Kochen–Specker theorem
Cooking - Specker theorem (KS theorem) is a set from the field of foundations of quantum mechanics, which proves the impossibility of non-contextual model with hidden variables in quantum mechanics. In addition to Bell's inequality, it is probably the second most popular so-called " no-go theorem " (impossibility proof ) of hidden variables in quantum mechanics. The KS theorem was formulated in its original form by Simon Cook and Ernst Specker in 1967.
Introduction
The debate about the completeness of quantum mechanics in terms of a realistic physical theory had its origin in a thought experiment in 1935 by Albert Einstein, Boris Podolsky and Nathan Rosen (EPR ) has been released. It became known as the Einstein - Podolsky-Rosen paradox ( EPR paradox). For a long time the plug end into the paradox criticism of quantum mechanics was considered undecidable. Mid-1960s this changed with a series of papers by John Bell. In them, he asked the questions raised by Einstein, Podolsky and Rosen in a general mathematical context. He also presented the eponymous Bell's inequality, which shows a way of experimental verifiability of the EPR paradox. The performance Bells can be described so that he translated the possibly misleading notion of reality from the EPR criticism in the context of a general theory of local hidden variables. From the always valid for such theories inequality can, as can be read in the accompanying article, inequalities for the expectation values of certain quantum mechanical observables are derived, which are clearly violated in the mathematical formulation of quantum mechanics.
Considering the experimental verification of the violation of Bell's inequality in quantum mechanics free from doubt (although this is still debated in the scientific community ), there remains only the choice between two paths:
- ( NL) Quantum mechanics allows for a realistic interpretation in terms of a hidden - variable model, this model is not local. This interpretation follows, for example, the Bohm'sche mechanics.
- (NR) Quantum mechanics is a local theory, and therefore leaves no realistic interpretation. This trail follows the particular operational interpretation that is at least recognized as minimal interpretation by virtually all scientists.
The Bell's inequalities thus forcing us if we are interested in a realistic interpretation ( and that's probably the majority of scientists ) to choose a non-local model. Although the non-locality of such a model, although no breach of the principle of causality (and thus, for example, against the theory of relativity ) implies in any operational sense, that is, you can build, for example, no devices, the information about instantaneously by transmit A to B, it's just the realistic interpretation of the model, which in turn prepares most scientists headache. The realistic interpretation states precisely that we, as real properties of this unique system to interpret the measurement results that we have obtained in a single measurement on an individual quantum system. Then, however, 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 perceived by the majority of scientists as against the spirit of the theory of relativity and therefore discarded, being but also members of numerous well-known physicist.
Cooking - Specker theorem restricts the possibility of a hidden - variable model of quantum mechanics, however, in a broader sense. While the discussion of EPR to Bell primarily related to the necessarily non-local aspects of such models, attacked cooking and Specker in 1967 a discussion on who had already initiated John von Neumann in 1932, and with the so-called contextuality of the models in terms focused on measurements on individual systems.
Von Neumann had already The mathematical foundations of quantum mechanics described at the time in his groundbreaking book, the incompatibility of quantum mechanics with hidden variables. His reasoning proved initially to be incomplete, but could cook and Specker using the insights of Andrew Gleason in 1957 and Bell in 1966 (but other than the above-mentioned publication ) specify the argument and pour into a mathematical sentence.
Contextuality
The predicted by the mathematical theory and the observed in the experiment statistics of measurement results on an ensemble of quantum systems agree undoubtedly agree well. In attempting to interpret these statistics in terms of a model with hidden variables, the assumption is fundamental that the measurement results of individual measurements have real meaning, that is actually shed light on the physical state of this single system. As used herein, the concept of physical state is therefore such that this state is completely determined by certain internal parameters (hidden variables ) of the single system and sets exactly in the context of measurement the measured value. For the relation between the hidden variables and the measured values obtained it appears a priori sensible and obvious to make the following assumptions:
- (WD ) ( value - definedness ) The measured values of a given observable in a single system are definite, that is, they are at all times will be concrete and determine a property of the individual system.
- (NK) (non- Contextuality ) When a single -quantum-system has a certain property, which leads to a certain measuring value, the system has this property, regardless of the context of the measurement, in particular the measurement value is independent of how the measurement is specifically constructed.
The assumption (WD ) arises naturally from our empirical knowledge about measurements assuming the reality of the measured property. This assumption and the assumption (NK ), however, are so modest they are therefore used initially, a significant limitation of the model. In fact, that is all existing hidden - variable models of quantum mechanics, such as the Bohm'sche mechanics, contextually, and not without reason: The KS theorem proves that a model with hidden variables in quantum mechanics, not so much (WD ) and (NK ) may be sufficient.
KS theorem
To prove the above statement, some technical advance is necessary. The core of the KS theorem is a rather unassuming sentence about the geometric structure of the quantum mechanical Hilbert space. His substantial force wins the theorem, however, by the discharge of certain rules for computing the values of the individual systems with respect to various observables, which can be derived from (WD ) and ( NK). We want to outline the derivation of these rules here:
The mathematical model of quantum mechanics describes a condition in terms of an ensemble by means of a density ρ operator, and ψ by a Hilbert space vector in the case of a pure state. Observables are described by self-adjoint operators whose eigenvalues are possible readings. For two observables A and B and an arbitrary state ρ the following calculation rule (linearity ) is valid for the expected values :
But since compatibility of observables in particular simultaneous measurability of observables conditional, applies for compatible observables A and B, that a joint measurement of two observables, a measurement of the observable C = A B and D = AB implies, by simply adding the measured values or multiplied. Assuming the value - definiteness now be each individual system of the ensemble values
Assigned to determine the measured values with a possible measurement. In particular, the values of the composite observables have the condition
Meet, as the measured values of these observables yes operationally straight can be determined and so these values exist without the context of the measurement. In order to get to the core of the theorem:
Assertion:
To carry out the proof of this assertion, we will construct a counterexample. It is sufficient to choose a finite dimensional Hilbert space and specify a finite number of specific observables with which one then brings about a contradiction to ( KSa ) and ( KSb ). In fact, the smallest the Hilbert space, in which the contradiction is possible, the three-dimensional case. In two-dimensional vector spaces over the complex numbers, the KS theorem, as one can easily show does not apply. This is not a problem for the general statement, because quantum mechanics used eventually in general higher-dimensional spaces. Because the space for which the counterexample with the fewest observables is known, is four-dimensional, the counterexample of A. Cabello appear to demonstrate appropriate here:
Consider this a four-dimensional vector space, with a orthogonal basis. The projector onto the subspace generated by a vector, have the eigenvalues 0 and 1 and belong to a " yes-no " measurement. The members of the base projectors to commute in pairs and are therefore compatible with each other. From ( KSb ) therefore follows for these operators
As the sum of the four projectors gives the identity operator which represents the observable, which always supplies the measured value of 1. The identity follows from the product rule in ( KSb ), since each observable R is compatible with 1 and is therefore considered. Similarly, one sees and must therefore be either 0 or 1. Consequently, exactly the above sum must be a term equal to 1 and the other three is equal to 0
Now Man 18 suitable vectors choose and form of four orthogonal vectors nine different bases:
Each column of this table represents a base of orthogonal vectors dar. Each of the 18 vectors is doing exactly twice before (same vectors are in the same color dyed ). For a given system, there are single assuming (KSA) for each vector in the table a measured value. Since each vector appears exactly twice and must be either 0 or 1, the sum of these values over the entire table is always an even integer. On the other hand, the sum of these values in each column must be 1, so that a total of 9 results. This is the contradiction that leads to the adoption of ( KSa ) and ( KSb ) is impossible.
Comments
As noted above, the cooking - Specker theorem excludes only a certain class of hidden - variables models, namely those which are not contextual. Contextual models that you can actually construct, therefore the demand for value - definiteness and non - contextuality not fulfill. An analysis of such models quickly shows where the contextuality of such models emerged: it will be introduced not only in the state space hidden variables, but also in the space of observables. A quantum mechanical observables is therefore within the scope of such a model on the space of hidden - variable states that model the individual systems, define a so-called unsharp observables ( fuzzy observables ). This fuzzy observables can in the analog sense as mixed from sharp variables ( with definite measured values) are seen as a mixed state is composed of pure states.