Problem on reconstruction of state of finite-dimension quantum information transfer channel, pure or mixed, by results of measurements of needed number of observables, is considered. It is shown that in general case it is needed to measure incompatible observables in number exceeding by one dimension of space of vectors of state. Each of incompatible observables is measured in its statistically valuable series of measurements. In special case, when one of observables is a non-demolition observable, measurement of the other observables is needed for realization of control of property of non-demolition. In case of paired channel it is shown that results of measurements of observables that do not demolish states of sub-channels are characterized by mutual distribution of probabilities while results of measurement of over-classical observables are characterized by mutual correlation only. This correlation vanishes completely in case of pure unentangled states.
Peculiarities of multiqubit measurement are for the most part similar to peculiarities of measurement for qudit — quantum object with finite-dimensional Hilbert space. Three different interpretations of measurement concept are analyzed. One of those is purely quantum and is in collection, for a given state of the object to be measured, of incompatible observable measurement results in amount enough for reconstruction of the state. Two others make evident the difference between the reduced density matrix and the density matrices of physical objects involved in the measurement. It is shown that the von Neumann projectors, in combination with the concept of qudit phase space, produce an idea of a phase portrait of qudit state as a set of mathematical expectations for projectors on the possible pure states. The phase portrait is not a probability distribution since the projectors on nonorthogonal states are incompatible observables. Along with that, the phase portrait includes probability distributions for all the resolutions of identity of the qudit observable algebra. Additional peculiarities of measurement of the qudit degenerated observables, caused by the possibility of independent measurement of the observables for the particles, make possible to distinguish the local reduction of the qudit particle states from the entanglement of the local measurement results. The phase portrait of a composite system comprised by a qudit pair generates local and conditional phase portraits of particles. The entanglement is represented by the dependence of the shape of conditional phase portrait on the properties of the observable used in the measurement for the other particle. Analysis of the properties of a conditional phase portrait of a multiqubit qubits shows that absence of the entanglement is possible only in the case of substantial restrictions imposed on the method of multiqubit decomposition into qubits. Such a special method for determination of particles exists for each multiqubit state.
Analysis of the process of accumulation of the results of measurements shows that the success of this process substantially depends on the possibility of coordination of actions of two participants of the process - preparator who prepares the series of the states being measured and registrator who chooses in each event of measurement one of incompatible observables. A new value, the complexity of measurement of a state, is used, this characterizes the number of the measurement events needed for solution of the reconstruction problem. By means of this value it is shown that the dependence of the upper limit of the needed number of measurements on the permissible error is a square law one, so a twofold decrease of permissible error corresponds to a fourfold increase of needed number of measurements.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
The problem of quantum state reduction in the process of measurement has attracted attention of almost everyone who created, developed or explained quantum physics to the students. Absence of a solution is the basis for the statement that the physical laws alone are not enough to describe the reduction of state of the measured particle. In the presented study, it is taken into account that the measurement of all properties of quantum particles can not be carried out in one physical process (the no-cloning theorem). For such a measurement several series of incompatible observations are needed, each series is to be carried out by means of a separate measuring instrument. Each event of measurement is a separate physical process; combination of different measurement events is performed by the Measurer - a subject capable of recognizing or not recognizing different events as similar depending on the purpose of the measurement, and capable as well of planning the measurements, of obtaining results and making conclusions. Several strategies for measuring the quantum state of photon polarization are considered; it is shown that modeling of a mixed state by a set of photon pure states is possible only if for this set in all the measurements the same measuring devices are used. It is also shown that the reduction of the quantum state is determined by the parameters of interaction of the measuring device with the measured particle. This fact removes the restrictions on the magnitudes of correlations of different observables, known as Bell's inequalities.
An initial unknown quantum state can be determined with a single measurement apparatus by letting it interact with an auxiliary, ancilla, system as proposed by Allahverdyan, Balian, and Nieuwenhuizen [Phys. Rev. Lett. 92, 120402 (2004)]. In the case of two qubits, this procedure allows one to reconstruct the initial state of the qubit of interest,
S, by measuring three commuting observables and therefore by means of a single apparatus, for the total system
S+A at a later time. The determinant of the matrix of the linear transformation connecting the measurements of three commuting observables at time
t>0 to the components of the polarization vector of
S at time
t=0 is used as an indicator of the reconstructability of the initial state of the system
S. We show that a connection between the entanglement entropy of the total system
S+A and such a determinant exists, and that for a pure state a vanishing entanglement individuates, without a need for any measurement, those intervals of time for which the reconstruction procedure is least efficient. This property remains valid for a generic dimension of
S. In the case of a mixed state, this connection is lost.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.