0.8664bits0.8664\phantom{\rule{0.3em}{0ex}}\text{bits}0.8664bits attained by an entangled measurement. This proves our main result, that not every ensemble of product states can be optimally distinguished by an unentangled measurement, if the measure of distinguishability is defined in the sense of Shannon. We also present results from numerical calculations and discuss briefly the case of parallel spins.
We give a complete analysis of covariant measurements on two spins. We consider the cases of two parallel and two antiparallel spins, and we consider both collective measurements on the two spins and measurements that require only local quantum operations and classical communication (LOCC). In all cases we obtain the optimal measurements for arbitrary fidelities. In particular, we show that if the aim is to determine as accurately as possible the direction in which the spins are pointing, it is best to carry out measurements on antiparallel spins (as already shown by Gisin and Popescu), second best to carry out measurements on parallel spins, and worst to be restricted to LOCC measurements. If the aim is to determine as accurately as possible a direction orthogonal to that in which the spins are pointing, it is best to carry out measurements on parallel spins, whereas measurements on antiparallel spins and LOCC measurements are both less suitable but equivalent.
Antiparallel spins are superior in orienteering to parallel spins. This intriguing phenomenon is tied to entanglement associated with quantum measurements rather than quantum states. Using photonic systems, we experimentally realize the optimal orienteering protocols based on parallel spins and antiparallel spins, respectively. The optimal entangling measurements for decoding the direction information from parallel spins and antiparallel spins are realized using photonic quantum walks, which is a useful idea that is of wide interest in quantum information processing and foundational studies. Our experiments clearly demonstrate the advantage of antiparallel spins over parallel spins in orienteering. In addition, entangling measurements can extract more information than local measurements even if no entanglement is present in the quantum states.