Weak versus approximate values in quantum state determination

Haapasalo, Erkka; Lahti, Pekka; Schultz, Jussi

doi:10.1103/physreva.84.052107

Cited by 47 publications

(56 citation statements)

References 29 publications

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“…These linear response formulas for the von Neumann measurement have also been obtained and discussed in the literature with varying degrees of generality and rigor (e.g., [1,3,12,13,15,[18][19][20][23][24][25][26][27]40]). However, our derivation has a conceptual advantage in that we see explicitly how the origins of the real and imaginary parts of the weak value differ with respect to the measurement ofÂ.…”

Section: Weak Valuementioning

confidence: 99%

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Significance of the imaginary part of the weak value

2012

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Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.

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Section: Weak Valuementioning

confidence: 99%

“…(See, also, [12,13,15,[18][19][20][21][23][24][25][26][27].) We can in such a case simplify the perturbative expansions (36) using the following identities and definitions:Â…”

Section: B Qubit Observablementioning

confidence: 99%

Significance of the imaginary part of the weak value

2012

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“…An alternative is our direct measurement method, which may have advantages over tomography, such as simplicity, versatility, and directness. A quantitative comparison of measures such as the signal to noise ratio, resolution, and fidelity, has not been undertaken but some limitations of the direct method have been identified in [4]. As compared to tomography, which works with mixed states, the most significant limitation of the direct measurement of the wavefunction is that it has only been shown to work with pure states.…”

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confidence: 99%

Procedure for Direct Measurement of General Quantum States Using Weak Measurement

2012

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Recent work [J.S. Lundeen et al. Nature, 474, 188 (2011)] directly measured the wavefunction by weakly measuring a variable followed by a normal (i.e. 'strong') measurement of the complementary variable. We generalize this method to mixed states by considering the weak measurement of various products of these observables, thereby providing the density matrix an operational definition in terms of a procedure for its direct measurement. The method only requires measurements in two bases and can be performed 'in situ', determining the quantum state without destroying it.PACS numbers: 03.65. Ta, 03.65.Wj, 42.50.Dv, The wavefunction Ψ is at the heart of quantum mechanics, yet its nature has been debated since its inception. It is typically relegated to being a calculational device for predicting measurement outcomes. Recently, Lundeen et al. proposed a simple and general operational definition of the wavefunction based on a method for its direct measurement: "it is the average result of a weak measurement of a variable followed by a strong measurement of the complementary variable [1,2]." By 'direct' it is meant that a value proportional to the wavefunction appears straight on the measurement apparatus itself without further complicated calculations or fitting. The 'wavefunction' referred to here is a special case of a general quantum state, known as a 'pure state.' The general case is represented by the density operator ρ, which can describe both pure and 'mixed' states. The latter incorporates both the effects of classical randomness (e.g., noise) and entanglement with other systems (e.g., decoherence). The density operator plays an important role in quantum statistics, quantum information, and the study of decoherence. Because of its generality and because it follows naturally from classical concepts of probability and measures, some consider ρ, rather than Ψ, to be the fundamental quantum state description. In this letter, we propose two methods to directly measure general quantum states, one of which directly gives the matrix elements of ρ.The standard method for experimentally determining the density operator is Quantum State Tomography [3]. In it, one makes a diverse set of measurements on an ensemble of identical systems and then determines the quantum state that is most compatible with the measurement results. An alternative is our direct measurement method, which may have advantages over tomography, such as simplicity, versatility, and directness. A quantitative comparison of measures such as the signal to noise ratio, resolution, and fidelity, has not been undertaken but some limitations of the direct method have been identified in [4]. As compared to tomography, which works with mixed states, the most significant limitation of the direct measurement of the wavefunction is that it has only been shown to work with pure states.Previous works have developed direct methods to measure quasi-probability distributions, such as the Wigner function [5], Husimi Q-function [6], and the GlauberSudarshan P-function [7]...

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“…is the generalised weak value of the Hamiltonian H S (0), given the initial state ρ, and postselected by outcome x of the POVM M [30][31][32][33].…”

Section: Energy Change Conditional On Measurement Outcomementioning

confidence: 99%

Conditional work statistics of quantum measurements

Mohammady

Romito

2019

Quantum

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In this paper we introduce a definition for conditional energy changes due to general quantum measurements, as the change in the conditional energy evaluated before, and after, the measurement process. By imposing minimal physical requirements on these conditional energies, we show that the most general expression for the conditional energy after the measurement is simply the expected value of the Hamiltonian given the post-measurement state. Conversely, the conditional energy before the measurement process is shown to be given by the real component of the weak value of the Hamiltonian. Our definition generalises well-known notions of distributions of internal energy change, such as that given by stochastic thermodynamics. By determining the conditional energy change of both system and measurement apparatus, we obtain the full conditional work statistics of quantum measurements, and show that this vanishes for all measurement outcomes if the measurement process conserves the total energy. Additionally, by incorporating the measurement process within a cyclic heat engine, we quantify the non-recoverable work due to measurements. This is shown to always be nonnegative, thus satisfying the second law, and will be independent of the apparatus specifics for two classes of projective measurements.Accepted in Quantum 2019-07-30, click title to verify. Published under CC-BY 4.0. arXiv:1809.09010v6 [quant-ph]

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Weak versus approximate values in quantum state determination

Cited by 47 publications

References 29 publications

Significance of the imaginary part of the weak value

Significance of the imaginary part of the weak value

Procedure for Direct Measurement of General Quantum States Using Weak Measurement

Conditional work statistics of quantum measurements

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