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Physical Review A

Optimal extraction of information from two spins

Lina Chang, Nan Li, Shunlong Luo, Hongting Song

Published Apr 18, 2014• 15 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.89.042110

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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 investigate the properties of two particular sets of separable two-qubit states with maximally mixed marginals, named parallel states and antiparallel states respectively. Although these two sets of states are prepared by the same type of quantum operations acting on classically correlated states with equal classical correlations, they share different properties. By comparing two fundamental figures of merit based on quantum correlations and quantum Fisher information for accessing information quality, the superiority of parallel states over antiparallel ones is investigated. This is different from the previous results that antiparallel state turns out to be more effective in the tasks such as sending unknown direction information. The amount of the information that can be extracted from the local measurements is also presented, which turns out to be the same for these two states. This work reveals new aspects of information encoding and extraction problem.
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 show that the Bloch vectors lying on any great circle comprise the largest set SL{S}_{L}SL​ for which the parallel states $|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉$ can always be exactly transformed into the antiparallel states $|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉.$ Thus more information about $\stackrel{\ensuremath{\rightarrow}}{n}$ is not extractable from $|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉$ than from $|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉$ by any measuring strategy, for $\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\in}{S}_{L}.$ Surprisingly this most general transformation reduces to just a flip operation on the second particle. We also show here that a probabilistic exact parallel to antiparallel transformation is not possible if the corresponding antiparallel states span the whole Hilbert space of the two qubits. These considerations allow us to generalize a conjecture of Gisin and Popescu [Phys. Rev. Lett. 83,83,83, 432 (1999)].
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.
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.
Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information about their direction can be obtained from a pair of anti-parallel spins compared to a pair of parallel spins, where the first member of the pair (which we call the pointer member) can point equally along any direction in the Bloch sphere. They argued that this was due to the difference in dimensionality spanned by these two alphabets of states. Here we consider similar alphabets, but with the first spin restricted to a fixed small circle of the Bloch sphere. In this case, the dimensionality spanned by the anti-parallel versus parallel alphabet is now equal. However, the anti-parallel alphabet is found to still contain more information in general. We generalize this to having N parallel spins and M anti-parallel spins. When the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension, yet in general, more directional information can be found for sets with smaller |N-M| for any fixed total number of spins. We find that the optimal POVMs for extracting directional information in these cases can always be expressed in terms of the Fourier basis. Our results show that dimensionality alone cannot explain the greater information content in anti-parallel combinations of spins compared to parallel combinations. In addition, we describe an LOCC protocol which extract optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied.