In 1989, Deutsch gave a basic physical explanation of why quantum-mechanical probabilities are squares of amplitudes. Essentially, a general state vector is transformed into a highly symmetric equal-amplitude superposition. The argument was recently elaborated and publicised by DeWitt. It has remained incomplete, however, inasmuch as both authors anticipate the usual normalization (sum of amplitudes squared) of state vectors. In the present paper, a thought experiment is devised in which Deutsch's idea is demonstrated independently of the normalization, exploiting further symmetries instead.
In 1989, Deutsch gave a basic physical explanation of why quantum-mechanical probabilities are squares of amplitudes. Essentially, a general state vector is transformed into a highly symmetric equal-amplitude superposition. The argument was recently elaborated and publicised by DeWitt. It has remained incomplete, however, inasmuch as both authors anticipate the usual normalization (sum of amplitudes squared) of state vectors. In the present paper, a thought experiment is devised in which Deutsch's idea is demonstrated independently of the normalization, exploiting further symmetries instead.
In a recent article (quant-ph/9906015) Deutsch, using concepts of decision theory, provided a strikingly simple argument to explain why quantum-mechanical probabilities are squares of amplitudes. The decisiontheoretic reasoning was criticized by Barnum et al. (quant-ph/9907024). Taking a more conventional approach, we reinterpret a crucial substitution in Deutsch’s paper as the time evolution of an apparatus, capable of separating internal states of an “atom ” into a suitable number of “cavities”. Permutational symmetries play a central role in this analysis. PACS numbers: 03.65.BzIn a recent paper, Deutsch [1] invoked the non-probabilistic part of classical decision theory to deduce from only non-probabilistic axioms of quantum mechanics the Statistical Axiom: The probability for measuring xi in a superposition state |ψ 〉 = ∑ i ψi|xi 〉 is |ψi | 2. The paper has attracted some vehement criticism [2] relating to its decision-theoretic setting, but also to priorities. As pointed out by Ref. [2], in 1957 already Gleason [3] gave a mathematically rigorous proof for the redundance of the Statistical Axiom, showing that if normalized probabilities are to be assigned to the eigenvectors of each bounded hermitian operator, then the only solution is the usual quantum mechanical prescription. Unfortunately for a result of such fundamental importance to quantum physics, the existing proofs of Gleason’s theorem are not easily received by many physicists, although mathematically elementary [4], synoptical [5], and even textbook [6] elaborations have been presented. Ref. [1] has a strong point in this respect, suggesting that the mathematics behind amplitudes being square-roots of probabilities essentially is the Pythagorean theorem in Hilbert space. Consider a superposition state of the special form |ψ 〉 =
In modern physics only relative quantities are considered to have physical significance. For example, position assigned to a system depends on the choice of coordinates, and only relative distances between different systems have physical relevance. On the other hand, in quantum theory the scenario where one system, A, is localised around certain position while another system B is in a spatial superposition is considered to be physically different from the scenario where A is in a superposition, while B is localised. Different physical effects are anticipated in the two scenarios especially when the two systems have widely different masses. Here we show that for any superposed amplitudes that are related by a symmetry transformation, the above scenarios are physically equivalent: probabilities for any measurement are the same whether we assign the superposition state to system A or to system B. More generally, linearity of quantum theory guarantees that if a theory is invariant under some symmetry transformations it is necessarily also invariant under their arbitrary `superpositions'. Thus the notion of a superposition turns out to be relative to the choice of coordinates, once it is realised that relations between coordinates do not need to be classical.
First steps are taken toward a formulation of quantum mechanics which avoids the use of probability amplitudes and is expressed entirely in terms of observable probabilities. Quantum states are represented not by state vectors or density matrices but by “probability tables,” which contain only the probabilities of the outcomes of certain special measurements. The rule for computing transition probabilities, normally given by the squared modulus of the inner product of two state vectors, is re-expressed in terms of probability tables. The new version of the rule is surprisingly simple, especially when one considers that the notion of complex phases, so crucial in the evaluation of inner products, is entirely absent from the representation of states used here.
Square roots of probabilities appear in several contexts, which suggests that they are somehow more fundamental than probabilities.Square roots of probabilities appear in expressions of the Fisher-Rao Metric and the Hellinger-Bhattacharyya distance.They also come into play in Quantum Mechanics via the Born rule where probabilities are found by taking the squared modulus of the quantum amplitude.Why should this be the case and why do these square roots not arise in the various formulations of probability theory?In this short, inconclusive exploration, I consider quantifying a logical statement with a vector defined by a set of components each quantifying one of the atomic statements defining the hypothesis space.I show that conditional probabilities (bi-valuations), such as P(x|y), can be written as the dot product of the two vectors quantifying the logical statements x and y each normalized with respect to the vector quantifying the conditional y.The components of the vectors are proportional to the square root of the probability.As a result, this formulation is shown to be consistent with a concept of orthogonality applied to the set of mutually exclusive atomic statements such that the sum rule is represented as the sum of the squares of the square roots of probability.
Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a very natural consistency constraint: if there are two different ways to compute the amplitude for a given process, the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. An immediate consequence is that the Schr\"odinger equation must be linear; i.e., nonlinear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve any probabilities, is used to obtain a proof of Born's statistical postulate: we show that the probability of a certain outcome in an experiment is given by the square of the modulus of the corresponding amplitude. Thus, consistency leads to indeterminism.
It is proved that: What the two-slit experiments have refuted is the preconceived idea that of the event that a micro particle passing through one slit arrives somewhere on the screen is irrelative to the condition that whether or not the other slit is open. Consideringaddition theorem of probabilities, the above idea can be expressed as follows:
(a)Assum that a beam passes through a partition with two slit and arrives a screen. The of the event that an indivedual particle falls on somewhere of the screen under the condition that two slit open simultaneously, is equal to the sum of two of the same event under that two slit open in turn.
Defining the experimental condition such as two slit open simultaneously or in turn as passage the proposition (a)is manifested as probabilities is independent of passage conditions.
On the other hand, in terms of the above, superposition principle of in quantum mechanics ought to be indicated as probability amplitudes is independent of passage conditions. Correspondingly, (a)is named hereof as superposition presumption of probability.
For lack of the conception of passage conditions, (a)is confused with theorem of probabilities. As a result, classical physics tacitly approve (a), and thereby it fails to comprehend the two-slit experiments. In reverse, to explain two-slit experiments, quantum mechanics spurns addition theorem of so that it meets with various difficults about explanation.
Any alternative of quantum mechanics must be able to explain two-slit experiments, therefore it must contain passage conditions or amplitudes. But hidden variable theory Bell used dissatisfies such a prerequisite. It comes thus into conflict with experiment facts inevitably. As a result, Lochak's argument that Bell's inequality is irrelative to locality is rediscovered.