Density matrix mixed states

The quantum-mechanical equivalent of phase density is known as the density matrix or density operator. It is best understood in the case of a mixed ensemble whose systems are not all in the same quantum state, as for a pure ensemble. 

In this case the entanglement of formation is given in terms of another entanglement measure, the concurrence C [42-44]. The entanglement of formation varies monotonically with the concurrence. From the density matrix of the two-spin mixed states, the concurrence can be calculated as follows ... 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

This has the advantage of treating all relevant states I at once, and allows them to mix over time as determined by the L-vN equation. The density operator can be represented by a density matrix (DM) p(t) by expanding the states rji(q, t) in an electronic basis set (<7), in which case the matrix L-vN equation for the electronic DM must be solved coupled to the hamiltonian equations for the nuclear motions. Details of this procedure have been given in a previous publication.[32]... 

The density matrix of a mixed state can be constructed even for a system which has no quantum mechanical states since it is not isolated and constitutes only a subsystem of a larger system. For example, the spin system of a molecule in a liquid is not isolated owing to the presence of interactions between the degrees of freedom of the spins and other, mostly rotational, degrees of freedom which are usually called the lattice. The state of the entire system may be represented by the basis set which is formed from the direct product of the basis sets of its subsystems ... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

The theoretical analysis of MEFs and CIDNP due to the S-Tq mixing can be carried out with Eqs. (11-57). With these equations, the theoretical calculation of MEFs and CIDNP due to the S-Tq mixing is possible. In Chapter 12, typical results of such an analysis of MFEs will be shown. The theoretical analysis of CIDEP due to the S-To mixing can also be carried out with the density matrix method as shown in the following part of this section [5]. The electron spin polarization (P ) of radical A in a nuclear state (a) can be written as... 

Let us start by defining what they are. A pure state may be written as a projection operator p, = mixed state, or so-called density matrix, is defined to be any convex combination of such projectors... 

Adiabatically rotating the mixing angles 6 from 0 to 7t/2 leads to a complete and reversible transfer of all photonic states to a collective atomic excitation if the maximum number of photons n + m is less than the number of atoms N. Let the initial quantum state of the light field be described by the density matrix... 

So how is a pure state different from a mixed state In the former the elements of the corresponding density matrix are related to each other in a specific way, Eq. (10.12), resulting from their association with the amplitudes of the expansion of the pure state in terms of the given basis. In a mixed state such a relationship does not exist. 

The breadth "of dispersion is measured by entropy, which ranges in value from zero for a pure state (idempotent density matrix) of a given energy, to a maximum for the stable equilibrium (thermodynamic equilibrium) mixed state of the same energy. 

To describe chemical reactions upon reencoimter, which in general will occur with different probabilities for the singlet and triplet states, one has to separate the variables qss TqTo again. However, because electron-spin polarization does not arise during a diffusive excursion and is randomized in the approach of the radicals to their contact distance, it suffices to retain the component 2Im(ps7 u) in the density matrix the component 2Re(ps7 u) is not needed. In this new basis, the mixing matrix M of Equation (17) becomes... 

---