FOUNDATIONS OF PROBABILITY AND PHYSICS  4

In Memoriam Walter Philipp
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A Mathematician’s Viewpoint to Bell’s theorem: In Memory of Walter Philipp
View Description Hide DescriptionIn this paper dedicated to the memory of Walter Philipp, we formalize the rules of classical→ quantum correspondence and perform a rigorous mathematical analysis of the assumptions in Bell’s NO‐GO arguments.

Chameleon Effect, the Range of Values Hypothesis and Reproducing the EPR‐Bohm Correlations
View Description Hide DescriptionWe present a detailed analysis of assumptions that J. Bell used to show that local realism contradicts QM. We find that Bell’s viewpoint on realism is nonphysical, because it implicitly assume that observed physical variables coincides with ontic variables (i.e., these variables before measurement). The real physical process of measurement is a process of dynamical interaction between a system and a measurement device. Therefore one should check the adequacy of QM not to “Bell’s realism,” but to adaptive realism (chameleon realism). Dropping Bell’s assumption we are able to construct a natural representation of the EPR‐Bohm correlations in the local (adaptive) realistic approach.

Concerning Dice and Divinity
View Description Hide DescriptionEinstein initially objected to the probabilistic aspect of quantum mechanics—the idea that God is playing at dice. Later he changed his ground, and focussed instead on the point that the Copenhagen Interpretation leads to what Einstein saw as the abandonment of physical realism. We argue here that Einstein’s initial intuition was perfectly sound, and that it is precisely the fact that quantum mechanics is a fundamentally probabilistic theory which is at the root of all the controversies regarding its interpretation. Probability is an intrinsically logical concept. This means that the quantum state has an essentially logical significance. It is extremely difficult to reconcile that fact with Einstein’s belief, that it is the task of physics to give us a vision of the world apprehended sub specie aeternitatis. Quantum mechanics thus presents us with a simple choice: either to follow Einstein in looking for a theory which is not probabilistic at the fundamental level, or else to accept that physics does not in fact put us in the position of God looking down on things from above. There is a widespread fear that the latter alternative must inevitably lead to a greatly impoverished, positivistic view of physical theory. It appears to us, however, that the truth is just the opposite. The Einsteinian vision is much less attractive than it seems at first sight. In particular, it is closely connected with philosophical reductionism.

Three Ways to Look at Mutually Unbiased Bases
View Description Hide DescriptionThis is a review of the problem of Mutually Unbiased Bases in finite dimensional Hilbert spaces, real and complex. Also a geometric measure of “mubness” is introduced, and applied to some explicit calculations in six dimensions (partly done by Björck and by Grassl). Although this does not yet solve any problem, some appealing structures emerge.

Rényi Entropy and the Uncertainty Relations
View Description Hide DescriptionQuantum mechanical uncertainty relations for the position and the momentum and for the angle and the angular momentum are expressed in the form of inequalities involving the Rényi entropies. These uncertainty relations hold not only for pure but also for mixed states. Analogous uncertainty relations are valid also for a pair of complementary observables (the analogs of x and p) in N‐level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized Rényi entropies. The mathematical proofs of all the inequalities discussed in this paper can be found in Phys. Rev. A 74, No. 5 (2006); arXiv:quant‐ph/0608116.

From Objective Amplitudes to Bayesian Probabilities
View Description Hide DescriptionWe review the Consistent Amplitude approach to Quantum Theory and argue that quantum probabilities are explicitly Bayesian. In this approach amplitudes are tools for inference. They codify objective information about how complicated experimental setups are put together from simpler ones. Thus, probabilities may be partially subjective but the amplitudes are not.

Security in Quantum Cryptography vs. Nonlocal Hidden Variables
View Description Hide DescriptionIn order to prove equivalence of quantum mechanics with nonlocal hidden‐variable theories of a Bohm type one assumes that all the possible measurements belong to a restricted class: (a) we measure only positions of particles and (b) have no access to exact values of initial conditions for Bohm’s trajectories. However, in any computer simulation based on Bohm’s equations one relaxes the assumption (b) and yet obtains agreement with quantum predictions concerning the results of positional measurements. Therefore a theory where (b) is relaxed, although in principle allowing for measurements of a more general type, cannot be experimentally falsified within the current experimental paradigm. Such generalized measurements have not been invented, or have been invented but the information is qualified, but we cannot exclude their possibility on the basis of known experimental data. Since the measurements would simultaneously allow for eavesdropping in standard quantum cryptosystems, the arguments for security of quantum cryptography become logically circular: Bohm‐type theories do not allow for eavesdropping because they are fully equivalent to quantum mechanics, but the equivalence follows from the assumption that we cannot measure hidden variables, which would be equivalent to the possibility of eavesdropping… Here we break the vicious circle by a simple modification of entangled‐state protocols that makes them secure even if our enemies have more imagination and know how to measure hidden‐variable initial conditions with arbitrary precision.

Operational Axioms for Quantum Mechanics
View Description Hide DescriptionThe mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of physical experiment and from five simple Postulates concerning experimental accessibility and simplicity. For the infinite dimensional case, on the other hand, a C^{*}‐algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand‐Naimark‐Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in Ref. [1]. The main ingredient of the axiomatization is the postulated existence of faithful states that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the transposed of a physical transformation. What is new in the present paper with respect to Ref. [1], is the operational deduction of an involution corresponding to the complex‐conjugation for effects, whose extension to transformations allows to define the adjoint of a transformation when the extension is composition‐preserving. The existence of such composition‐preserving extension among possible extensions is analyzed.

Stochastic Models of Quantum Mechanics — A Perspective
View Description Hide DescriptionA subjective survey of stochastic models of quantum mechanics is given along with a discussion of some key radiative processes, the clues they offer, and the difficulties they pose for this program. An electromagnetic basis for deriving quantum mechanics is advocated, and various possibilities are considered. It is argued that only non‐local or non‐causal theories are likely to be a successful basis for such a derivation.

Fractal States for Quantum Information Processing
View Description Hide DescriptionQuantum information processing typically involves states of significant numbers of qubits. In particular, the use of decoherence‐free subspaces for error mitigation and the use of error correction codes in practical quantum computing increases the number of qubits beyond the number considered in the simple treatments often given the literature. When concatenated quantum codes are used for these purposes, self‐similar quantum states arise that are fractal in character. Recently, a definition of fractal states for discrete systems in quantum information processing was introduced to capture and formalize this phenomenon. Examples of fractal quantum states for quantum communications and quantum computing applications are shown to accord with this definition. The possibility of fractal quantum states of linear qubit arrays appearing in relation to quantum phase transitions is also noted here.

Bohr‐Heisenberg Reality and System‐free Quantum Mechanics
View Description Hide DescriptionMotivated by Heisenberg’s assertion that electron trajectories do not exist until they are observed, we present a new approach to quantum mechanics in which the concept of observer independent system under observation is eliminated. Instead, the focus is only on observers and apparatus, the former describing the latter in terms of labstates. These are quantum states over time‐dependent Heisenberg nets, which are quantum registers of qubits representing information gateways accessible to the observers. We discuss the motivation for this approach and lay down the basic principles and mathematical notation.

Quantum Mechanics for Military Officers
View Description Hide DescriptionWe present a trivial probabilistic illustration for representation of quantum mechanics as an algorithm for approximative calculation of averages.

Can you do quantum mechanics without Einstein?
View Description Hide DescriptionThe present form of quantum mechanics is based on the Copenhagen school of interpretation. Einstein did not belong to the Copenhagen school, because he did not believe in probabilistic interpretation of fundamental physical laws. This is the reason why we are still debating whether there is a more deterministic theory. One cause of this separation between Einstein and the Copenhagen school could have been that the Copenhagen physicists thoroughly ignored Einstein’s main concern: the principle of relativity. Paul A. M. Dirac was the first one to realize this problem. Indeed, from 1927 to 1963, Paul A. M. Dirac published at least four papers to study the problem of making the uncertainty relation consistent with Einstein’s Lorentz covariance. It is interesting to combine those papers by Dirac to make the uncertainty relation consistent with relativity. It is shown that the mathematics of two coupled oscillators enables us to carry out this job. We are then led to the question of whether the concept of localized probability distribution is consistent with Lorentz covariance.

Security Aspects of the Authentication Used in Quantum Cryptography
View Description Hide DescriptionUnconditionally secure message authentication is an important part of Quantum Cryptography (QC). We analyze security effects of using a key obtained from QC in later rounds of QC. It has been determined earlier that partial knowledge of the key in itself does not incur a security problem. However, by accessing the quantum channel used in QC, the attacker can change the message to be authenticated. This, together with partial knowledge of the key does incur a security weakness of the authentication. We suggest a simple solution to this problem, and stress usage of this or an equivalent extra security measure in QC.

Conditional Density Operators and the Subjectivity of Quantum Operations
View Description Hide DescriptionAssuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the Choi‐Jamiołkowski isomorphism in terms of “conditional” density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.

Models of Measurement for Quantum Fields and for Classical Continuous Random Fields
View Description Hide DescriptionA quantum field model for an experiment describes thermal fluctuations explicitly and quantum fluctuations implicitly, whereas a comparable continuous random field model would describe both thermal and quantum fluctuations explicitly. An ideal classical measurement does not affect the results of later measurements, in contrast to ideal quantum measurements, but we can describe the consequences of the thermal and quantum fluctuations of classically non‐ideal measurement apparatuses explicitly. Some details of continuous random fields and of Bell inequalities for random fields will be discussed.

Nonequivalence of Inertial Mass and Active Gravitational Mass
View Description Hide DescriptionFor the Kerr‐Newman black hole, it is pointed out that the active gravitational mass equals twice the inertial mass. This remains valid when the solution is applied as a model for the electron. The factor 2 also occurs in a classical electrodynamics model for the electron proposed recently, where gravity is negligible. On the basis of this, one may assume that for all matter the active gravitational mass equals twice the inertial mass. Newtonian physics will remain unchanged provided the gravitational constant is reduced by a factor 2. In cosmology, effects will appear.

Prediction and Repetition in Quantum Mechanics: The EPR Experiment and Quantum Probability
View Description Hide DescriptionThe article considers the implications of the experiment of A. Einstein, B. Podolsky, and N. Rosen (EPR), and of the exchange (concerning this experiment) between EPR and Bohr concerning the incompleteness, or else nonlocality, of quantum mechanics for our understanding of quantum phenomena and quantum probability. The article specifically argues that in the case of quantum phenomena, including those involved in the experiments of the EPR type, the probabilistic considerations are important even when the predictions concerned can be made with certainty, due to the impossibility, in general, to repeat any given quantum experiment with the same outcome. The article argue that this fact, not properly considered or taken into account by EPR, makes it difficult and ultimately impossible to sustain their argument, which it is consistent with Bohr’s counterargument to EPR and with his view of quantum phenomena and quantum mechanics.