Negative and Complex Probability in Quantum Information. Vasil Penchev 1

Transcription

1 Negative and Complex Probability in Quantum Information Vasil Penchev 1 Abstract: Negative probability in practice: quantum communication. Very small phase space regions turn out to be thermodynamically analogical to those of superconductor. Macrobodies or signals might exist in coherent or entangled state. Such physical objects having shockingly extraordinary properties could be in the base of quantum communicative channels or even material ones Questions and a few answers about negative probability: Why does it appear in quantum mechanics? It appears in phase-space formulated quantum mechanics; next, in quantum correlations and for wave-corpuscular dualism. Its meaning, mathematically: a ratio of two measures (of sets), which are not collinear; physically: of the ratio of the measuring of two physical quantities, which are not simultaneously measurable. The main innovation is about the mapping between phase and Hilbert space since both are sums. Phase space is a sum of cells, and Hilbert space is a sum of qubits. The mapping is reduced to the mapping of a cell into a qubit and vice versa. Negative probability helps quantum mechanics to be represented quasi-statistically by quasi-probabilistic distributions. The states of negative probability cannot happen, but they, where they are, decrease the sum probability of the integrally positive regions of the distributions. They reflect the immediate interaction (interference) of probabilities in quantum mechanics. The much more extended text and presentation, from which the present article is a very small part, can be found on Key words: negative probability, quantum correlation, phase space, transformation between phase and Hilbert space, entanglement, Bell s inequalities I. Why does negative probability appear in quantum mechanics? 1. Negative or complex probability appears there where the measure of two parts or of a part and the whole forms any angle. It can happen when the probability is geometric. 2. Negative or complex probability cannot be excluded from considering in quantum mechanics since any quantum object consists of two parts : wave and corpuscular one. 3. That s why the effects of relative rotated measures are observed in any separated quantum object as well as in the systems of quantum objects. They reflect the immediate interaction of probabilities without any hidden parameters. 1 Institute of Philosophical Research, Bulgarian Academy of Science; vasildinev@gmail.com;

2 The meaning of negative probability expressed mathematically is the following: a ratio of two measures (of sets), which are not collinear; or physically: a ratio of the measuring of two physical quantities, which are not simultaneously measurable. II. The appearing of negative probability in the phase-space formalism of quantum mechanics can be represented schematically by a correspondence with its ordinary Hilbert-space formalism or by a similar correspondence between their atoms : a phase-space sell and a Hilbert-space qubit. The problem is the transformation between the twodimensional cell of phase space and the qubit of Hilbert space. The base is the Wigner function (1932) in a contemporary view. Hermann Weyl s paper (1927) is historically the first one. He considered abstractly and mathematically the transformation, but did not discuss the fact that it can obtain negative values. We are going to mention the time-frequency (= time-energy) reformulation of Wigner function accomplished by Ville (1948). It is especially important as to quantum information looking at all physical processes like at informational ones not alone transmitting, but being themselves signals. The necessary condition of such a viewpoint is the reciprocity of time and energy (coordinates and momentums) being implied by the Heisenberg uncertainty, and consequently, by the quantum, discrete character of mechanical motion. That hints at Cohen s generalization of Wigner function, which was subsequently realized (Cohen 1966: 782; 1995: 136). Further is obviously that if straight Wigner function can be generalized, then analogically the reverse transform of Weyl can be, and that was done by Cohen in a later paper: (Cohen 2008: 260). The manner of Cohen s thought is well to keep in mind discussing Groenewold s statistical ideas (1946). The main ideas of Moyal s statistical approach (1949) would be represented by a few significant his textual citations: Classical statistical mechanics is, however, only a special case in the general theory of dynamical statistical (stochastic) processes. In the general case, there is the possibility of diffusion of the probability fluid, so that the transformation with time of the probability distribution need not be deterministic in the classical sense. In this paper, we shall attempt to interpret quantum mechanics as a form of such a general statistical dynamics (Moyal 1949: 99). According to Moyal, symmetry (or antisymmetry) conditions introduce a probability dependence between any two particles in B. E. (or F. D.) ensembles even in the absence of any energy interaction. It is this dependence which gives rise to the 'exchange energy' between the particles when they interact (Moyal 1949: 116).

3 III. Negative probability in quantum correlations: In the light of quantum information and the studied by it quantum correlations, the battle for or against hidden parameters in quantum mechanics can be interpreted and reformulated as local hidden parameters (causality) against nonlocal ones (quantum correlations). The beginning should be sought for in 1935: the famous paradox (or argument, in fact and essence) of Einstein, Podolsky, and Rosen and Schrödinger s cat paradox paper (1935). The deducted and discussed in those articles quantum correlation implies negative probability in the last analysis. Negative probability appears effectively, i.e. by the restriction of the degrees of freedom (DOM) as to any correlating quantum object by the others. The mechanism of such transformation is discussed bellow. The quantum allegory of Schrödinger s alive-and-dead cat helps us to understand the actual state by restricting DOM of any possible states, and consequently, in reverse way, quantum correlations to be suggested between the states forming any quantum superposition, and hence, as the mechanism of the mystic collapse of wave packet during the real process of measurement. Von Neumann s theorem (1932) about the absence of hidden parameters in quantum mechanics underlies both quantum correlation and quantum superposition. Its conclusion is: There are no ensembles which are free from dispersion. There are homogeneous ensembles (Neumann 1932: 170) 2. Consequently, there are no homogeneous ensembles, i.e. for example those of a single quantity being free of dispersion. Any quantity has dispersion, which is not due to any cause, to any hidden variable. What is the connection between von Neumann s theorem and negative probabilities? By introducing negative probability, expectation is not additive one in general, the premises of the theorem are not fulfilled, and the deduction is not valid. Here are a few equivalent expressions for the boundary of the validity of von Neumann s theorem: 1. Non-negative probability 2. Orthogonal possible states 3. Separated worlds 4. An isolated quantum system 2 Here and bellow the translation of the textual citations from von Neumann s book from German to English is according to: J. von Neumann Mathematical Foundations of Quantum Mechanics. Princeton: University Press.

4 5. The additivity of expectation Bell s criticism (1966) about von Neumann s theorem partly rediscovered Grete Hermann s objections (1935) is too important for clearing the connection between causality, quantum correlation, and negative probability. The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. It is urged that in further examination of this problem an interesting axiom would be that mutually distant systems are independent of one another (Bell 1966: 447). His essential assumption is: Any real linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values is the expectation value of the combination. This is true for quantum mechanical states; it is required by von Neumann of the hypothetical dispersion free states also (Bell 1966: ). The idea of Bell s inequalities (1964) can be expressed by negative probability. Since von Neumann s theorem is valid only about nonnegative probability (expectation additivity), and quantum mechanics permits negative probability, the idea is the domain of the theorem validity to be described by an inequality of the expectation of two quantities (the spin of two particles) according to the EPR conditions: In the light of von Neumann s theorem, Bell s paper (1966) reveals that an eventual violation of the inequality above would require generalizing the notion of hidden parameter introducing nonlocal one. The main steps are: 1. Von Neumann s theorem as well as the theories of hidden parameters interprets them as local ones implicitly. 2. Bell s inequalities discuss the distinction between local and nonlocal parameter because quantum mechanics allows nonlocal variables. The notion of non-local hidden parameter suggests the notion of the externality of a system considering it as a nonstandard, namely ambient, or environmental part of the system. Not any external neighborhood, but only a small one (of the order of a few ħ), at that correspondingly, only in phase space. In the corresponding small neighborhood in Minkowski space, Lorentz invariance is not valid. We can displays the appearing of negative probability when quantum mechanics is reformulated from the standard language of Ψ-functions into that of a standard statistical description containing however nonstandard, or external towards the whole, parts of it.

5 The notion of effective probability aids us in bridging physical interaction and mathematical probability. It is effective probability that exerts impact from a given probability to another. Effective probability is a probabilistic force by means of which both (or more) the probabilities interact. Since Dirac (1942: 8), negative probability has been thought just rather as an effective probability than a real probability of whatever. Negative probability has been likened to money since it assists the balance, and being non-existing at that. Following after Mermin (1998), we could at least put question as to or if the existence of negative probabilities or of probability at all as a new, nonstandard, but maybe omnipresent, universal and all-embracing substance including matter and energy, which are ordinary and generally accepted substances in physics. The necessary and sufficient condition of immediate probabilistic interaction is to be shared common possible states of nonzero probability. The same equivalence refers to the superposition of all the states of an isolated system. The halo of negative probability pushes away the system itself, any of the both, to the states of positive probability, too. They become relatively more probable. Consequently, relativity (as Skolemian as Einsteinian one) by means of negative probability reveals the intimate mechanism of any physical interaction manifesting itself always in the final analysis by being restricted DOF of any system participating in the interaction in question. Negative probability is only the one of two ways to be represented physical reality which way corresponds rather to the Boltzmann statistical consideration than to the Gibbs one. Respectively, there exist two kinds of their ontological projections: The statistical formalism permits to be calculated together the simultaneously immeasurable quantities of the standard formalism. Kochen-Specker s theorem displays that homogenous quantities have dispersion even also in quantities simultaneously measurable in mathematical sense, and respectively, there cannot be hidden parameters. Kochen Specker s theorem can be discussed also as a generalization of von Neumann s theorem in the following way: Von Neumann s theorem (1932: ) covers isolated systems and simultaneously immeasurable quantities. Bell s theorem (or inequalities, 1964) clears up the influence of the absence of hidden parameters in interacting systems.

6 Kochen Specker s theorem covers isolated systems and simultaneously measurable quantities; the dispersion of homogenous quantities is conditioned by quantum leaps. The two authors formalized the notion of simultaneous measurability: Kochen and Specker (1967: 63 ) interpreted simultaneous measurability as the availability of a common measure which required the measure of the set of discontinuity points (quantum leaps) to be zero, i.e. simultaneously measurable quantities not to be continuous, but to be almost continuous. They proved on such a condition that homogenous quantities have necessary dispersion, i.e. there are not hidden parameters. IV. Negative probability for wave-corpuscular dualism We should return to Einstein s paper (1905I) about mass and energy. Next, we are going to juxtapose it with a not less famous paper of the same year (Einstein: 1905Ü) about quanta and energy, by means of which to search for the connection between mass and quanta. In contemporary designations its content may be abstracted by the formula: Е=ħν. Light is discrete (corpuscular) in its interaction with matter, but it is continuous (wave) as a medium by itself which is propagated in space. Quantum mechanics transferred initially that property formulated about electromagnetic radiation to all the quanta, and in fact, to all the physical objects. Afterwards the separation of different relations has been abandoned replacing it by the Ψ-function description. The hypothesis of hidden parameters from such a viewpoint, in fact, conserves the original and already clearly formulated opinion of Einstein (1905) to be separated the two contradicting aspects of continuity and discreteness, thus to be separated in different relations. Einstein reformulated the principle of relativity in mathematical language as the invariance of physical laws towards diffeomorfisms. Discrete transformations were excluded in such a way. However they are the essential subject of quantum mechanics. Correspondingly, the uniting of quantum mechanics and relativity can be researched as generalization of invariance towards a wider class of morphisms, which should include discrete ones. The wave-corpuscular dualism considered as a definite type of generalization about the principle of relativity introduces negative probability. The following hypothesis can be advanced: that any relevant generalization of the principle of relativity which includes discrete morphisms should introduce negative probability explicitly or implicitly.

7 There exists a link between any possible generalization of relativity and a rejection (or a generalization) of so-called (by Einstein) Mach s principle. (Einstein 1918: ). It seems that the colligation of general relativity (introducing negative probabilities) implies restricting the validity or generalization of Mach s principle: The generated by negative probabilities restriction of the degrees of freedom of a part of a system can be equivalently equated with energy and hence, with mass. For the formula Е=ħν the frequency ν should interpret as the density of informational exchange between the parts of the system (i.e. bits per time). Very interesting is Pauli s (2000: 71-72) consideration about negative probability. In the spirit of a Skolemian relativity (Skolem 1970: 138; Penchev 2009: ), any system of entangled subsystem can be represented equivalently both as an independent, isolated, indivisible system and as an arbitrary operator transforming a point of the Hilbert space of the one system into that of another system. The conclusion made above implies particularly (entanglement = 0) that any measured value of a quantity in the system an apparatus a quantum object is the objective value of that quantity of the quantum object: no hidden parameter which states deterministically among its random values. The Hermitian character of any physical quantity generalizes the requirement that the value is in an exactly given point of time also about discrete functions. Ψ-function represents that a quantum quantity can obtain non-zero values only on areas whose common measure is zero. The absence of hidden parameter is due of the zero measure of any area with non-zero values. Instead of conclusions (as the necessary conclusions have mentioned in the abstract and the beginning of the paper), a few questions: 1. Whether is negative probability only a mathematical construction, or there exist the physical objects of negative probability? 2. Whether are negative probability and pure relation (such one which cannot be reduced to predications) equivalent expressing the same case in different ways? 3. Whether does negative probability imply the physical existence of probability? 4. Can probabilities interact immediately (i.e. without any physical interaction of the things, phenomena, or events possessing those probabilities)? 5. Whether physical existing information is equivalent to the interaction of probabilities?

8 LITERATURE: Bell, J. (1964). On the Einstein Podolsky Rosen paradox. Physics (New York), 1, Bell, J. (1966). On the Problem of Hidden Variables in Quantum Mechanics. Reviews of Modern Physics. 38, Cohen, L. (1966). Generalized phase-space distribution functions. Journal of Mathematical Physics. 7, Cohen, L. (1995). Time-Frequency Analysis. New York: Prentice-Hall. Cohen, L. (2008). The Weyl transformation and its generalization. Rendiconti del Seminario Matematico. Università Politecnico di Torino, 66, Dirac, P. (1942). Bakerian Lecture. The Physical Interpretation of Quantum Mechanics. Proceedings of the Royal Society of London, A, 180, Einstein, A. (1905). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik. 18, Einstein, A. (1905). Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik. 17, Einstein, A. (1918). Prinziplelles zur allgemeinen Relativitätstheorie. Annalen der Physik. 55, Einstein, A., B. Podolsky and N. Rosen. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, Groenewold, H. (1946). On the Principles of Elementary Quantum Mechanics. Physica. XII, Kochen, S., E. Specker. (1967). The problem of hidden variables in quantum mechanics. Physical Review A. 17, Mermin, D. (1998). What Is Quantum Mechanics Trying to Tell Us? American Journal of Physics. 66, Moyal, J. (1949). Quantum mechanics as a statistical theory. Proceedings of the Cambridge Philosophical Society. 45, 1, Neumann, J. von. (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Verlag von Julius Springer. Pauli, W. (2000). Pauli lectures on physics. Vol. 6. Selected topics in filed quantization. New York: Courier Dover Publications. Penchev, V. (2009). Philosophy of Quantum Information. Sofia: IPhR-BAS (in Bulgarian). Schrödinger, E. (1935). Die gegenwärtige situation in der Quantenmechanik. Die Naturwissenschaften, 48, ; 49, , 50, Skolem, T. (1970). Selected works in logic. Oslo: Univforlaget. Ville, J. (1948). Théorie et Applications de la Notion de Signal Analytique. Cables et Transmission, 2A, Weyl, H. (1927). Quantenmechanik und Gruppentheorie. Zeitschrift für Physik. 46, Wigner, E. (1932). On the Quantum Correction For Thermodynamic Equilbrium. Physical Review. 40,