We find a probability of determining an unknown quantum state if some information about it is known.Namely, we know that the state is one of N quantum states.These states are represented as eigenstates of operator of projection of a spin on N directions, corresponding to positive eigenvalues.The probability of determining the unknown state with the smallest number of measurements and for some number of measurements n is obtained.Dependence of these probabilities on the angles between the vectors defining the N directions is studied.
We find a probability of determining an unknown quantum state if some information about it is known.Namely, we know that the state is one of N quantum states.These states are represented as eigenstates of operator of projection of a spin on N directions, corresponding to positive eigenvalues.The probability of determining the unknown state with the smallest number of measurements and for some number of measurements n is obtained.Dependence of these probabilities on the angles between the vectors defining the N directions is studied.
The unknown state rho of a quantum system S is determined by letting it interact with an auxiliary system A, the initial state of which is known. A one-to-one mapping can thus be realized between the density matrix rho and the probabilities of the occurrence of the eigenvalues of a single and factorized observable of S+A, so that rho can be determined by repeated measurements using a single apparatus. If S and A are spins, it suffices to measure simultaneously their z components after a controlled interaction. The most robust setups are determined in this case for an initially pure or a completely disordered state of A. They involve an Ising or anisotropic Heisenberg coupling and an external field.
A measurement procedure on a spin 1/2 system is described, whose effect on the (possibly unknown) state of the system can be reversed, by means of other similar measurements, with a sizable probability of success Prev (e.g., 21<Prev<1, depending on the state). Repeated such measurements followed by reversal, on a single system, allow us to determine its (a priori unknown) state operator with a probability of success which is nonzero, though extremely small (because Prev<1).
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.
This is the first of two papers showing that the quantum problem of measurement remains unsolved even when the initial state of the apparatus is described by a statistical operator and when the results of measurement have a small probability of being erroneous. A realistic treatment of the measurement of observables of microscopic objects (e.g., the position or the spin of an electron) by means of observables of macroscopic apparatus (e.g., the position of a spot on a photographic plate) requires the consideration of errors. The first paper considers measurement procedures of the following type: An initial eigenstate of the object observable leads to a final statistical operator of the object plus apparatus which describes a mixture of “approximate” eigenstates of the apparatus observable. It is proved that each of a large class of initial states leads to a final statistical operator which does not describe any mixture containing even one “approximate” eigenstate of the apparatus observable. INTRODUCTION Several writers have tried to solve the quantum-mechanical problem of measurement through one or both of the following proposals: (a) describthe initial state of the measuring apparatus by a projection onto a subspace of the associated Hilbert space or by a statistical operator, thus taking into account the practical impossibility of knowing the exact quantum state of a macroscopic object; (b) recognizing that there may be some physical inaccuracy, such as a small error in the position of a pointer needle, in the final registration of the outcome of the measurement by the apparatus.
We consider the problem of optimally identifying the state of a probe qudit, prepared with given prior probability in a pure state belonging to a finite set of possible states which together span a D-dimensional subspace of the d-dimensional Hilbert space the qudit is defined in. It is assumed that we do not know some or all of the states in the set, but for each unknown state we are given a reference qudit into which this state is encoded. We show that from the measurement for optimal state identification with d=D one can readily determine the optimal figure of merit for qudits with d>D, without solving a new optimization problem. This result is applied to the minimum-error identification and to the optimal unambiguous identification of two qudit states with $d\ensuremath{\ge}2$, where either one or both of the states are unknown, and also to the optimal unambiguous identification of N equiprobable linearly independent unknown pure qudit states with $d\ensuremath{\ge}N$. In all cases the optimal figure of merit, averaged over the unknown states, increases with growing dimensionality d of the qudits.
The concept of objectification Weak and strong objectification Let S be a quantum system that is prepared in a pure state φ and A a discrete nondegenerate observable with eigenvalues a i and eigenstates φ a i . If the preparation φ is not an eigenstate of A , then quantum mechanics does not provide any information about the value of the observable A . The pair 〈φ, A 〉 merely defines a probability distribution p (φ, a i ), the experimental meaning of which is given by the statistical interpretation of quantum mechanics: the real positive number p (φ a i ) is the probability of obtaining the result a i after a measurement process of the observable A . This means that if one were to perform a large series of N measurements of this kind, then the relative frequency f N (φ, a i ) of the result a i would approach for almost all test series the probability p (φ, a i ), for N → ∞ (cf. chapter 3). One could, however, in addition to these well-established results tentatively assume that a certain value a i or even an eigenstate φ a i of A pertains objectively to the system S but that this value or state is subjectively unknown to the observer, who knows only the probability p (φ, a i ) and hence the distribution of possible measurement outcomes. The probability would then express the subjective ignorance (or knowledge) about an objectively decided value or eigenstate of A . The hypothetical attribution of a certain value or eigenstate of A to the system S will be called objectification .