On the role of complex phases in the quantum statistics of weak measurements

Quantum Stud.: Math. Found.

2017

One of the most difficult problems in quantum mechanics is the analysis of the measurement processes. In this paper, we point out that many of these difficulties originate from the different roles of measurement outcomes and observable quantities, which cannot simply be identified with each other. Our analysis shows that the Hilbert space formalism itself describes a fundamental separation between quantitative properties and qualitative outcomes that needs to be taken into account in an objective description of quantum measurements. We derive fundamental relations between the statistics of measurement outcomes and the values of physical quantities that explain how the objective properties of a quantum system appear in the context of different measurement interactions. Our results indicate that non-classical correlations can be understood in terms of the actual role of physical properties as quantifiable causes of the external effects observed in a quantum measurement.

Section: Error Free Measurementmentioning

confidence: 99%

On the relation between measurement outcomes and physical properties

Nii

Iinuma

Quantum Stud.: Math. Found.

2017

“…The consistency of these results strongly suggests that the statistics of weak measurements is a fundamental element of the Hilbert space formalism. In particular, it is possible to develop a consistent explanation for weak measurement statistics in terms of complex conditional and joint probabilities [22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

“…In the following, I will take a closer look at this relation between different joint probabilities. It is shown that deterministic transformations are described by the complex conditional probabilities p(c|a, b) that also characterize weak measurement statistics [25]. Reversibility of the transformation requires that the information about a can be recovered completely from the information about c after the transformation.…”

Section: Introductionmentioning

confidence: 99%

Complex joint probabilities as expressions of reversible transformations in quantum mechanics

2012

New J. Phys.

Self Cite

149

The density operator of a quantum state can be represented as a complex joint probability of any two observables whose eigenstates have nonzero mutual overlap. Transformations to a new basis set are then expressed in terms of complex conditional probabilities that describe the fundamental relation between precise statements about the three different observables. Since such transformations merely change the representation of the quantum state, these conditional probabilities provide a state-independent definition of the reversible and therefore effectively deterministic relations between the outcomes of different quantum measurements, including measurements of the same property performed at different times. In this paper, it is shown how classical reality emerges as an approximation to the fundamental laws of quantum determinism expressed by complex conditional probabilities. The quantum mechanical origin of phase spaces and trajectories is identified and implications for the interpretation of quantum measurements are considered. It is argued that the transformation laws of quantum determinism provide a fundamental description of the measurement dependence of empirical reality.

“…It may be worth noting that dispersion effects appear as phase shifts in the frequency representation of the photon state, so that the above result can provide a prescription for the active compensation of unintended dispersion effects. In particular, it is possible to compress the pulse, corresponding to an optimization of the overlap between the photon state and the short-time pulse at t by a unitary transform conserving the frequency ω [36].…”

Section: Wavefunction Measurement and Evaluation Of Entanglementmentioning

confidence: 99%

Direct observation of temporal coherence by weak projective measurements of photon arrival time

Ren

2013

Phys. Rev. A

We show that a weak projective measurement of photon arrival time can be realized by controllable two photon interferences with photons from short-time reference pulses at a polarization beam splitter. The weak value of the projector on the arrival time defined by the reference pulse can be obtained from the coincidence rates conditioned by a specific output measurement. If the weak measurement is followed by a measurement of frequency, the coincidence counts reveal the complete temporal coherence of the single photon wavefunction. Significantly, the weak values of the input state can also be obtained at higher measurement strengths, so that correlations between weak measurements on separate photons can be observed and evaluated without difficulty. The method can thus be used to directly observe the non-classical statistics of time-energy entangled photons.