Conditional Density Matrix Research Articles

On the description of subsystems in relativistic hypersurface Bohmian mechanics.

A candidate for a realistic relativistic quantum theory is the hypersurface Bohm-Dirac model. Its formulation uses a foliation of space-time into space-like hypersurfaces. In order to apply the theory and to make contact with the usual quantum formalism, one needs a framework for the description of subsystems. The presence of spin together with the foliation renders the subsystem description more complicated than in the non-relativistic case with spin. In this paper, we provide such a framework in terms of an appropriate conditional density matrix and an effective wave function as well as clarify their relation, thereby generalizing previous subsystem descriptions in the non-relativistic case.

Weak measurement and Bohmian conditional wave functions

It was recently pointed out and demonstrated experimentally by Lundeen et al. that the wave function of a particle (more precisely, the wave function possessed by each member of an ensemble of identically-prepared particles) can be “directly measured” using weak measurement. Here it is shown that if this same technique is applied, with appropriate post-selection, to one particle from a perhaps entangled multi-particle system, the result is precisely the so-called “conditional wave function” of Bohmian mechanics. Thus, a plausibly operationalist method for defining the wave function of a quantum mechanical sub-system corresponds to the natural definition of a sub-system wave function which Bohmian mechanics uniquely makes possible. Similarly, a weak-measurement-based procedure for directly measuring a sub-system’s density matrix should yield, under appropriate circumstances, the Bohmian “conditional density matrix” as opposed to the standard reduced density matrix. Experimental arrangements to demonstrate this behavior–and also thereby reveal the non-local dependence of sub-system state functions on distant interventions–are suggested and discussed.

Spin and the Thermal Equilibrium Distribution of Wave Functions

Consider a quantum system S weakly interacting with a very large but finite system B called the heat bath, and suppose that the composite S∪B is in a pure state Ψ with participating energies between E and E+δ with small δ. Then, it is known that for most Ψ the reduced density matrix of S is (approximately) equal to the canonical density matrix. That is, the reduced density matrix is universal in the sense that it depends only on S’s Hamiltonian and the temperature but not on B’s Hamiltonian, on the interaction Hamiltonian, or on the details of Ψ. It has also been pointed out that S can also be attributed a random wave function ψ whose probability distribution is universal in the same sense. This distribution is known as the “Scrooge measure” or “Gaussian adjusted projected (GAP) measure”; we regard it as the thermal equilibrium distribution of wave functions. The relevant concept of the wave function of a subsystem is known as the “conditional wave function.” In this paper, we develop analogous considerations for particles with spin. One can either use some kind of conditional wave function or, more naturally, the “conditional density matrix,” which is in general different from the reduced density matrix. We ask what the thermal equilibrium distribution of the conditional density matrix is, and find the answer that for most Ψ the conditional density matrix is (approximately) deterministic, in fact (approximately) equal to the canonical density matrix.

Conditional density matrix: Entangled states in the Heisenberg representation

The application of the conditional density matrix in the Heisenberg representation to the processes caused by entanglement allows elimination of any misunderstanding of these phenomena. We consider quantum entanglement of two- and four-particle systems, quantum correlations different from entanglement, and their causes.

Analysis on Behaviors of Controlled Quantum Systems via Quantum Entropy

In this paper, we investigate the essential properties of finite dimensional measurement-based quantum feedback control systems using a kind of quantum entropy, or so-called linear entropy. We show how the terms appear in the stochastic master equation affect the purity of the conditional density matrix of the system, and clarify a limitation of control action via Hamiltonian. Moreover, applying the stochastic version of LaSalle's invariance theorem, we derive a sufficient condition under which the conditional density matrix converges in probability to the set of all pure states for any control input. The result shows a class of measurement which assures preparation of a pure state.

Quantum phenomena via complex measure: Holomorphic extension

AbstractThe complex measure theoretic approach proposed earlier is reviewed and a general version of density matrix as well as conditional density matrix is introduced. The holomorphic extension of the complex measure density (CMD) is identified to be the Wigner distribution function of the conventional quantum mechanical theory. A variety of situations in quantum optical phenomena are discussed within such a holomorphic complex measure theoretic framework. A model of a quantum oscillator in interaction with a bath is analyzed and explicit solution for the CMD of the coordinate as well as the Wigner distribution function is obtained. A brief discussion on the assignment of probability to path history of the test oscillator is provided.

On the Role of Density Matrices in Bohmian Mechanics

It is well known that density matrices can be used in quantum mechanics to represent the information available to an observer about either a system with a random wave function (“statistical mixture”) or a system that is entangled with another system (“reduced density matrix”). We point out another role, previously unnoticed in the literature, that a density matrix can play: it can be the “conditional density matrix,” conditional on the configuration of the environment. A precise definition can be given in the context of Bohmian mechanics, whereas orthodox quantum mechanics is too vague to allow a sharp definition, except perhaps in special cases. In contrast to statistical and reduced density matrices, forming the conditional density matrix involves no averaging. In Bohmian mechanics with spin, the conditional density matrix replaces the notion of conditional wave function, as the object with the same dynamical significance as the wave function of a Bohmian system.

Entanglement and relative phase created by decay

We discuss an exactly solvable model for the creation of entanglement between two subsystems by the observation of decay products. The system consists of two identical decaying boson modes, and the decay channels are observed through a beam splitter. For a reasonable class of initial states the decay process is completely decoupled from the buildup of the relative phase. Exact expressions are derived for the distribution over the two output channels, and for the conditional density matrix after a given detection history.

Characterization of states of infinite boson systems

In the present paper we deal with the problem of existence and uniqueness of the conditional reduced density matrix (c.r.d.m.) corresponding to a locally normal state of a boson system. The c.r.d.m. was introduced in [3] (Part I of the present series of papers). In order to characterize the class of states possessing a c.r.d.m. we will introduce the family of conditional states of a locally normal state, and we will discuss the relation between the conditional states, the c.r.d.m. and the conditional distribution of the position distribution of the state.

Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes

A correspondence ${a}_{j}\ensuremath{\leftrightarrow}{\ensuremath{\alpha}}_{j}$ between operators $\mathrm{a}=[{a}_{1}, {a}_{2}, \ensuremath{\cdots}{a}_{f}]$ and $c$ numbers $\ensuremath{\alpha}=[{\ensuremath{\alpha}}_{1}, {\ensuremath{\alpha}}_{2}, \ensuremath{\cdots}{\ensuremath{\alpha}}_{f}]$ together with an arbitrary ordering rule $\mathcal{C}$ (e.g., in sequence from 1 to $f$) permit an association $M(\mathrm{a})=\mathcal{C}{M}^{(c)}(\ensuremath{\alpha})$ between a general operator $M(\mathrm{a})$ and an associated $c$ number function ${M}^{(c)}(\ensuremath{\alpha})$. A quasiprobability $P(\ensuremath{\alpha}, t)$ is then defined so that a general ensemble average can be written as an ordinary integration: $〈M(\mathrm{a}(t))〉=\ensuremath{\int}{M}^{(c)}(\ensuremath{\alpha})d\ensuremath{\alpha}P(\ensuremath{\alpha}, t)$. The equation for $\frac{\ensuremath{\partial}P(\ensuremath{\alpha}, t)}{\ensuremath{\partial}t}$ suggests that the $\ensuremath{\alpha}$ obeys a classical Markoff process. If this classical Markoff process is taken literally, multitime classical averages can be computed. Do these correspond to appropriate quantum averages? For the case of field operators such that $[b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}]=1$, important in discussing laser statistics, we show that with ${a}_{1}={b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${a}_{f}=b$, the classical multitime average is equivalent to the average of the corresponding quantum operators written in time-ordered, normal-ordered sequence. For the atomic operators in a laser problem, we obtain the desired correspondence, but find that the more complicated commutation rules necessarily lead to derivative correction terms when multitime averages are taken. Our derivation of multitime averages is based on the quantum regression theorem. We show that this theorem is equivalent to assuming the quantum system to be Markoffian, by showing that it leads to an appropriate factorization of a multitime density matrix and to a Chapman-Kolmogoroff-like condition on the conditional density matrix.