The brief works of Peierls on the role of the observer in quantum mechanics are examined, interpreted and expanded to widen accessibility and understanding of these works. The approach followed here is very much in the spirit adopted by Peierls who eschewed a `rigorous axiomatic' and aimed at using `logic at the level of the working physicist'. The fundamental tenet of his work is that the wavefunction or density matrix represents the knowledge of an observer and that two observers of the same system may well have different knowledge and will use different density matrices to describe it. Essential to the understanding of Peierls's approach is the demonstration given here that the density matrix generally can be expressed entirely in terms of probabilities of observable outcomes and that such probabilities are subjective in the first instance.The process by which the knowledge of two observers may be combined is discussed in detail for some simple cases and, by using Bayes's theorem, the progress to a common understanding of the correct density matrix for a given situation demonstrated. Peierls gave a criterion to ensure that the amalgamation of data from two observers can never lead to violation of the uncertainty principle : that the density matrices of two observers must commute. It is pointed out that it is essential that the density matrices of two observers commute under all circumstances no matter how inaccurate or even fanciful the data or beliefs used to construct them. Once this is appreciated it can be shown that Peierls's density matrix commutation criterion is fully equivalent to the standard result in quantum mechanics that two observables must commute if they are to be known precisely and simultaneously,but now extended to imperfectly known mixed states rather than a single eigenstate.
The brief works of Peierls on the role of the observer in quantum mechanics are examined, interpreted and expanded to widen accessibility and understanding of these works. The approach followed here is very much in the spirit adopted by Peierls who eschewed a rigorous axiomatic' and aimed at using logic at the level of the working physicist'. The fundamental tenet of his work is that the wavefunction or density matrix represents the knowledge of an observer and that two observers of the same system may well have different knowledge and will use different density matrices to describe it. Essential to the understanding of Peierls's approach is the demonstration given here that the density matrix generally can be expressed entirely in terms of probabilities of observable outcomes and that such probabilities are subjective in the first instance.The process by which the knowledge of two observers may be combined is discussed in detail for some simple cases and, by using Bayes's theorem, the progress to a common understanding of the correct density matrix for a given situation demonstrated. Peierls gave a criterion to ensure that the amalgamation of data from two observers can never lead to violation of the uncertainty principle : that the density matrices of two observers must commute. It is pointed out that it is essential that the density matrices of two observers commute under all circumstances no matter how inaccurate or even fanciful the data or beliefs used to construct them. Once this is appreciated it can be shown that Peierls's density matrix commutation criterion is fully equivalent to the standard result in quantum mechanics that two observables must commute if they are to be known precisely and simultaneously,but now extended to imperfectly known mixed states rather than a single eigenstate.
Sir Rudolph Peierls, in a reply to John Bell's last critique of the state of our understanding of quantum mechanics, maintained that it is easy to give an acceptable account of the physical significance of the quantum theory. The key is to recognize that all the density matrix characterizing a physical system ever represents is knowledge about that system. In answer to Bell's implicit rejoinder "Whose knowledge?" Peierls offered two simple consistency conditions that must be satisfied by density matrices that convey the knowledge different people might have about one and the same physical system: their density matrices must commute and must have a non-zero product. I describe a simple counterexample to his first condition, but show that his second condition, which holds trivially if the first does, continues to be valid in its absence. It is an open question whether any other conditions must be imposed.
Sir Rudolph Peierls, in a reply to John Bell's last critique of the state of our understanding of quantum mechanics, maintained that it is easy to give an acceptable account of the physical significance of the quantum theory. The key is to recognize that all the density matrix characterizing a physical system ever represents is knowledge about that system. In answer to Bell's implicit rejoinder Whose knowledge? Peierls offered two simple consistency conditions that must be satisfied by density matrices that convey the knowledge different people might have about one and the same physical system: their density matrices must commute and must have a non-zero product. I describe a simple counterexample to his first condition, but show that his second condition, which holds trivially if the first does, continues to be valid in its absence. It is an open question whether any other conditions must be imposed.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.
A quantum state, via the Born Rule, assigns probabilities to the different possible values of observables of a quantum system. It is only in virtue of these probabilities — or, alternatively, expectation values — that quantum mechanics is empirically testable. Since assigning a state to a system means assigning probabilities to the values of observables, it may seem natural to read Peierls’ claim that the state ‘represents our knowledge of the system’ (Peierls [1991], p. 19) as a shorthand for saying that the state represents our knowledge of these probabilities. The latter view is sometimes even straightforwardly identified with the epistemic conception of states, for example by Marchildon, who claims that ‘[i]n the epistemic view [of quantum states], the state vector (or wave function or density matrix) does not represent the objective state of a microscopic system […], but rather our knowledge of the probabilities of outcomes of future measurements’ (Marchildon [2004], p. 1454). However, according to an argument due to Fuchs and endorsed by Timpson, 31 the idea that quantum states represent our knowledge about quantum probabilities should not be regarded as the core claim of the epistemic conception of quantum states, simply because, as they argue, this idea is in fact incompatible with the latter. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract The fundamentals of the theory of density matrices are reviewed. The density matrix is introduced first in terms of state vectors for pure states over which a probability distribution is given. It is then shown that the applications of the density matrix do not require knowledge of the particular probability distribution over pure states that may have been used for its original determination. It is stressed that density matrices, like state vectors, describe ensembles rather than single systems, as determinism in quantum theory (except for some conservation laws) is of a statistical character only. The entropy of a quantum‐mechanical system is defined as the average per system of the entropy of an ensemble of systems described by the quantum‐mechanical (pure or mixed) state. For pure states this entropy is zero, in agreement with the fact that state vectors constitute the maximal information that can be given about a quantum‐mechanical system. As the entropy should be independent of the particular probability distribution over pure states that may have been used to form the mixed state, it is calculated from statistical considerations about an imaginary ensemble in which the eigenvalues of the density matrix are probabilities over pure states given by the eigenfunctions of the density matrix. The entropy is calculated for a mixture of two nonorthogonal states. The time evolution of the density matrix, in general, is not a unitary transformation, as observable systems are not closed and therefore in the Hamiltonian contain unknown external parameters that are not the same for the entire mixture. Therefore transitions between pure states and mixed states take place. Those from pure to mixed states take place spontaneously and irreversibly under increase of entropy. Transitions from mixed states to pure states can be fabricated; an example is given. As Wigner's thought experiment of mixing coherently the two beams from a Stern‐Gerlach apparatus according to estimates of Bohm should be impossible by loss of coherence in the gradient magnetic field, it is indicated how Wigner's ideas may be realized by using, instead, polarized light beams from birefringent crystals. After a brief review of the quantum theory of measurements and the various arguments usually given for the reducibility of state after the measurement, Wigner's argument against blind use of these arguments is interpreted as a limit of validity of the familiar probability rule of quantum theory, which should not be applied after an unsuccessful measurement. By splitting light beams coherently by glass plates, one can by the same apparatus have measurements both of the relative intensities of two coherent beams, and of interference effects between the two beams. This shows that complementarity applies to the individual systems in the beam, as they reach the glass plate where a choice must be made between two complementary possibilities of future measurement. Finally, by describing Schrödinger's cat by a density matrix and asking about its time dependence, it is shown that, if it were possible to measure the interference effects between living and dead cats used by some people as a justification for writing down a state vector for the cat, then this measurement could be used to revive some dead cats.