The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can find a particular orientation of an SG machine such that the state is in its + or − port. This also means that the state is an eigenvector of some operator, or is always a linear combination of the computational basis states.
The formalism for quantum systems developed so far applies to what are called pure states. A system in a pure state is completely specified by the state vector. A complete set of experimental tests will determine the system state fully: we have maximal knowledge of the system. For example, for a spin system, we can find a particular orientation of an SG machine such that the state is in its + or − port. This also means that the state is an eigenvector of some operator, or is always a linear combination of the computational basis states.
Beyond state vectors We cannot always assign a definite state vector |ψ〉 to a quantum system Q. It may be that Q is part of a composite system RQ that is in an entangled state |Ψ (RQ) 〉. Or it may be that our knowledge of the preparation of Q is insufficient to determine a particular state |ψ〉. Consider, for instance, a qubit sent from Alice to Bob during the BB84 key distribution protocol from Section 4.4. The state of this qubit could be |0〉, |1〉, |+〉 or |−〉, each with equal likelihood. In either case – whether Q is a subsystem of an entangled system, or the state of Q is determined by a probabilistic process – we cannot specify a quantum state vector |ψ〉 for Q. Nevertheless, in either case we are in a position to make statistical predictions about the outcomes of measurements on Q. In this chapter we describe the mathematical machinery for doing this. Mixtures of states Suppose the state of Q arises by a random process, so that the state |ψ α 〉 is prepared with probability p α . The possible states |ψ α 〉 need not be orthogonal (as you can see in the BB84 example above). We call this situation a mixture of the states |ψ α 〉, or a mixed state for short. One way to interpret a mixed state is to return to the idea of an ensemble of systems, which we introduced in Section 3.3.
The state of a quantum mechanical system is represented by a nonzero vector in a separable Hilbert space. The word ‘represented’ means that knowledge of the state vector gives complete information on the properties of the system. In quantum mechanics, the state vectors are denoted using Dirac's notation. Several quantum systems possess a property called spin. Spin is a purely quantum mechanical feature which has no direct classical counterpart. The set of density matrices is a convex subset of the space of Hermitian operators whose ‘extreme’ points represent pure states. In quantum mechanics, observed quantities are called observables. A continuous spectrum is such that it contains no isolated points. The chapter discusses the case of measurement of observables with discrete spectrum.
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.
So far we have dealt with quantum systems in a well-defined state, namely, systems that can be fully described by a single state vector https://www.w3.org/1998/Math/MathML" display="inline"> | ψ ⟩ . Such states are called pure . Here we will learn how to describe situations where knowing the state vector is not enough because the system is under the influence of another, typically larger, system which we know little about. This kind of interaction degrades the “quantumness” of the system we are interested to describe. The mathematics used to described this phenomenon is also suitable for establishing the degree of entanglement present in a quantum system, being it in a pure state or not.
In quantum physics, we often need to consider ensembles that are not pure cases, that is, that cannot be described by state vectors. This is so, for instance, when an ensemble is composed of several subensembles that are described by nonequivalent state vectors. It also happens in most cases in which we consider ensembles of definite subsystems of some larger systems. The density matrix—or "statistical operator"—formalism is one that makes it possible to deal with such situations swiftly and efficiently. The rules apply only to ensembles that can be described by state vectors. These rules entail as consequences new expressions for quantum probabilities and mean values, which can be called the—derived— "statistical operator rules." However, it is also possible to formulate the statistical operator rules directly instead. The Heisenberg "uncertainty" relations imply that in general f(x,p) cannot be interpreted as really being a joint probability distribution.
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.