Cooking-Specker Theorem

The Koch-Specker theorem (KS theorem) is a sentence from the field of the fundamentals of quantum mechanics , which proves the impossibility of a non-contextual model with hidden variables of quantum mechanics. In addition to Bell's inequality , it is probably the second best-known so-called “no-go theorem” (proof of impossibility) about hidden variables in quantum mechanics. The KS theorem was proven by John Stewart Bell in 1966 and formulated by Simon Cooking and Ernst Specker in 1967.

introduction

The debate about the completeness of quantum mechanics in terms of a realistic physical theory originated in a thought experiment published in 1935 by Albert Einstein , Boris Podolsky and Nathan Rosen (EPR). It became known as the Einstein-Podolsky-Rosen paradox (EPR paradox). For a long time the critique of quantum mechanics contained in the paradox was considered undecidable. This changed in the mid-1960s with a series of publications by John Bell . In them he put the questions raised by Einstein, Podolsky and Rosen in a general mathematical context. In addition, he presented Bell's inequality , named after him , which shows a way to experimentally test the EPR paradox. Bell's performance can be described in such a way that he translated the possibly misleading concept of reality from the EPR criticism into the context of a general theory with local hidden variables. From the inequality that is always valid for such theories, as can be read in the associated article , inequalities for the expected values of certain quantum mechanical observables can be derived, which are clearly violated in the mathematical formulation of quantum mechanics .

If one considers the experimental verification of the violation of Bell's inequality in quantum mechanics to be beyond doubt (even if this is still discussed in the scientific community), then there is only one choice between two options:

(NL) Quantum mechanics allows a realistic interpretation in the sense of a hidden variable model, but this model is not local. This interpretation follows z. B. Bohm's mechanics .
(NR) Quantum mechanics is a local theory and therefore does not allow a realistic interpretation. This approach is followed in particular by the operational interpretation, which is at least recognized as a minimal interpretation by practically all scientists.

If we are interested in a realistic interpretation, Bell's inequalities force us to choose a non-local model. Although the non-locality of such a model does not imply any violation of the principle of causality (and thus e.g. of the theory of relativity) in any operational sense, i. i.e., you can e.g. For example, if you do not build devices that transmit information instantaneously from A to B, it is precisely the realistic interpretation of the model that in turn gives most scientists a headache. The realistic interpretation says that we interpret the measurement results that we have received in a single measurement on an individual quantum system as real properties of this individual system. But then we are forced to consider the instantaneous change in the properties of a distant system as a real effect (even if this is not measurable). This spooky action at a distance is seen by the majority of scientists as contrary to the spirit of the theory of relativity and is therefore rejected, although it is at the same time represented by numerous well-known physicists.

The Koch-Specker theorem restricts the possibility of a hidden variable model of quantum mechanics in a further direction. While the discussion from EPR to Bell mainly related to the necessarily non-local aspects of such models, in 1967 Koch and Specker took up a discussion that had already been initiated by John von Neumann in 1932 and related to the so-called contextuality of the models dealt with measurements on individual systems.

Von Neumann had already described the incompatibility of quantum mechanics with hidden variables in his groundbreaking book The Mathematical Foundations of Quantum Mechanics . His argumentation turned out to be sketchy at first, but with the help of the findings of Andrew Gleason from 1957 and Bell from 1966 (albeit a different publication than the above-mentioned publication) , Koch und Specker were able to specify the argument more precisely and cast it into a mathematical sentence .

Contextuality

The statistics of measurement results on an ensemble of quantum systems predicted by mathematical theory and those determined in the experiment undoubtedly agree very well. When trying to interpret these statistics in the sense of a model with hidden variables, the basic assumption is that the measurement results of the individual measurements have real meaning , i.e. actually provide information about the physical state of this individual system. The term “ physical state ” used here is therefore such that this state is completely determined by certain internal parameters (hidden variables) of the individual system and precisely defines the measured value in the context of the measurement. For the relationship between the hidden variables and the measured values obtained, it seems sensible and obvious a priori to make the following assumptions:

(WD) (Value-Definedness) The measured values of a certain observable on a single system are definite , i.e. That is, they are concretely fixed at all times and determine a property of the individual system.
(NK) (non-contextuality) If an individual quantum system has a certain property that leads to a certain measured value, the system has this property regardless of the context of the measurement , in particular the measured value is independent of how the measurement is specifically structured .

The assumption (WD) naturally arises from our empirical knowledge of measurements under the assumption of the reality of the property being measured. This assumption and the assumption (NK), however modest they may seem at first, are a major limitation of the model. In fact, all existing hidden variable models of quantum mechanics, such as Bohm's mechanics, are contextual, and not without reason: The KS theorem proves that a model with hidden variables of quantum mechanics is not both (WD) and (NK) can suffice.

The KS theorem

There is no such thing as a non-contextual, hidden variable model of quantum mechanics.

In order to prove the above statement, some technical lead is required. The core of the KS theorem is a rather inconspicuous sentence about the geometric structure of the quantum mechanical Hilbert space . However, the theorem gains its essential strength from the derivation of certain calculation rules for the measured values of the individual systems with regard to various observables, which can be derived from (WD) and (NK). We want to sketch the derivation of these rules here:

The mathematical model of quantum mechanics describes a state in the sense of an ensemble by a density operator ρ, or by a Hilbert space vector ψ in the case of a pure state. Observables are described by self-adjoint operators whose eigenvalues are possible measured values. For two observables A and B and any state ρ, the following calculation rule (linearity) applies to the expected values:

{\ displaystyle \ mathbb {E} _ {\ rho} (\ lambda \ mathbf {A} + \ mu \ mathbf {B}) = \ lambda \ mathbb {E} _ {\ rho} (\ mathbf {A}) + \ mu \ mathbb {E} _ {\ rho} (\ mathbf {B}).}

But since the compatibility of observables in particular requires the simultaneous measurability of these observables, for compatible observables A and B , a common measurement of the two observables also implies a measurement of the observables C = A + B and D = AB by simply adding the measured values , or multiplied. Assuming the definiteness of values, each individual system of the ensemble now becomes values

{\ displaystyle {\ text {(KSa)}} \ qquad \, \ nu (\ mathbf {A}), \, \ nu (\ mathbf {B}), \, \ nu (\ mathbf {C}) \ , {\ text {and}} \ nu (\ mathbf {D})}

assigned, which determine the measured values in a possible measurement. In particular, the values of the composite observables must satisfy the condition

{\ displaystyle {\ text {(KSb)}} \ qquad \, \ nu (\ mathbf {C}) = \ nu (\ mathbf {A}) + \ nu (\ mathbf {B}) \, {\ text {and}} \, \ nu (\ mathbf {D}) = \ nu (\ mathbf {A}) \ cdot \ nu (\ mathbf {B})}

because the measured values of these observables can just be determined operationally and these values exist free of the context of the measurement. This brings us to the core of the theorem:

Claim:

The two requirements from (KSb) can not be fulfilled in quantum mechanics for any pairs of compatible observables A and B that define the values from (KSa).

To prove this claim we will construct a counterexample. It is sufficient to choose a finite-dimensional Hilbert space and to specify a finite number of concrete observables with which one then contradicts (KSa) and (KSb). In fact, the smallest Hilbert space in which this contradiction is possible is the three-dimensional case. As one can easily show, the KS theorem does not apply in two-dimensional vector spaces over the complex numbers . But this is not a problem for the general statement, because quantum mechanics generally uses higher-dimensional spaces. Since the space for which the counterexample with the fewest observables is known is four-dimensional, the counterexample by A. Cabello seems suitable for demonstration: ${\ displaystyle \ mathbb {C}}$

To do this, consider a four-dimensional vector space over , with an orthogonal basis . The projector on the subspace generated by a vector has the eigenvalues 0 and 1 and belongs to a “yes-no” measurement. The projectors to belonging to the base commute in pairs and are therefore compatible with one another. From (KSb) it follows for these operators ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle (u_ {1}, u_ {2}, u_ {3}, u_ {4})}$ ${\ displaystyle P_ {u}}$ ${\ displaystyle u}$ ${\ displaystyle P_ {u_ {1}}}$ ${\ displaystyle P_ {u_ {4}}}$

{\ display style \ nu (P_ {u_ {1}}) + \ nu (P_ {u_ {2}}) + \ nu (P_ {u_ {3}}) + \ nu (P_ {u_ {4}}) = \ nu (\ mathbf {1}) = 1,}

because the sum of the four projectors results in the unity operator, which represents the observable, which always delivers the measured value 1. The identity also follows from the product rule in (KSb), since every observable R is compatible with 1 and therefore holds. You can also see and therefore either 0 or 1 must be. It follows that in the above sum, exactly one term must be 1 and the other three must be 0. ${\ displaystyle \ nu (\ mathbf {1}) = 1}$ ${\ displaystyle \ nu (\ mathbf {R}) \, \ nu (\ mathbf {1}) = \ nu (\ mathbf {R} \ cdot \ mathbf {1}) = \ nu (\ mathbf {R}) }$ ${\ displaystyle \ nu (P_ {u}) = \ nu (P_ {u} ^ {2}) = \ nu (P_ {u}) ^ {2}}$ ${\ displaystyle \ nu (P_ {u})}$

Now choose 18 suitable vectors and form nine different bases from four orthogonal vectors: ${\ displaystyle \ mathbb {R} ^ {4}}$

${\ displaystyle u_ {1}}$	(0,0,0,1)	(0,0,0,1)	(1, -1,1, -1)	(1, -1,1, -1)	(0,0,1,0)	(1, -1, -1.1)	(1.1, -1.1)	(1.1, -1.1)	(1,1,1, -1)
${\ displaystyle u_ {2}}$	(0,0,1,0)	(0,1,0,0)	(1, -1, -1.1)	(1,1,1,1)	(0,1,0,0)	(1,1,1,1)	(1,1,1, -1)	(-1,1,1,1)	(-1,1,1,1)
${\ displaystyle u_ {3}}$	(1,1,0,0)	(1,0,1,0)	(1,1,0,0)	(1.0, -1.0)	(1,0,0,1)	(1,0,0, -1)	(1, -1,0,0)	(1,0,1,0)	(1,0,0,1)
${\ displaystyle u_ {4}}$	(1, -1,0,0)	(1.0, -1.0)	(0,0,1,1)	(0,1,0, -1)	(1,0,0, -1)	(0.1, -1.0)	(0,0,1,1)	(0,1,0, -1)	(0.1, -1.0)

Each column of this table represents a basis of orthogonal vectors. Each of the 18 vectors occurs exactly twice (the same vectors are colored in the same color). Let us consider any single system in the state ρ. We can obtain a projection operator from every vector in the table above . The application of this operator to the state ρ corresponds to the decision of a yes / no question, represented by the possible measured values 0 and 1. Quantum mechanics only predicts with certain probabilities which value will actually be measured. According to the assumption (KSa), on the other hand, the value is measured with certainty (the value may depend on the state ρ, but since we are only looking at one state, we do not need to note this separately). Since every vector occurs exactly twice in the table and must be either 0 or 1, the sum of these values over the entire table is an even natural number. On the other hand, the sum of these values - as derived from (KSb) above - must be 1 in each individual column, resulting in a total of 9. This is the contradiction that makes the assumption of (KSa) and (KSb) impossible. ${\ displaystyle u}$ ${\ displaystyle P_ {u}}$ ${\ displaystyle \ nu (P_ {u})}$ ${\ displaystyle u}$ ${\ displaystyle \ nu (P_ {u})}$

Remarks

As noted above, the Koch-Specker theorem only excludes a certain class of hidden-variable models, namely those that are not contextual. Contextual models that can actually be constructed therefore do not meet the requirements for value-definiteness and non-contextuality. An analysis of such models quickly shows where the contextuality of such models comes from: not only variables hidden in the state space are introduced, but also in the space of the observables. A quantum mechanical observable is in the context of such a model, therefore, a so-called fuzzy Observable ( fuzzy Observable define) on the space of hidden variables states modeling the individual systems. These fuzzy observables can be seen in the analogous sense as a mixture of sharp variables (with definite measured values), just as a mixed state is composed of pure states.

swell

↑ S. Cooking, E. Specker: The Problem of Hidden Variables in Quantum Mechanics. In: Journal of Mathematics and Mechanics. Volume 17, No. 1, July 1967, pp. 59-87.
^ Max Born, Albert Einstein: Albert Einstein, Max Born. Correspondence 1916–1955. Nymphenburger, Munich 1955, p. 210.
^ J. Bell: On the Problem of Hidden Variables in Quantum Mechanics. In: Reviews of Modern Physics. 38, 1966, pp. 447-452.
^ Carsten Held: The Cooking-Specker Theorem. In: Edward N. Zalta (Ed.): The Stanford Encyclopedia of Philosophy. Winter 2003 Edition,
↑ A. Cabello: Proof with 18 Vectors of the Bell-Cooking-Specker Theorem. In: New Developments on Fundamental Problems in Quantum Physics. Kluwer Ac. Press (see online version on arXiv.org )
^ RD Gill, MS Keane :: A Geometric Proof of the Cooking-Specker No-Go Theorem. In: J. Phys. A: Math. Gen. 29, 1996.

Web links

[1] S. Cooking, E. Specker: The Problem of Hidden Variables in Quantum Mechanics. In: Journal of Mathematics and Mechanics. Volume 17, No. 1, July 1967, pp. 59-87.

[Einstein1955-2] Max Born, Albert Einstein: Albert Einstein, Max Born. Correspondence 1916–1955. Nymphenburger, Munich 1955, p. 210.

[3] J. Bell: On the Problem of Hidden Variables in Quantum Mechanics. In: Reviews of Modern Physics. 38, 1966, pp. 447-452.

[4] Carsten Held: The Cooking-Specker Theorem. In: Edward N. Zalta (Ed.): The Stanford Encyclopedia of Philosophy. Winter 2003 Edition,

[5] A. Cabello: Proof with 18 Vectors of the Bell-Cooking-Specker Theorem. In: New Developments on Fundamental Problems in Quantum Physics. Kluwer Ac. Press (see online version on arXiv.org )

[6] RD Gill, MS Keane :: A Geometric Proof of the Cooking-Specker No-Go Theorem. In: J. Phys. A: Math. Gen. 29, 1996.