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Does Anti-Parallel Spin contain more Information ?

Sibasish Ghosh A, Anirban Roy, Ujjwal Sen

Published Apr 18, 2000• 0 citations
OpenAlex

Abstract

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)).

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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 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)].
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 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 study the problem of information masking through nonzero linear operators that distribute information encoded in single qubits to the correlations between two qubits. It is shown that a nonzero linear operator cannot mask any nonzero measure set of qubit states. We prove that the maximal maskable set of states on the Bloch sphere with respect to any masker is the ones on a spherical circle. Any states on a spherical circle on the Bloch sphere are maskable, which also proves the conjecture on maskable qubit states given by Modi et al. [Phys. Rev. Lett. 120, 230501 (2018)]. we provide explicitly operational unitary maskers for all maskable sets. As applications, different protocols for secret sharing are introduced.
An ability to describe quantum states directly by average values of measurement outcomes is provided by the Bloch vector. For an informationally complete set of measurements one can construct unique Bloch vector for any quantum state. However, not every Bloch vector corresponds to a quantum state. It seems that only for two dimensional quantum systems it is easy to distinguish proper Bloch vectors from improper ones, i.e. the ones corresponding to quantum states from the other ones. I propose an alternative approach to the problem in which more than one vector is used. In particular, I show that a state of the qutrit can be described by the three qubit-like Bloch vectors.
An ability to describe quantum states directly by average values of measurement outcomes is provided by the Bloch vector. For an informationally complete set of measurements one can construct unique Bloch vector for any quantum state. However, not every Bloch vector corresponds to a quantum state. It seems that only for two-dimensional quantum systems it is easy to distinguish proper Bloch vectors from improper ones, i.e. the ones corresponding to quantum states from the other ones. I propose an alternative approach to the problem in which more than one vector is used. In particular, I show that a state of the qutrit can be described by the three qubit-like Bloch vectors.