Geometric description of modular and weak values in discrete quantum systems using the Majorana representation

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Résumé

We express modular and weak values of observables of three- and higher- level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N−1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N−1 contributions. Their modulus is determined by the product of N−1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N−1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three box paradox [1] from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre-and postselected.
langueAnglais
Numéro d'article305302
Nombre de pages26
journalJournal of Physics A: Mathematical and Theoretical
Volume50
Les DOIs
étatPublié - 29 juin 2017

Empreinte digitale

Discrete Systems
Quantum Systems
Paradox
Qubit
paradoxes
Polar Form
Solid angle
Geometric Approach
Entanglement
Observer
Discontinuity
Modulus
Express
Projection
Singularity
boxes
discontinuity
projection
products

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title = "Geometric description of modular and weak values in discrete quantum systems using the Majorana representation",
abstract = "We express modular and weak values of observables of three- and higher- level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N−1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N−1 contributions. Their modulus is determined by the product of N−1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N−1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three box paradox [1] from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre-and postselected.",
author = "Mirko Cormann and Yves Caudano",
year = "2017",
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doi = "10.1088/1751-8121/aa7639",
volume = "50",
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TY - JOUR

T1 - Geometric description of modular and weak values in discrete quantum systems using the Majorana representation

AU - Cormann,Mirko

AU - Caudano,Yves

PY - 2017/6/29

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N2 - We express modular and weak values of observables of three- and higher- level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N−1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N−1 contributions. Their modulus is determined by the product of N−1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N−1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three box paradox [1] from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre-and postselected.

AB - We express modular and weak values of observables of three- and higher- level quantum systems in their polar form. The Majorana representation of N-level systems in terms of symmetric states of N−1 qubits provides us with a description on the Bloch sphere. With this geometric approach, we find that modular and weak values of observables of N-level quantum systems can be factored in N−1 contributions. Their modulus is determined by the product of N−1 ratios involving projection probabilities between qubits, while their argument is deduced from a sum of N−1 solid angles on the Bloch sphere. These theoretical results allow us to study the geometric origin of the quantum phase discontinuity around singularities of weak values in three-level systems. We also analyze the three box paradox [1] from the point of view of a bipartite quantum system. In the Majorana representation of this paradox, an observer comes to opposite conclusions about the entanglement state of the particles that were successfully pre-and postselected.

UR - https://arxiv.org/abs/1612.07023v2

U2 - 10.1088/1751-8121/aa7639

DO - 10.1088/1751-8121/aa7639

M3 - Article

VL - 50

JO - Journal of Physics A: Mathematical and Theoretical

T2 - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

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