arXiv subjects

P. K. Aravind

Publications and source records attributed to P. K. Aravind.

31 records · Page 2Linked to original sources

arXiv2010

Isomorphism between the Peres and Penrose proofs of the BKS theorem in three dimensions

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.

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arXiv2010

Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem

Aravind and Lee-Elkin (1997) gave a proof of the Bell-Kochen-Specker theorem by showing that it is impossible to color the 60 directions from the center of a 600-cell to its vertices in a certain way. This paper refines that result by showing that the 60 directions contain many subsets of 36 and 30 directions that cannot be similarly colored, and so provide more economical demonstrations of the theorem. Further, these subsets are shown to be critical in the sense that deleting even a single direction from any of them causes the proof to fail. The critical sets of size 36 and 30 are shown to belong to orbits of 200 and 240 members, respectively, under the symmetries of the polytope. A comparison is made between these critical sets and other such sets in four dimensions, and the significance of these results is discussed.

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arXiv2007

Variation on a theme by GHZM

In the Greenberger-Horne-Zeilinger-Mermin (GHZM) proof of Bell's theorem, a source periodically emits an entangled state of three particles whose properties are analyzed by three distant observers and used to prove Bell's nonlocality theorem. This paper analyzes a somewhat different gedanken experiment involving only two observers that nevertheless makes indirect use of the GHZ states to prove Bell's theorem. The relationship of the GHZM proof to the present one is discussed, and it is pointed out that the latter provides an interesting new view of the connection between the "two theorems of John Bell".

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arXiv2005

Quantum Kaleidoscopes and Bell's theorem

A quantum kaleidoscope is defined as a set of observables, or states, consisting of many different subsets that provide closely related proofs of the Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes prove the BKS theorem through a simple parity argument, which also doubles as a proof of Bell's nonlocality theorem if use is made of the right sort of entanglement. Three closely related kaleidoscopes are introduced and discussed in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a 60-state kaleidoscope. The close relationship of these kaleidoscopes to a configuration of 12 points and 16 lines known as Reye's configuration is pointed out. The "rotations" needed to make each kaleidoscope yield all its apparitions are laid out. The 60-state kaleidoscope, whose underlying geometrical structure is that of ten interlinked Reye's configurations (together with their duals), possesses a total of 1120 apparitions that provide proofs of the two Bell theorems. Some applications of these kaleidoscopes to problems in quantum tomography and quantum state estimation are discussed.

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arXiv2005

Decoherence of a Measure of Entanglement

We demonstrate by an explicit model calculation that the decay of entanglement of two two-state systems (two qubits) is governed by the product of the factors that measure the degree of decoherence of each of the qubits, subject to independent sources of quantum noise. This demonstrates an important physical property that separated open quantum systems can evolve quantum mechanically on time scales larger than the times for which they remain entangled.

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arXiv2004

Best conventional solutions to the King's Problem

In the King's Problem, a physicist is asked to prepare a d-state quantum system in any state of her choosing and give it to a king who measures one of (d+1) sets of mutually unbiased observables on it. The physicist is then allowed to make a control measurement on the system, following which the king reveals which set of observables he measured and challenges the physicist to predict correctly all the eigenvalues he found. This paper obtains an upper bound on the physicist's probability of success at this task if she is allowed to make measurements only on the system itself (the "conventional" solution) and not on the system as well as any ancillary systems it may have been coupled to in the preparation phase, as in the perfect solutions proposed recently. An optimal conventional solution, with a success probability of 0.7, is constructed in d = 4; this is to be contrasted with the success probability of 0.902 for the optimal conventional solution in d = 2. The gap between the best conventional solution and the perfect solution grows quite rapidly with increasing d.

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arXiv2003

The generalized Kochen-Specker theorem

A proof of the generalized Kochen-Specker theorem in two dimensions due to Cabello and Nakamura is extended to all higher dimensions. A set of 18 states in four dimensions is used to give closely related proofs of the generalized Kochen-Specker, Kochen-Specker and Bell theorems that shed some light on the relationship between these three theorems.

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arXiv2003

A simple demonstration of Bell's theorem involving two observers and no probabilities or inequalities

This paper describes a device, consisting of a central source and two widely separated detectors with six switch settings each, that provides a simple gedanken demonstration of Bell's theorem without relying on either statistical effects or the occurrence of rare events. The mechanism underlying the operation of the device is revealed for readers with a knowledge of quantum mechanics.

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arXiv2002

Solution to the King's Problem in prime power dimensions

It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1-5 (2001)].

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arXiv2002

Bell's theorem without inequalities and only two distant observers

A proof of Bell's theorem without inequalities and involving only two observers is given by suitably extending a proof of the Bell-Kochen-Specker theorem due to Mermin. This proof is generalized to obtain an inequality-free proof of Bell's theorem for a set of n Bell states (with n odd) shared between two distant observers. A generalized CHSH inequality is formulated for n Bell states shared symmetrically between two observers and it is shown that quantum mechanics violates this inequality by an amount that grows exponentially with increasing n.

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arXiv2002

A pseudo-simulation of Shor's quantum factoring algorithm

This paper presents a computer program, written in Maple, that allows a user to simulate certain aspects of Shor's quantum factoring algorithm on a desktop or laptop computer. The program does not simulate the unitary operations carried out by a quantum computer but does faithfully mimic its output at the crucial "readout" step of the order-finding process. The program reqires only two inputs from the user - the number to be factored (which can be up to 10 digits long) and the number of qubits to be used in the factoring (for which a helpful hint is given). The program then returns a detailed history of all its attempts at factoring the number, beginning with its various unsuccessful attempts and ending with the final successful attempt that leads to the correct factors. The structure of the simulation is described, a typical output produced by it is shown, and the factors limiting its performance are discussed. The purpose of this simulation is to provide the user with some "hands-on" experience of how quantum factoring works on integers somewhat larger than can be handled by today's quantum computers.

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arXiv2001

Optimal quantum measurements for spin-1 and spin-3/2 particles

Positive operator valued measures (POVMs) are presented that allow an unknown pure state of a spin-1 particle to be determined with optimal fidelity when 2 to 5 copies of that state are available. Optimal POVMs are also presented for a spin-3/2 particle when 2 or 3 copies of the unknown state are available. Although these POVMs are optimal they are not always minimal, indicating that there is room for improvement.

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arXiv2001

Mermin's pentagram and Bell's theorem

It is shown how a proof of the Bell-Kochen-Specker (BKS) theorem given by Kernaghan and Peres can be experimentally realized using a scheme of measurements derived from a related proof of the same theorem by Mermin. It is also pointed out that if this BKS experiment is carried out independently by two distant observers who repeatedly make measurements on a specially correlated state of six qubits, it provides an inequality-free demonstration of Bell's theorem as well.

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