Full characterization of modular values for finite-dimensional systems

Ho, Le Bin; Imoto, Nobuyuki

doi:10.1016/j.physleta.2016.05.005

Cited by 18 publications

(22 citation statements)

References 30 publications

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“…r,w = [1 + tan θ ( √ 2 sin e jχ 2 + cos e jχ 1 )] −1 . The set of three states |ψ i , |ψ r and |ψ f is transformed to the specific set (16) by applying the unitary operator:…”

Section: Singularities In Weak Valuesmentioning

confidence: 99%

“…Using a symbolic computation package, it is straightforward to show that tan Ω = tan(Ω 1 + Ω 2 ) = (tan Ω 1 + tan Ω 2 )/(1 − tan Ω 1 tan Ω 2 ) and that the angles are defined in the proper quadrants. The values given above for Ω 1 and Ω 2 result directly from the definitions of 16)…”

Section: Appendix A3 Qubit Unitary Operatormentioning

confidence: 99%

“…Modular values form a similar class of complex numbers describing pre-and postselected measurements of the observableÂ. Modular values appear in quantum-gate type interactions [15,16,17] and are the counterparts of weak values for unitary operators:…”

Section: Introductionmentioning

confidence: 99%

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

Cormann¹,

Caudano²

2017

J. Phys. A: Math. Theor.

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Abstract. We express modular and weak values of observables of three-and higherlevel 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 preand postselected.arXiv:1612.07023v2 [quant-ph]

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“…r,w = [1 + tan θ ( √ 2 sin e jχ 2 + cos e jχ 1 )] −1 . The set of three states |ψ i , |ψ r and |ψ f is transformed to the specific set (16) by applying the unitary operator:…”

Section: Singularities In Weak Valuesmentioning

confidence: 99%

Section: Appendix A3 Qubit Unitary Operatormentioning

confidence: 99%

See 1 more Smart Citation

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

Cormann¹,

Caudano²

2017

J. Phys. A: Math. Theor.

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“…Here, the constant g can take any real value. Weak and modular values are related to each other through g [5,6]. Both of them are found to be useful in explaining many phenomena such as quantum paradoxes [7][8][9][10][11][12][13][14], quantum ergodicity [15], and many applications in amplification and precision metrology [16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Quantum weak and modular values in enlarged Hilbert spaces

Imoto

2018

Phys. Rev. A

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We introduce an enlarged state, which combines both pre-and post-selection states at a given time t in between the pre-and post-selection. Based on this form, quantum weak and modular values can be completely interpreted as expectation values of a linear combination of given operators in the enlarged Hilbert space. This formalism thus enables us to describe and measure the weak and modular values at any time dynamically. A protocol for implementing an enlarged Hamiltonian has also been proposed and applied to a simple example of a single spin under an external magnetic field. In addition, the time-dependent weak and modular values for pre-and post-selection density matrices mapping onto an enlarged density matrix are also discussed.

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“…In this Letter, we discuss modular-value-based measurements [29][30][31][32][33] with spin-j coherent pointers [34,35]. They are different from the weak-value-based ones and can allow arbitrary strength interactions between the pointers and sensors.…”

mentioning

confidence: 99%

Modular-value-based metrology with spin coherent pointers

Kondo

2019

Physics Letters A

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Modular values are quantities that described by pre-and postselected states of quantum systems like weak values but are different from them: The associated interaction is not necessary to be weak. We discuss an optimal modular-value-based measurement with a spin coherent pointer: A quantum system is exposed to a field in which strength is to be estimated through its modular value. We consider two cases, with a two-dimensional and a higher-dimensional pointer, and evaluate the quantum Fisher information. The modular-value-based measurement has no merit in the former case, while its sensitivity can be enhanced in the latter case. We also consider the pointer under a phase-flip error. Our study should motivate researchers to apply the modular-value-based measurements for quantum metrology.Introduction.-In the concept of quantum sensing, a physical quantity on a small scale can be measured indirectly via a quantum system, a quantum property, or a quantum phenomenon [1]. The principle of measurements is (i) preparing quantum systems, hereafter called sensors, of which number is L, (ii) exposing to a field of which strength is to be measured for a period of t, and (iii) obtaining a state change before and after the exposure. The change is a measure of the strength of the field. This procedure is repeated T /t times in a total measurement time T . If the sensors are independent, the effective total measurement number N is given as N = LT /t. For fixed T and t, the uncertainty of the estimation is proportional to 1/ √ L, which is known as the standard quantum limit or the shot noise limit [2][3][4][5]. However, if the uncertainty scales as 1/L then it is called the Heisenberg limit [6] which is a fundamental limit.

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Full characterization of modular values for finite-dimensional systems

Cited by 18 publications

References 30 publications

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

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

Quantum weak and modular values in enlarged Hilbert spaces

Modular-value-based metrology with spin coherent pointers

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