Modular Values and Weak Values of Quantum Observables

Kedem, Yaron; Vaidman, Lev

doi:10.1103/physrevlett.105.230401

Cited by 76 publications

(102 citation statements)

References 24 publications

(62 reference statements)

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“…This shows the equivalence of modular and weak values ofσ n (see also [27]). We can thus apply our scheme to determine an arbitrary weak value of the Pauli operator in its polar representation.…”

supporting

confidence: 54%

“…Nevertheless, they characterize all projective couplings between the probe and meter systems, where they generalize weak values in a non-perturbative way. They typically describe quantumgate type interactions [27] and quantum interference experiments [28][29][30], but appear also in photon trajectory measurements [17] for example. In the following, we relate the physical interpretation of modular values to the visibility and the phase of interferometric experiments.…”

mentioning

confidence: 99%

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Revealing geometric phases in modular and weak values with a quantum eraser

et al. 2016

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In this letter, we present a new procedure to determine completely the complex modular values of arbitrary observables of pre-and post-selected ensembles, which works experimentally for all measurement strengths and all post-selected states. This procedure allows us to discuss the physics of modular and weak values in interferometric experiments involving a qubit meter. We determine both the modulus and the argument of the modular value for any measurement strength in a single step, by controlling simultaneously the visibility and the phase in a quantum eraser interference experiment. Modular and weak values are closely related. Using entangled qubits for the probed and meter systems, we show that the phase of the modular and weak values has a topological origin. This phase is completely defined by the intrinsic physical properties of the probed system and its time evolution. The physical significance of this phase can thus be used to evaluate the quantumness of weak values.In 1988, Aharonov, Albert, and Vaidman (AAV) introduced the weak value of a quantum observableÂ from an extension of the von Neumann measurement scheme [1]. They pointed out that the result of a measurement involving a weak coupling between a meter and the observableÂ of a system with a pre-selected initial state |ψ i , and a post-selected final state |ψ f depends directly on the weak value:an unbounded complex number. In particular, they showed that the shift of the average detected position due to post-selection is proportional to the real part of the weak value. Since for weak measurements in the absence of post-selection, this shift is proportional to the average of the observable ψ i |Â|ψ i / ψ i |ψ i , a direct but bold physical interpretation of the weak value assumes it represents somehow the average ofÂ in the pre-and post-selected ensemble. They also related the imaginary part of the weak value to the shift of the average impulsion. Beside the AAV approach, weak values may also appear using a meter strongly coupled to the observablê A [2][3][4][5][6][7]. In these instances, the effective weak interaction is achieved by selecting particular initial states of the meter system, so that the probability of actually measurinĝ A is low and the probed system is left unperturbed most of the time. Therefore, both methods transform the standard von Neumann procedure to a weak measurement with a high incertitude. Weak values and weak measurements proved useful in many fields of physics and chemistry [8][9][10][11][12][13][14][15][16][17][18][19][20]. Nevertheless, the proper physical interpretation of weak values remains highly debated. For example, weak values were used to develop a time-symmetrized approach to standard quantum theory, the two-state vector formalism [21], where they appear as purely quantum objects.Oppositely, a purely classical view of the occurrence of unbounded, real weak values -and possibly of complex ones -was proposed recently [22] (which is criticizable though [23-25]).In this letter, we uncover a physical interpretati...

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supporting

confidence: 54%

mentioning

confidence: 99%

Revealing geometric phases in modular and weak values with a quantum eraser

et al. 2016

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“…Y. Kedem and L. Vaidman, however, recently considered the interaction between a system and a meter qubit [15], where the system (not necessary be a qubit but could be in a higher dimensional Hilbert space) is conditioned by an initial-and a final-state vectors |ψ and |φ [16], and the state of the meter qubit is initially prepared to be γ|0 m +γ|1 m (γ andγ are real numbers satisfying γ 2 +γ 2 = 1), withγ ≪ 1. The interaction Hamiltonian is written aŝ H = g(t)ÂΠ m , t t0 g(t)dt = gδ(t − t 0 ) .…”

Section: Introductionmentioning

confidence: 99%

“…Let us give an example of a spin operatorsσ x ,σ y andσ z and coupling constant g = − π 2 . We have [15]:…”

Section: Introductionmentioning

confidence: 99%

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Full characterization of modular values for finite-dimensional systems

Imoto

2016

Physics Letters A

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Vaidman pointed out the importance of modular values, and related the modular value of a Pauli spin operator to its weak value for specific coupling strengths [Phys. Rev. Lett. 105, 230401 (2010)]. It would be useful if this relationship is generalized since a modular value, which assumes a finite strength of the measurement interaction, is sometimes more practical than a weak value, which assumes an infinitesimally small interaction. In this paper, we give a general expression that relates the weak value and the modular value of an arbitrary observable in the 2-dimensional Hilbert space for an arbitrary coupling strength. Using this expression, we show the "failure of sum rule" for modular values, which has a resemblance to the "failure of product rule" for weak values. We give examples of "failure of sum rule" for some interesting cases, i.e., paradoxes based on nonlocality, which include EPR paradox, Hardy's paradox, and Cheshire cat experiment.

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Materials Innovations for Quantum Technology Acceleration: A Perspective

et al. 2023

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A broad perspective of quantum technology state of the art is provided and critical stumbling blocks for quantum technology development are identified. Innovations in demonstrating and understanding electron entanglement phenomena using bulk and low‐dimensional materials and structures are summarized. Correlated photon‐pair generation via processes such as nonlinear optics is discussed. Application of qubits to current and future high‐impact quantum technology development is presented. Approaches for realizing unique qubit features for large‐scale encrypted communication, sensing, computing, and other technologies are still evolving; thus, materials innovation is crucially important. A perspective on materials modeling approaches for quantum technology acceleration that incorporate physics‐based AI/ML, integrated with quantum metrology is discussed.

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Modular Values and Weak Values of Quantum Observables

Cited by 76 publications

References 24 publications

Revealing geometric phases in modular and weak values with a quantum eraser

Revealing geometric phases in modular and weak values with a quantum eraser

Full characterization of modular values for finite-dimensional systems

Materials Innovations for Quantum Technology Acceleration: A Perspective

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