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.
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.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.
The ability to transfer information securely is essential in the modern world we live in. While the field of cryptography has long researched ways to do this, events over the last couple of years have brought the issues of secure information transfer squarely into the public eye. As a consequence, security and cryptography have quickly become issues of social and political importance, sparking debates about the values of privacy and the role of government. An essential ingredient of modern cryptography is the generation of randomness: cryptographic techniques are built on the premise that one has access to random bits. It is well known, however, that computers cannot produce algorithmically random sequences – that is, sequences with maximal algorithmic information content – but are doomed to produce ‘pseudorandomness’. This lack of randomness can be, and has been, exploited to compromise security, see [4]. The active field of quantum information theory has proposed approaches to provide supposedly ‘unbreakable’ security by exploiting various quantum phenomena. This security unfortunately relies on assumptions about the nature of quantum measurements and their ability to generate random bits. Anton Zeilinger summarises this by postulating that the simplest quantum systems, qubit, can hold only one bit of classical information [6]. This foundational principle is in line with a wider paradigm shift to view quantum information as an extension of classical information, but it is nonetheless unsatisfying to simply postulate this principle, given its importance in determining the practical advantages of quantum information theory. In order to understand better how quantum mechanics can help generate meaningful information, we instead look to relate the outcomes of quantum measurements to formal properties of the system based on more fundamental assumptions. Indeed, we have shown that a) some of the postulated properties of quantum information follow from the formal structure of the theory and b) a purely formal notion of information within a quantum world can generate, via measurement, meaningful and useful information in the macroscopic world (for example, in cryptography). The indeterminism of quantum measurements can be formalised via the notion of value indefiniteness. To explain this concept, let us consider an arbitrary quantum system and ask whether the outcome of a measurement of any observable quantity A (such as the energy of the system, its angular momentum (spin), etc.) is determined prior to the measurement. If this is the case, then we say the observable is value definite with value v(A); otherwise, the observable is value indefinite and v(A) is undefined. Mathematically, one can reduce all observable quantities to so-called projection observables, which can only take the values 0 or 1. Thus, the question of whether the outcome of several measurements can be simultaneously determined in advance can be rephrased in terms of the information ‘carried’ by a particular system in a definite quantum state. Classically, one expects that all quantities are determined in advance, and hence all observables are value definite. Quantum mechanically, however, the belief is that this is not the case, and the information content of quantum systems is limited. Formulating carefully the notion of value indefiniteness allows us to formalise the notion of (quantum) indeterminism; however, this doesn’t help clarify whether quantum systems are indeed value indefinite or not. Staying in this formal framework, the Kochen-Specker theorem [5] provides a first positive result, showing that at least some observables must be value indefinite if one makes the assumption known as non-contextuality, which states that any definite values that exist must be independent of other compatible measurements that may or may not be performed on the system. Under the same assumption, this theorem can be strengthened to show that only one single observable can have the definite value 1 (see [2]). Furthermore, only observables that can be measured simultaneously with this single one can have the definite value 0; the rest must all be value indefinite. Since the preparation of a system involves precisely ensuring that, usually via measurement, the system is in a definite state with respect to some desired observable, this result shows that no other incompatible observable can be value definite. That is, preparing a system in a definite state by making the ‘preparation’ observable value definite specifies completely the information content of the quantum system. This is an example of syntactical quantum information acquiring meaning at the level of the quantum itself. Can the syntactical information at the level of the quantum generate meaning outside the quantum, that is, at the macroscopic level? The results cited above hold in the Hilbert-space framework of quantum mechanics and are formulated only in terms of a syntactical notion of information. Their real importance becomes evident when one interprets them in the context of quantum measurements. Specifically, if we prepare a quantum system in a known state, they allow us to ‘locate’ observables which we can measure, but which are value indefinite; that is, observables whose measurement results are not specified by any pre-existing property of the quantum system. Furthermore, with respect to a mathematical model of unpredictability which we developed in [3], the results of these measurements can be shown to be absolutely unpredictable. In this way we get a mathematical explanation and justification of the largely accepted intuition that quantum mechanics is inherently unpredictable, and that this unpredictability arises from the phenomenon of indefiniteness within the quantum world. Furthermore, if one considers a hypothetical infinite sequence generated by the repeated measurement of a quantum value indefinite observable, one can prove that these sequences must be strongly incomputable, technically ‘bi-immune’ [1]. Such sequences cannot be generated by any Turing machine or classical computer, showing that value indefiniteness leads to a clear classical/quantum split in a purely algorithmic context. Importantly, bi-immunity is a property of observed, macroscopic quantities, quite separate from the quantum framework in which the value indefiniteness is formalised. This form of macroscopic meaning created from the lack of syntactic information is precisely the scenario that quantum random number generators try to create, and which is essential for the development and certification of quantum cryptographic systems. The fact that the macroscopic information created goes beyond anything classically obtainable serves as a valuable practical resource, outside of and removed from the quantum formalism. To conclude, quantum information creates meaning within the quantum and via measure- ment, the lack of information within the quantum, creates meaning and valuable information at the macroscopic level. Acknowledgments The second author thanks Prof. S. Marcus for useful conversations on quantum information theory. This work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. References and Notes Abbott, A.A.; Calude, C.S.; Svozil, K. Strong Kochen-Specker theorem and incomputability of quantum randomness. Physical Review A 2012, 86, 062109. Abbott, A.A.; Calude, C.S.; Svozil, K. Value-indefinite observables are almost everywhere. Physical Review A 2013, 89, 032109. Abbott, A.A.; Calude, C.S.; Svozil, K. On the unpredictability of individual quantum measurement outcomes. CDMTCS Research Report 2014, 458. Calude, C.S. Quantum randomness & cryptology. CyberTalk 2013, 3, 42–43. Kochen, S.; Specker, E. The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 1967, 17, 59–87. Zeilinger, A. A foundational principle for quantum mechanics. Foundations of Physics 1999, 29(4),631–643.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.
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.
Knowing and guessing, these are two essential epistemological pillars in the theory of quantum-mechanical measurement. As formulated quantum mechanics is a statistical theory. In general, a priori unknown states can be completely determined only when measurements on infinite ensembles of identically prepared quantum systems are performed. But how one can estimate (guess) quantum state when just incomplete data are available (known)? What is the most reliable estimation based on a given measured data? What is the optimal measurement providing only a finite number of identically prepared quantum objects are available? These are some of the questions we address. We present several schemes for a reconstruction of states of quantum systems from measured data: (1) We show how the maximum entropy (MaxEnt) principle can be efficiently used for an estimation of quantum states on incomplete observation levels. (2) We show how Bayesian inference can be used for reconstruction of quantum states when only a finite number of identically prepared systems are measured. (3) We describe the optimal generalized measurement of a finite number of identically prepared quantum systems which results in the estimation of a quantum state with the highest fidelity.
A classical state specifies the quantitative values of all the state variables relevant to a given system. In some situations, we can know the overall state of the system without being sure about the individual states of the constituent parts. On top of the problems inherent in picking out the crucial details of a system, there is another issue with trying to specify an initial state precisely: there is always some uncertainty connected with measurement. The quantum state of a system contains a series of complex numbers related to the probability that the system will collapse into a new state when a measurement takes place. Each of these new states represents a possible result of the measurement, which might be a path or a specific quantitative value of some physical variable.