# Ensemble interpretation

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The ensemble interpretation, or statistical interpretation of quantum mechanics, is an interpretation that can be viewed as a minimalist interpretation; it is a quantum mechanical interpretation that claims to make the fewest assumptions associated with the standard mathematical formalization. At its heart, it takes to the fullest extent the statistical interpretation of Max Born for which he won the Nobel Prize in Physics.[1] The interpretation states that the wave function does not apply to an individual system – or for example, a single particle – but is an abstract mathematical, statistical quantity that only applies to an ensemble of similarly prepared systems or particles. Probably the most notable supporter of such an interpretation was Albert Einstein:

The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems.

—Albert Einstein[2]

To date, probably the most prominent advocate of the ensemble interpretation is Leslie E. Ballentine, Professor at Simon Fraser University, and writer of the graduate-level textbook "Quantum Mechanics, A Modern Development".[3]

The ensemble interpretation, unlike many other interpretations of quantum mechanics, 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 is simply a statement as to the manner of wave function interpretation.

## Meaning of "Ensemble" and "System"

The "ensemble" of the ensemble interpretation is identified by an ensemble of setting up and performing the (essentially) same experiment many times. This is referred to as an ensemble of systems. It is not, for example, an "ensemble" of performing one single experiment on a simultaneous set "ensemble" of particles. A group of particles as in a gas, is not the "ensemble" of the "ensemble interpretation", although it is possible that a repeated set of ensemble experiments may involve a large "ensemble" of particles as its system.

According to Ballentine, the distinguishing difference between the Copenhagen interpretation (CI) and the ensemble interpretation (EI) is the following:

CI: A pure state $|y\rangle$ provides a complete and exhaustive description on an individual system. A dynamical variable represented by the operators $\operatorname{Q}$ has a value ($q$, say) if and only if $\operatorname{Q}|y\rangle=q|y\rangle$.

EI: A pure state describes the statistical properties of an ensemble of similarly prepared systems.

Ballentine emphasis 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.[4] For example:

$| \psi \rangle = a|\chi_1 \rangle + b|\chi_2\rangle$

Specifies that a mixed state is a description only of the probabilities, $\mathcal{P}(\chi_1)$ and $\mathcal{P}(\chi_2)$ of positions, not a description of actual individual positions. That is, $a|\chi_1 \rangle \rightarrow \mathcal{P}(\chi_1)$ and not $a|\chi_1 \rangle \rightarrow \chi_1$. In this way it is noted that a mixed state is a mixture of probabilities of physical states, not a mixture of actual physical states.

## Ensemble interpretation applied to single systems

The statement that the quantum mechanical wave function itself does not apply to a single system in one sense does not imply that the ensemble interpretation itself does not apply to single systems in the sense meant by the ensemble interpretation. The condition is that there is not a direct one to one correspondence of the wave function with an individual system that might imply, for example, that an object might physically exist in two states simultaneously. The ensemble interpretation may well be applied to a single system or particle, and predict what is the probability that that single system will have for a value of one of its properties, on repeated measurements.

Consider the throwing of two dice simultaneously on a craps table. The system in this case would consist of only the two dice. There are probabilities of various results, e.g. two fives, two twos, a one and a six etc. Throwing the pair of dice 100 times, would result in an ensemble of 100 trials. Classical statistics would then be able predict what typically would be the number of times that certain results would occur. However, classical statistics would not be able to predict what definite single result would occur with a single throw of the pair of dice. That is, probabilities applied to single one off events are, essentially, meaningless, except in the case of a probability equal to 0 or 1. It is in this way that the ensemble interpretation states that the wave function does not apply to an individual system. That is, by individual system, it is meant a single experiment or single throw of the dice, of that system.

The Craps throws could equally well have been of only one die, that is, a single system or particle. Classical statistics would also equally account for repeated throws of this single dice. It is in this manner, that the ensemble interpretation is quite able to deal with "single" or individual systems on a probabilistic basis. The standard Copenhagen Interpretation (CI) is no different in this respect. A fundamental principle of QM is that only probabilistic statements may be made, whether for individual systems/particles, a simultaneous group of systems/particles, or a collection (ensemble) of systems/particles. An identification that the wave function applies to an individual system in standard CI QM, does not defeat the inherent probabilistic nature of any statement that can be made within standard QM. To verify the probabilities of quantum mechanical predictions, however interpreted, inherently requires the repetition of experiments, i.e. an ensemble of systems in the sense meant by the ensemble interpretation. QM cannot state that a single particle will definitely be in a certain position, with a certain momentum at a later time, irrespective of whether or not the wave function is taken to apply to that single particle. In this way, the standard CI also "fails" to completely describe "single" systems.

However, it should be stressed that, in contrast to classical systems and older ensemble interpretations, the modern ensemble interpretation as discussed here, does not assume, nor require, that there exist specific values for the properties of the objects of the ensemble, prior to measurement.

## Measurement and collapse

The attraction of the ensemble interpretation is that it immediately dispenses with the metaphysical issues associated with reduction of the state vector, Schrödinger cat states, and other issues related to the concepts of multiple simultaneous states. As the ensemble interpretation postulates that the wave function only applies to an ensemble of systems, there is no requirement for any single system to exist in more than one state at a time, hence, the wave function is never physically required to be "reduced". This can be illustrated by an example:

Consider a classical die. If this is expressed in Dirac notation, the "state" of the die can be represented by a "wave" function describing the probability of an outcome given by:

$| \psi \rangle = \frac {|1\rangle + |2\rangle + |3\rangle + |4\rangle + |5\rangle + |6\rangle} {\sqrt{6}}$

Where it should be noted that the "+" sign of a probabilistic equation is not an addition operator, it is a standard probabilistic or Boolean logical OR operator. The state vector is inherently defined as a probabilistic mathematical object such that the result of a measurement is one outcome OR another outcome.

It is clear that on each throw, only one of the states will be observed, but it is also clear that there is no requirement for any notion of collapse of the wave function/reduction of the state vector, or for the die to physically exist in the summed state. In the ensemble interpretation, wave function collapse would make as much sense as saying that the number of children a couple produced, collapsed to 3 from its average value of 2.4.

The state function is not taken to be physically real, or be a literal summation of states. The wave function, is taken to be an abstract statistical function, only applicable to the statistics of repeated preparation procedures, similar to classical statistical mechanics. It does not directly apply to a single experiment, only the statistical results of many.

### Criticism

David Mermin sees the Ensemble interpretation as being motivated by an adherence ("not always acknowledged") to classical principles.

"[...] the notion that probabilistic theories must be about ensembles implicitly assumes that probability is about ignorance. (The 'hidden variables' are whatever it is that we are ignorant of.) But in a non-deterministic world probability has nothing to do with incomplete knowledge, and ought not to require an ensemble of systems for its interpretation".

However, according to Einstein and others, a key motivation for the ensemble interpretation is not about any alleged, implicitly assumed probabilistic ignorance, but the removal of "…unnatural theoretical interpretations…". A specific example being the Schrödinger Cat problem stated above, but this concept applies to any system where there is an interpretation that postulates, for example, that an object might exist in two positions at once.

Mermin also emphasises the importance of describing single systems, rather than ensembles.

"The second motivation for an ensemble interpretation is the intuition that because quantum mechanics is inherently probabilistic, it only needs to make sense as a theory of ensembles. Whether or not probabilities can be given a sensible meaning for individual systems, this motivation is not compelling. For a theory ought to be able to describe as well as predict the behavior of the world. The fact that physics cannot make deterministic predictions about individual systems does not excuse us from pursuing the goal of being able to describe them as they currently are."[5]

## Single particles

According to proponents of this interpretation, no single system is ever required to be postulated to exist in a physical mixed state so the state vector does not need to collapse.

It can also be argued that this notion is consistent with the standard interpretation in that, in the Copenhagen interpretation, statements about the exact system state prior to measurement cannot be made. That is, if it were possible to absolutely, physically measure say, a particle in two positions at once, then quantum mechanics would be falsified as quantum mechanics explicitly postulates that the result of any measurement must be a single eigenvalue of a single eigenstate.

### Criticism

Arnold Neumaier finds limitations with the applicability of the ensemble interpretation to small systems.

"Among the traditional interpretations, the statistical interpretation discussed by Ballentine in Rev. Mod. Phys. 42, 358-381 (1970) is the least demanding (assumes less than the Copenhagen interpretation and the Many Worlds interpretation) and the most consistent one. It explains almost everything, and only has the disadvantage that it explicitly excludes the applicability of QM to single systems or very small ensembles (such as the few solar neutrinos or top quarks actually detected so far), and does not bridge the gulf between the classical domain (for the description of detectors) and the quantum domain (for the description of the microscopic system)".

(spelling amended) [6]

However, the "ensemble" of the ensemble interpretation is not directly related to a real, existing collection of actual particles, such as a few solar neutrinos, but it is concerned with the ensemble collection of a virtual set of experimental preparations repeated many times. This ensemble of experiments may include just one particle/one system or many particles/many systems. In this light, it is arguably, difficult to understand Neumaier's criticism, other than that Neumaier possibly misunderstands the basic premise of the ensemble interpretation itself.[citation needed]

## Schrödinger's cat

The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. That being the case, the state vector would not apply to individual cat experiments, but only to the statistics of many similar prepared cat experiments. Proponents of this interpretation state that this makes the Schrödinger's cat paradox a trivial non-issue. However, the application of state vectors to individual systems, rather than ensembles, has claimed explanatory benefits, in areas like single-particle twin-slit experiments and quantum computing (see Schrödinger's cat applications). As an avowedly minimalist approach, the ensemble interpretation does not offer any specific alternative explanation for these phenomena.

## The frequentist probability variation

The claim that the wave functional approach fails to apply to single particle experiments cannot be taken as a claim that quantum mechanics fails in describing single-particle phenomena. In fact, it gives correct results within the limits of a probabilistic or stochastic theory.

Probability always requires a set of multiple data, and thus single-particle experiments are really part of an ensemble — an ensemble of individual experiments that are performed one after the other over time. In particular, the interference fringes seen in the double-slit experiment require repeated trials to be observed.

## The quantum Zeno effect

Main article: Quantum Zeno effect

Leslie Ballentine promoted the ensemble interpretation in his book Quantum Mechanics, A Modern Development. In it,[7] he described what he called the "Watched Pot Experiment". His argument was that, under certain circumstances, a repeatedly measured system, such as an unstable nucleus, would be prevented from decaying by the act of measurement itself. He initially presented this as a kind of reductio ad absurdum of wave function collapse.[8]

The effect has been shown to be real. Ballentine later wrote papers claiming that it could be explained without wave function collapse.[9]

## Earlier Classical Ensemble ideas

Early proponents of statistical approaches regarded quantum mechanics as an approximation to a classical theory. John Gribbin writes:

"The basic idea is that each quantum entity (such as an electron or a photon) has precise quantum properties (such as position or momentum) and the quantum wavefunction is related to the probability of getting a particular experimental result when one member (or many members) of the ensemble is selected by an experiment"

However, hopes for turning quantum mechanics back into a classical theory were dashed. Gribbin continues:

"There are many difficulties with the idea, but the killer blow was struck when individual quantum entities such as photons were observed behaving in experiments in line with the quantum wave function description. The Ensemble interpretation is now only of historical interest."[10]

Willem de Muynck describes an "objective-realist" version of the ensemble interpretation featuring counterfactual definiteness and the "possessed values principle", in which values of the quantum mechanical observables may be attributed to the object as objective properties the object possesses independent of observation. He states that there are "strong indications, if not proofs" that neither is a possible assumption.[11]

## References

1. ^ "The statistical interpretation of quantum mechanics" (PDF). Nobel Lecture. December 11, 1954.
2. ^ Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York)
3. ^ Leslie E. Ballentine (1998). Quantum Mechanics: A Modern Development. World Scientific. Chapter 9. ISBN 981-02-4105-4.
4. ^ Quantum Mechanics, A Modern Development, p.48
5. ^ Mermin, N.D. The Ithaca interpretation
6. ^ Arnold Neumaier's FAQ
7. ^ Leslie E. Ballentine. Quantum Mechanics: A Modern Development. p. 342. ISBN 981-02-4105-4.
8. ^ "Like the old saying "A watched pot never boils", we have been led to the conclusion that a continuously observed system never changes its state! This conclusion is, of course false. The fallacy clearly results from the assertion that if an observation indicates no decay, then the state vector must be |y_u>. Each successive observation in the sequence would then "reduce" the state back to its initial value |y_u>, and in the limit of continuous observation there could be no change at all. Here we see that it is disproven by the simple empirical fact that [..] continuous observation does not prevent motion. It is sometimes claimed that the rival interpretations of quantum mechanics differ only in philosophy, and can not be experimentally distinguished. That claim is not always true. as this example proves". Ballentine, L. Quantum Mechanics, A Modern Development(p 342)
9. ^ "The quantum Zeno effect is not a general characteristic of continuous measurements. In a recently reported experiment [Itano et al., Phys. Rev. A 41, 2295 (1990)], the inhibition of atomic excitation and deexcitation is not due to any collapse of the wave function, but instead is caused by a very strong perturbation due to the optical pulses and the coupling to the radiation field. The experiment should not be cited as providing empirical evidence in favor of the notion of wave-function collapse." Physical Review
10. ^ John Gribbin. Q is for Quantum. ISBN 978-0684863153.
11. ^ Quantum Mechanics the Way I see it