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

Abstract

Weak measurements performed between quantum state preparation and post-selection result in complex values for self-adjoint operators, corresponding to complex conditional probabilities for the projections on specific eigenstates. In this paper, it is shown that the complex phases of these weak conditional probabilities describe the dynamic response of the system to unitary transformations. Quantum mechanics thus unifies the statistical overlap of different states with the dynamical structure of transformations between these states. Specifically, it is possible to identify the phase of weak conditional probabilities directly with the action of a unitary transform that maximizes the overlap of initial and final states. This action provides a quantitative measure of how much quantum correlations can diverge from the deterministic relations between physical properties expected from classical physics or hidden variable theories. In terms of quantum information, the phases of weak conditional probabilities thus represent the logical tension between sets of three quantum states that is at the heart of quantum paradoxes.

16 pages, including 2 figures, added explanations on physics and relevance for recent experiments, final version for publication in NJP

References in corpus (11)

• Direct Measurement of the Quantum Wavefunction
• Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox
• Complex weak values in quantum measurement
• Weak values and the Leggett-Garg inequality in solid-state qubits
• Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair
• A double-slit `which-way' experiment on the complementarity--uncertainty debate
• Strange Weak Values
• Pre- and post-selection, weak values, and contextuality
• Uncertainty limits for quantum metrology obtained from the statistics of weak measurements
• Weak measurement of photon polarization by back-action induced path interference
• Geometrical aspects of weak measurements and quantum erasers

Cited by in corpus (34)

• Direct measurement of general quantum states using weak measurement
• Negative quasiprobabilities enhance phase estimation in quantum-optics experiment
• Complex joint probabilities as expressions of determinism in quantum mechanics
• Kirkwood-Dirac quasiprobability approach to the statistics of incompatible observables
• Conditions tighter than noncommutation needed for nonclassicality
• How weak values emerge in joint measurements on cloned quantum systems
• Weak values are universal in von Neumann measurements
• Quasiprobabilities in quantum thermodynamics and many-body systems
• Classical Field Approach to Quantum Weak Measurements
• Sequential measurements of non-commuting observables with quantum controlled interactions
• Derivation of quantum mechanics from a single fundamental modification of the relations between physical properties
• How quantum paradoxes originate from the non-classical statistics of physical properties related to each other by half-periodic transformations
• Observing the "quantum Cheshire cat" effect with noninvasive weak measurement
• Non-Abelian transport distinguishes three usually equivalent notions of entropy production
• On the fundamental role of dynamics in quantum physics
• Observation of non-classical correlations in sequential measurements of photon polarization
• Contextuality, Coherences, and Quantum Cheshire Cats
• Direct observation of temporal coherence by weak projective measurements of photon arrival time
• Reasonable conditions for joint probabilities of non-commuting observables
• Weak values in a classical theory with an epistemic restriction
• On the relation between measurement outcomes and physical properties
• Kirkwood-Dirac classical pure states
• Advantages of the Kirkwood-Dirac distribution among general quasi-probabilities for finite-state quantum systems
• Dependence of measurement outcomes on the dynamics of quantum coherent interactions between the system and the meter
• Snapshotting Quantum Dynamics at Multiple Time Points
• Complex counterpart of variance in quantum measurements for pre- and post-selected systems
• Uncertainty relations for the support of quantum states
• Derivation of the statistics of quantum measurements from the action of unitary dynamics
• Quantum causality relations and the emergence of reality from coherent superpositions
• What does the operator algebra of quantum statistics tell us about the objective causes of observable effects?
• Statistical signatures of quantum contextuality
• Weak value as an indicator of back-action
• Quantum coherence and negative quasi probabilities in a contextual three-path interferometer
• Quantum states as complex probabilities: The physics behind direct observations of photon wavefunctions in weak measurements