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arXiv:quant-ph/0004071 · Physical Review A

Antiparallel spin does not always contain more information

Sibasish Ghosh A, Anirban Roy, Ujjwal Sen

Published Dec 8, 2000• 12 citations
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Abstract

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)].

DOI
10.1103/physreva.63.014301

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We show that the Bloch vectors lying on any great circle comprise the largest set SL{S}_{L}SL​ for which the parallel states ∣n\ensuremath→,n\ensuremath→〉|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉∣n\ensuremath→,n\ensuremath→〉 can always be exactly transformed into the antiparallel states ∣n\ensuremath→,\ensuremath−n\ensuremath→〉.|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉.∣n\ensuremath→,\ensuremath−n\ensuremath→〉. Thus more information about n\ensuremath→\stackrel{\ensuremath{\rightarrow}}{n}n\ensuremath→ is not extractable from ∣n\ensuremath→,\ensuremath−n\ensuremath→〉|\stackrel{\ensuremath{\rightarrow}}{n},\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{n}〉∣n\ensuremath→,\ensuremath−n\ensuremath→〉 than from ∣n\ensuremath→,n\ensuremath→〉|\stackrel{\ensuremath{\rightarrow}}{n},\stackrel{\ensuremath{\rightarrow}}{n}〉∣n\ensuremath→,n\ensuremath→〉 by any measuring strategy, for n\ensuremath→\ensuremath∈SL.\stackrel{\ensuremath{\rightarrow}}{n}\ensuremath{\in}{S}_{L}.n\ensuremath→\ensuremath∈SL​. 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 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)).
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 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.
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 explore the joint measurability of incompatible qubit observables on ensembles of parallel and antiparallel spin-1/2 pairs. In parallel configuration, both spins are prepared in the same state, whereas in antiparallel case, each spin is paired with its flipped counterpart. We show that the antiparallel configuration uniquely enables exact simultaneous prediction of three mutually orthogonal spin components-an advantage not achievable with parallel states. Extending beyond three observables, we examine joint measurability for larger sets of spin measurements and further generalize our analysis to state configurations beyond the parallel and antiparallel cases. As we show, our results reveal a deep connection to the "mean King retrodiction task" proposed by Vaidman, Aharonov, and Albert and have implications for a cryptographic protocol introduced by Jeffrey Bub. We further demonstrate how the enhanced compatibility in the antiparallel configuration can facilitate efficient estimation of unknown measurement devices. Finally, we discuss prospects for experimentally realizing the enhanced measurement compatibility in antiparallel configuration by analyzing the effect on finite subensembles of states.