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.
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.
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.
We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement. Measuring a single qubit A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit. The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α| 2 and |β| 2 , respectively.
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.
Abstract We describe and realize an experimental procedure for assessing the incompatibility of two qubit measurements. The experiment consists in a state discrimination task where either measurement is used according to some partial intermediate information. The success statistics of the task provides an upper bound for the amount of incompatibility of the two measurements, as it is quantified by means of their incompatibility robustness. For a broad class of unbiased and possibly noisy qubit measurements, one can make this upper bound coincide with the true value of the robustness by suitably tuning the preparation of the experiment. We demonstrate this fact in an optical setup, where the qubit states are encoded into the photons’ polarization degrees of freedom, and incompatibility is directly accessed by virtue of a refined control on the amplitude, phase and purity of the final projection stage of the measurements. Our work thus establishes the practical feasibility of a recently proposed method for the detection of quantum incompatibility.
The problem of physical reality and measurability of observables is discussed on the example of phenomena revealed by the author. The essence of the phenomena is as follows. After measurement of the coordinate, the variance of the momentum can be smaller than the initial one and depends on the result of the coordinate measurement even if the coordinate and the momentum are uncorrelated. This refers to any pair of conjugate observables, in particular, to the number of quanta and the phase. In a state with a minimum product of uncertainties of the coordinate and the momentum, the mean value of the observable $${{\left\{ {\hat x^2 ,\hat p^2 } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\hat x^2 ,\hat p^2 } \right\}} 2}} \right. \kern-\nulldelimiterspace} 2}$$ , which corresponds to the classical product of the squares of the coordinate and the momentum, is negative.
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.