### 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.

Original language | English |
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Article number | 305302 |

Number of pages | 26 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 50 |

Issue number | 30 |

DOIs | |

Publication status | Published - 29 Jun 2017 |

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### Keywords

- geometric phase
- Majorana representation
- modular value
- three-box paradox
- weak measurement
- Weak value

### Cite this

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**Geometric description of modular and weak values in discrete quantum systems using the Majorana representation.** / Cormann, Mirko; Caudano, Yves.

Research output: Contribution to journal › Article

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

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 30

M1 - 305302

ER -