Can a negative quantity be deemed a probability?

Sadri Hassani

With the discovery of the quantum theory (describing the very small) and the special theory of relativity (describing the very fast), it was natural to seek a theory that described very fast subatomic particles. Such a theory had to combine STR and the quantum theory, resulting in an equation that would replace the Schro¨dinger equation. Early attempts yielded results, which, although deemed accurate later on, were considered nonsensical at the time. For example, one formulation appeared to yield negative probability.1 The negative sign of the probability can be traced to Equation (28.7), which contains the square of the energy, making it possible for the energy to be negative (keeping its square positive). This negative energy led to what was interpreted as negative probability.

Yu.I. Alimov, Yu. A. Kravtsov

D. Aleksandrov, E. Yu. Nikolskii, B.G. Novatskii +1 more

Vasil Dinev Penchev

The concepts of choice, negation, and infinity are considered jointly. The link is the quantity of information interpreted as the quantity of choices measured in units of elementary choice: a bit is an elementary choice between two equally probable alternatives. “Negation” supposes a choice between it and confirmation. Thus quantity of information can be also interpreted as quantity of negations. The disjunctive choice between confirmation and negation as to infinity can be chosen or not in turn: This corresponds to set-theory or intuitionist approach to the foundation of mathematics and to Peano or Heyting arithmetic. Quantum mechanics can be reformulated in terms of information introducing the concept and quantity of quantum information. A qubit can be equivalently interpreted as that generalization of “bit” where the choice is among an infinite set or series of alternatives. The complex Hilbert space can be represented as both series of qubits and value of quantum information. The complex Hilbert space is that generalization of Peano arithmetic where any natural number is substituted by a qubit. “Negation”, “choice”, and “infinity” can be inherently linked to each other both in the foundation of mathematics and quantum mechanics by the meditation of “information” and “quantum information”.

Yuri G. Gurevich, Vladimir Vovk

We explain, on the example of Wigner's quasiprobability distribution, how negative probabilities may be used in the foundations of probability.

Abdullah M. Almarashi, S. Abdel‐Khalek, Debasis Kundu

In this paper, a quantum scheme for a two-qubit system (2QS) and field initially prepared in the excited negative binomial distribution is presented. The field photon statistics is detected from the evolution of the Mandel parameter, while the evolution of von Neumann entropy detects the nonlocal correlation between the 2QS and radiation field. The concurrence is used to detect the qubit-qubit entanglement during the time evolution. The dynamical properties of single-qubit and two-qubit quantum Fisher information are investigated. We visualize the number of photon excitations on the field in negative binomial states with influence of photon success probability. A connection is provided between the dynamical behaviors of these statistical quantities. We have found that the proposed quantities are strongly influenced by the number of excited photons of the field in negative binomial states and photon success probability.