Wigner–Weyl correspondence in quantum mechanics for continuous and discrete systems—a Dirac-inspired view

Chaturvedi, S.; Ercolessi, Elisa; Marmo, Giuseppe; Morandi, G.; Mukunda, N.; Simon, Richard

doi:10.1088/0305-4470/39/6/014

Cited by 77 publications

(94 citation statements)

References 60 publications

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“…In various guises it has been investigated periodically during last half-century [38]. In optics, variations of the Dirac distribution have been used widely, appearing in Walther's definition of the radiance function in radiometry [39] and Wolf's specific intensity [40] (as pointed out in [36]). If O = ρ, the Dirac distribution is a representation of the quantum state of a system.…”

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confidence: 99%

“…For instance, the joint weak measurement of a position x and a momentum p (i.e. S xp ≡ p p|x x ) on a mixed state ρ gives the phase-space version of the Dirac distribution, S ρ (x, p), which, although it is complex, shares many of the desired features of a quasi-probability distribution [36]. In our weak measurement, if one scans a and b, so as to directly measure the Dirac distribution over all values of (a, b), one completely determines the density operator.…”

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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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Procedure for Direct Measurement of General Quantum States Using Weak Measurement

2012

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“…However, here the parameter s takes only discrete values (s = −1, 0, 1). These kernels are normalized and covariant under transformations of the generalized Pauli group They can be then conveniently represented aŝ 24) which is invertible, so that…”

Section: Quasidistribution Functionsmentioning

confidence: 99%

“…The transition operator (4.17) turns out to bê 24) which is nothing but a matrix representation of the CNOT operator.…”

Section: ) Andmentioning

confidence: 99%

Discrete coherent and squeezed states of many-qudit systems

2009

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We consider the phase space for a system of n identical qudits (each one of dimension d, with d a primer number) as a grid of d n × d n points and use the finite field GF(d n ) to label the corresponding axes. The associated displacement operators permit to define s-parametrized quasidistribution functions in this grid, with properties analogous to their continuous counterparts. These displacements allow also for the construction of finite coherent states, once a fiducial state is fixed. We take this reference as one eigenstate of the discrete Fourier transform and study the factorization properties of the resulting coherent states. We extend these ideas to include discrete squeezed states, and show their intriguing relation with entangled states between different qudits.

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“…Without elaborating much on these aspects (we refer to [14] for details) we simply state that the ⋆-product we have defined, when considered on phase space, becomes the standard Moyal product.…”

Section: Ehrenfest Formalismmentioning

confidence: 99%