The ensemble interpretation of quantum mechanics considers the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing that it exhaustively represents an individual physical system.
The advocates of the ensemble interpretation of quantum mechanics claim that it is minimalist, making the fewest physical assumptions about the meaning of the standard mathematical formalism. It proposes to take to the fullest extent the statistical interpretation of Max Born, for which he won the Nobel Prize in Physics in 1954. On the face of it, the ensemble interpretation might appear to contradict the doctrine proposed by Niels Bohr, that the wave function describes an individual system or particle, not an ensemble, though he accepted Born's statistical interpretation of quantum mechanics. It is not quite clear exactly what kind of ensemble Bohr intended to exclude, since he did not describe probability in terms of ensembles. The ensemble interpretation is sometimes, especially by its proponents, called "the statistical interpretation", but it seems perhaps different from Born's statistical interpretation.
As is the case for "the" Copenhagen interpretation, "the" ensemble interpretation might not be uniquely defined. In one view, the ensemble interpretation may be defined as that advocated by Leslie E. Ballentine, Professor at Simon Fraser University. His interpretation does not attempt to justify, or otherwise derive, or explain quantum mechanics from any deterministic process, or make any other statement about the real nature of quantum phenomena; it intends simply to interpret the wave function. It does not propose to lead to actual results that differ from orthodox interpretations. It makes the statistical operator primary in reading the wave function, deriving the notion of a pure state from that.
History In his 1926; English translation by Gunter Ludwig, pp. 206–225, 'On the quantum mechanics of collisions', in Wave Mechanics (1968), Pergamon, Oxford UK. paper introducing the concept of quantum scattering theory Max Born proposed to view "the motion of the particle follows the laws of probability, but the probability itself propagates in accord with causal laws", where the causal laws are the Schrödinger equations. As related in his 1954 Nobel Prize in Physics lecture Born viewed the statistical character of quantum mechanics as an empirical observation with philosophical implications.
Einstein maintained consistently that the quantum mechanics only supplied a statistical view. In 1936 he wrote ""the function does not in any way describe a condition which could be that of a single system; it relates rather to many systems, to 'an ensemble of systems' in the sense of statistical mechanics." However Einstein did not provide a detailed study of the ensemble, ultimately because he considered quantum mechanics itself to be incomplete primarily because it was only an ensemble theory. Einstein believed quantum mechanics was correct in the same sense that thermodynamics is correct, but that it was insufficient as means of unifying physics.
Also in the years around 1936, Karl Popper published philosophical studies countering the work of Heisenberg and Bohr. Popper considered their work as essentially subjectivist, unfalsifiable, and thus unscientific. He held that the quantum state represented statistical assertions which have no predictive power for individual particles. Popper described "propensities" as the correct notion of probability for quantum mechanics.
Although several other notable physicists championed the ensemble concept, including John C. Slater, Edwin C. Kemble, and Dmitry Blokhintsev, Leslie Ballentine's 1970 paper 'The statistical interpretation of quantum mechanics" and his textbook have become the main sources. Ballentine followed up with axiomatic development of propensity theory, analysis of decoherence in the ensemble interpretation and other papers spanning 40 years.
States, systems, and ensembles
Perhaps the first expression of an ensemble interpretation was that of Max Born. In a 1968 article, he used the German words 'gleicher Haufen', which are often translated into English, in this context, as 'ensemble' or 'assembly'. The atoms in his assembly were uncoupled, meaning that they were an imaginary set of independent atoms that defines its observable statistical properties. Born did not develop a more detailed specification of ensembles to complete his scattering theory work.
Although Einstein described quantum mechanics as clearly an ensemble theory he did not present a formal definition of an ensemble.Home, D. (1997). Conceptual Foundations of Quantum Physics: An Overview from Modern Perspectives, Springer, New York, , p. 362: "Einstein's references to the ensemble interpretation remained in general rather sketchy." Einstein sought a theory of individual entities, which he argued was not quantum mechanics.
Ballentine distinguish his particular ensemble interpretation the Statistical Interpretation. According to Ballentine, the distinguishing difference between many of the Copenhagen-like interpretations (CI) and the Statistical Interpretation (EI) is the following:
CI: A pure state provides a complete description of an individual system, e.g. an electron.
EI: A pure state describes the statistical properties of an ensemble of identically prepared systems.
Ballentine defines a quantum state as an ensemble of similarly prepared systems. For example, the system may be a single electron, then the ensemble will be "the set of all single electrons subjected to the same state preparation technique." He uses the example of a low-intensity electron beam prepared with a narrow range of momenta. Each prepared electron is a system, the ensemble consists of many such systems.
Ballentine emphasizes that the meaning of the "Quantum State" or "State Vector" may be described, essentially, by a one-to-one correspondence to the probability distributions of measurement results, not the individual measurement results themselves.Quantum Mechanics, A Modern Development, p. 48. A mixed state is a description only of the probabilities, and of positions, not a description of actual individual positions. A mixed state is a mixture of probabilities of physical states, not a coherent superposition of physical states.
Probability; propensity Quantum observations are inherently statistical. For example, the electrons in a low-intensity double slit experiment arrive at random times and seemingly random places and yet eventually show an interference pattern.
The theory of quantum mechanics offer only statistical results. Given that we have prepared a system in a state , the theory predicts a result as a probability distribution: . Different approaches to probability can be applied to connect the probability distribution of theory to the observed randomness.
Popper,Popper, Karl R. "The Propensity Interpretation of Probability." The British Journal for the Philosophy of Science, vol. 10, no. 37, 1959, pp. 25–42. JSTOR, http://www.jstor.org/stable/685773. Accessed 27 Aug. 2023. Ballentine, Paul Humphreys, and othersBerkovitz, Joseph. "The Propensity Interpretation of Probability: A Re-Evaluation." Erkenntnis (1975-) 80 (2015): 629–711. http://www.jstor.org/stable/24735118. point to propensity as the correct interpretation of probability in science. Propensity, a form of causality that is weaker than determinism, is the tendency of a physical system to produce a result. Thus the mathematical statement
means the propensity for event to occur given the physical scenario is . The physical scenario is viewed as weakly causal condition.
The weak causation invalidates Bayes' theorem and correlation is no longer symmetric. As noted by Paul Humphreys, many physical examples show the lack of reciprocal correlation, for example, the propensity for smokers to get lung cancer does not imply lung cancer has a propensity to cause smoking.
Propensity closely matches the application of quantum theory: single event probability can be predicted by theory but only verified by repeated samples in experiment. Popper explicitly developed propensity theory to eliminate subjectivity in quantum mechanics.
Preparative and observing devices as origins of quantum randomness
An isolated quantum mechanical system, specified by a wave function, evolves in time in a deterministic way according to the Schrödinger equation that is characteristic of the system. Though the wave function can generate probabilities, no randomness or probability is involved in the temporal evolution of the wave function itself. This is agreed, for example, by Born,: 630: "We have accustomed ourselves to abandon deterministic causality for atomic events; but we have still retained the belief that probability spreads in space (multi-dimensional) and time according to deterministic laws in the form of differential equations." Dirac,: "One can suppose that the initial state of a system determines definitely the state of the system at any subsequent time. ... The notion of probabilities does not enter into the ultimate description of mechanical processes." von Neumann, Translated as P. 349: "... the time dependent Schrödinger differential equation ... describes how the system changes continuously and causally." London & Bauer,London, F., Bauer, E. (1939). La Théorie de l'Observation dans la Mécanique Quantique, issue 775 of Actualités Scientifiques et Industrielles, section Exposés de Physique Générale, directed by Paul Langevin, Hermann & Cie, Paris, translated by Shimony, A., Wheeler, J.A., Zurek, W.H., McGrath, J., McGrath, S.M. (1983), at pp. 217–259 in Wheeler, J.A., Zurek, W.H. editors (1983). Quantum Theory and Measurement, Princeton University Press, Princeton NJ; p. 232: "... the Schrödinger equation has all the features of a causal connection." Messiah,Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, p. 61: "... specifying at a given initial instant uniquely defines its entire later evolution, in accord with the hypothesis that the dynamical state of the system is entirely determined once is given." and Feynman & Hibbs.Feynman, R.P., Hibbs, A. (1965). Quantum Mechanics and Path Integrals, McGraw–Hill, New York, p. 22: "the amultitudes are solutions of a completely deterministic equation (the Schrödinger equation)." An isolated system is not subject to observation; in quantum theory, this is because observation is an intervention that violates isolation.