Research
Explantion of quantum phenomena
☆Complex probabilities
In
conventional quantum mechanics, a quantum state can be described by
a
state vector or a density matrix, but these are merely mathematical
expressions. What physics is hidden behind these mathematical
expressions? In experiment, we do not see "superpositions" or "
interferences" between alternate realities. What we do see is the
statistics of outcomes, where the outcomes represent the actual
physical propeties of the system. The quantum formalism must
therefore
be understood as a statistical theory that describes the relation
between physical properties in terms of non-classical correlations.
Within the formalism, the different physical properties are also
connected by
unitary transformations, which often represent physically meaningful
implementations of reversible transformations, such as rotations or
accellerations. In the conventional operator formalism, the relation
between statistics and transformations is given by the expectation
values of commutation relations. Recent developments in weak
measurements show that non-commutativity is not just an abstract
concept.
In weak
measurements, it is possible to obtain the imaginary part of the
weak
value by replacing the weak measurement with a weak unitary. The
combination of the real and the imaginary part of a weak measurement
then define a complex-valued probability distribution that takes
over
the role of the classical phase space distribution in the extreme
quantum limit. Our research shows that complex joint probabilities
provide a complete and consistent description of quantum physics,
where
non-commutativity indicates imaginary correlations that represent
transformations between the physical properties defined by the
initial
state and possible measurements.
For more on the relation between complex probabilities and the
operator
algebra, see H. F. Hofmann, "Complex joint probabilities as
expressions
of reversible tranformations in quantum mechanics," New Journal of
Physics 14, 043031 (2012).
(http://iopscience.iop.org/
Quantum mechanics defines the relation between physical properties in a fundamentally different way, and this is the reason for using complex probabilities instead of real and positive ones. Importantly, complex probabilities explain why there are no joint realities of physical properties when these physical properties are connected by their dynamics. Specifically, the phases of complex conditional probabilities P(m|a,b) describe the action that is necessary to transform a into b along trajectories of constant m. Therefore, these conditional probabilities cannot be zero, even when there seems to be a classical distance between a and b for that particular value of m. In the quantum limit, this distance turns into a phase gradient, corresponding to the classical gradient of action that defines a transformation distance.
Classical physics emerges whenever the resolution of m is so low that the action phase rotates by several periods in the interval of m that is resolved. It is therefore possible to attribute a joint reality to a,b and m if the resolution of the three properties is much lower than the limit given by the action-phase ratio hbar. At higher resolution, no joint reality is possible, and imaginary and negative probabilities emerge instead. The latter effect can explain why quantum paradoxes are possible - and even necessary.
For more on action phases (and the "logical tension" of quantum paradoxes), see H. F. Hofmann, "On the role of complex phases in the quantum statistics of weak measurements," New Journal of Physics 13,
103009 (2011).
(http://iopscience.iop.org/
☆Quantum ergodicity as fundamental law of quantum physics
To understand how complex probabilities are possible, it is necessary to remember why we cannot observe joint probabilities of non-commuting properties in quantum mechanics. We experience physical objects by interactions, so "reality" is always the reality of an effect, not of a separate "thing". Quantum mechanics tells us that a precise measurement requires an interaction that completely randomizes the dynamics of the system generated by the observed property. Therefore, the most precise control we can get over quantum systems is limited to dynamical averages, and quantum statistics expresses the probability distributions associated with these dynamical averages. Using the term coined by Boltzmann to refer to the equivalence between dynamic averages and phase space averages in thermodynamics, we refer to these probability distributions as ergodic averages, to distinguish them from the more
fundamental complex probabilities. Quantum ergodicity is the principle by which actual measurement interactions convert the fundamental relations between physical properties, which are given by complex probabilities, into the real and positive probabilities that describe the relative frequencies of actual measurement outcomes. We can therefore explain the complete formalism of quantum mechanics in terms of experimentally accessible real-world phenomena.
For more on the law of quantum ergodicity, see H.F. Hofmann, "Derivation of quantum mechanics from a single fundamental modification of the relations between physical properties", Phys. Rev. A 89, 042115 (2014).
(free online, https://journals.aps.org/pra/
| Overview
of our research |
||
| What is so special about quantum physics ? | ||
| Optical quantum networks | Non-classical correlations | Explanation of quantum phenomena |