We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g. spin = 1/2) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain.
We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g. spin = 1/2) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain.
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
We consider one-particle spin states. We show that all the information contained in the state can be written in an equivalent form using three completely decohered mixed states. The density matrix of such a system has the form of tensor product of three diagonal matrices. The unitary operators defined in the space of one-particle spin states are represented by some transformation of tensor products of diagonal matrices. We determine the explicit form of the unitary transformations acting on the representing three-qubit system and corresponding basic operators acting in the space of the initial qubit states.
In this paper we review an approach to the dynamics of open quantum systems based of non-Hermitian Hamiltonians. Non-Hermitian Hamiltonians arise naturally when one wish to study a subsystem interacting with a continuum of states. Moreover, quantum subsystems with probability sinks or sources are naturally described by non-Hermitian Hamiltonians. Herein, we discuss a non-Hermitian formalism based on the density matrix. We show both how to derive the equations of motion of the density matrix and how to define statistical averages properly. It turns out that the laws of evolution of the normalized density matrix are intrinsically non-linear. We also show how to define correlation functions and a non-Hermitian entropy with a non zero production rate. The formalism has been generalized to the case of hybrid quantum-classical systems using a partial Wigner representation. The equations of motion and the statistical averages are defined analogously to the pure quantum case. However, the definition of the entropy requires to introduce a non-Hermitian linear entropy functional.
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.
In this paper we consider density matrices operator related to non-Hermitian Hamiltonians. In particular, we analyze two natural extensions of what is usually called a density matrix operator (DM), of pure states and of the entropy operator: we first consider those operators which are simply similar to a standard DM, and then we discuss those which are intertwined with a DM by a third, non invertible, operator, giving rise to what we call Riesz Density Matrix operator (RDM). After introducing the mathematical framework, we apply the framework to a couple of applications. The first application is related to a non-Hermitian Hamiltonian describing gain and loss phenomena, widely considered in the context of PT-quantum mechanics. The second application is related to a finite-dimensional version of the Swanson Hamiltonian, never considered before, and addresses the problem of deriving a milder version of the RDM when exceptional points form in the system.
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.