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.
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.
According to quantum theory, pure physical states correspond to equivalence classes of state vectors, where any two members of one class differ by a complex factor. The point is that such a factor does not change the probability for the occurrence of any measurement result as computed within the formalism of quantum mechanics. In the formalism to be presented here, the state vector does not only determine the probabilities for the occurrence of measurement results, rather it determines the result itself. The information which guides the selection process is obtained from the state vector in a way that is not invariant against multiplication by a complex factor. It can be seen as a kind of global phase of the state vectors. This global phase then is a perfectly hidden variable. For this presentation of quantum mechanics we have a remarkable parallelism to classical statistical mechanics: If we would know exactly the initial state of a system, we could compute the outcomes for all experiments to be done later on this system. In the quantum case the initial state is to be represented by its state vector with specified phase, e.g. as a concrete complex-valued wave function.
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 quantum state contains all accessible information about a system. However, for various reasons, the information is less complete than we might expect given the classical states we are used to. This suggests the possibility that the state itself contains a measure of entropy (lack of information), which could potentially arise in two ways. Firstly, quantum states contain an inherent uncertainty with respect to the measurement outcomes of conjugate variables. Secondly, due to limitations in the preparation process, we may not be sure which state a system is occupying. The best we can do in this situation is to assign each potential state a classical probability and construct a specialist mathematical representation to deal with the situation. This is known as the density matrix. It turns out to be a rather handy tool in general and crucial for the exploration of quantum entropy. So, in this chapter, we are going to extend our technical understanding of quantum theory and the representation of states. This will arm us sufficiently to venture into the physics of the measurement problem and the entropy of quantum systems, which we will tackle in Chapter 12 .
The nature of measurements, and their place within quantum theory, has engaged physicists and philosophers for generations [Bra92; Sch03]. Much of that interest centered on variables such as position and momentum of free particles, whose values form a continuum. The present monograph deals with discrete quantum states; the measurements are those required to specify as completely as possible a particular discrete quantum state Ψ or, more generally, a density matrix ρ defined within a finite-dimensional Hilbert space. General remarks General system . At the outset we assume that the possible quantum states are a small number – the N essential states used in formulating the time-dependent Schrödinger equation or specifying the dimensions of the density matrix. To completely characterize a density matrix for such a system we require the N 2 elements. Of these the N diagonal elements are real valued, while the off-diagonal elements of the upper right side are complex conjugates of those on the lower left. Thus with allowance for the requirement of unit trace a total of N 2 – 1 real numbers suffice to completely specify the density matrix. These values must be consistent with the constraints discussed in Sec. 16.6.3. Pure state . If it is known that the system is in a pure state, we require the magnitude and phase of N probability amplitudes. These are constrained by normalization, and so only 2 N –1 real numbers are needed. Out of these 2 N – 1 parameters the overall phase of the statevector is usually not of interest (but see Chap. 20).
In classical physics, the ‘state’ of a particle, at any particular time, is taken to involve a definite position in space and a definite motion; and the state of a system of particles is taken to involve a definite position in space and a definite motion for each of the particles. The situation in relation to quantum mechanics is completely different. As in classical physics, calculations can be made concerning the position and motion of a particle. However, this representation is not by means of symbols for scalar or vector quantities: it is by a more complex mathematical object. This quantum state does not indicate or represent position or motion in any direct way: what it does in substance is to enable calculation of the probability of the particle in question being found to have a position or momentum if an appropriate measurement is made.
Time relates to actualization, which is physically interpreted as a quantum measurement. The measurement has a dual structure consisting of the reality being measured (externality) and the reality by which the measurement takes place (an agent or ontolon). These two constituents refer to the two types of time: physical time measured by clocks, and internal time, defined by Bergson as la durée (duration). The latter holds the coherent internal quantum state of entangled potentialities, while the former represents the collapse of this state, thereby flowing through quantum transactions. This view unifies the Everett (Many-worlds) and the Copenhagen (Wave function collapse) interpretations of the quantum measurement. Time as a duration can be estimated via the energy-time uncertainty relation, which links the precision of the measurement result to the value of energy dissipation during it and to its span. The precision of the measurement result enables the process of quantum computation, which emerges in the systems characterized by internal closure, i.e., autopoietic (living) systems operating at the state of sustainable non-equilibrium. Energy dissipation in the process of measurement occurs as the emission of quanta that trigger the collapse of the coherent entangled state. These quanta are recognized in the system, enabling the performance of higher-level control over elementary computational actions through their integration within the complex system. It is concluded that the energy-time uncertainty relation represents the fundamental internal quantum constraint of natural computation, the process that spans from enzymatic catalysis to complex regulatory and adaptive activities, and finally to reflexive consciousness.