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 (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen. 24 2315-28) 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
Numéro30
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 (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen. 24 2315-28) 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.",
    keywords = "geometric phase, Majorana representation, modular value, three-box paradox, weak measurement, Weak value",
    author = "Mirko Cormann and Yves Caudano",
    year = "2017",
    month = "6",
    day = "29",
    doi = "10.1088/1751-8121/aa7639",
    language = "English",
    volume = "50",
    journal = "Journal of Physics A: Mathematical and Theoretical",
    issn = "1751-8113",
    publisher = "IOP Publishing Ltd.",
    number = "30",

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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

    Y1 - 2017/6/29

    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 (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen. 24 2315-28) 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 (Aharonov and Vaidman 1991 J. Phys. A: Math. Gen. 24 2315-28) 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.

    KW - geometric phase

    KW - Majorana representation

    KW - modular value

    KW - three-box paradox

    KW - weak measurement

    KW - Weak value

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

    UR - http://www.scopus.com/inward/record.url?scp=85022021507&partnerID=8YFLogxK

    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

    SN - 1751-8113

    IS - 30

    M1 - 305302

    ER -