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arXiv:quant-ph/0412101 · Journal of Physics A Mathematical and Theoretical

Estimation of pure qubits on circles

Samuel L. Braunstein, Sibasish Ghosh A, Simone Severini B

Published Feb 6, 2007• 5 citations

Abstract

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.

DOI: 10.1088/1751-8113/40/8/009

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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.

We show that the Bloch vectors lying on any great circle is the largest set S(L) for which the parallel states |n, n> can always be exactly transformed into the anti-parallel states |n, -n>. Thus more information about the direction of the Bloch vector "n" is not extractable from |n, -n> than from |n, n> by any measuring strategy, for "n" belonging to S(L). Surprisingly, the largest set of Bloch vectors for which the corresponding qubits can be flipped is again S(L). We then show that probabilistic exact parallel to anti-parallel transformation is not possible if the corresponding anti-parallels 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, 432 (1999)).

We show that the Bloch vectors lying on any great circle comprise the largest set

{S}_{L}

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,

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-

1/2

particles, whose physical representation decomposes as

0\oplus1

, 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

83,

. In this case the signal coherently occupies all sectors

\ell=0,\ldots,2j

of

j\otimes 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.

While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (

d>2

) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.