scholarly journals Hyperpolarization and the physical boundary of Liouville space

2021 ◽  
Vol 2 (1) ◽  
pp. 395-407
Author(s):  
Malcolm H. Levitt ◽  
Christian Bengs
Keyword(s):  
Master Equation ◽  
Density Operator ◽  
Spin Dynamics ◽  
Physical Region ◽  
State Density ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Liouville Space ◽  
Density Operators ◽  
Physical Boundary

Abstract. The quantum state of a spin ensemble is described by a density operator, which corresponds to a point in the Liouville space of orthogonal spin operators. Valid density operators are confined to a particular region of Liouville space, which we call the physical region and which is bounded by multidimensional figures called simplexes. Each vertex of a simplex corresponds to a pure-state density operator. We provide examples for spins I=1/2, I=1, I=3/2 and for coupled pairs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation.

scholarly journals Hyperpolarization and the Physical Boundary of Liouville Space

2021 ◽  
Author(s):  
Malcolm H. Levitt ◽  
Christian Bengs
Keyword(s):  
Master Equation ◽  
Density Operator ◽  
Spin Dynamics ◽  
Physical Region ◽  
State Density ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Liouville Space ◽  
Density Operators ◽  
Physical Boundary

Abstract. The quantum state of a spin ensemble is described by a density operator, which corresponds to a point in the Liouville space of orthogonal spin operators. Valid density operators are confined to a particular region of Liouville space, which we call the physical region, and which is bounded by multidimensional figures called simplexes. Each vertex of a simplex corresponds to a pure-state density operator. We provide examples for spins I = 1 / 2, I = 1, I = 3 / 2, and for coupled pairs of spins-1/2. We use the von Neumann entropy as a criterion for hyperpolarization. It is shown that the inhomogeneous master equation for spin dynamics leads to non-physical results in some cases, a problem that may be avoided by using the Lindbladian master equation.

Projected evolution superoperators and the density operator: theory and applications to inelastic scattering

1982 ◽  
Vol 60 (10) ◽  
pp. 1371-1386 ◽  
Author(s):  
R. E. Turner ◽  
J. S. Dahler ◽  
R. F. Snider
Keyword(s):  
Inelastic Scattering ◽  
Density Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Interaction Picture ◽  
Von Neumann ◽  
First Order ◽  
Density Operators ◽  
Projection Operator Method ◽  
Time Dependent Solution

The projection operator method of Zwanzig and Feshbach is used to construct the time dependent density operator associated with a binary scattering event. The formula developed to describe this time dependence involves time-ordered cosine and sine projected evolution (memory) superoperators. Both Sehrödinger and interaction picture results are presented. The former is used to demonstrate the equivalence of the time dependent solution of the von Neumann equation and the more familiar, frequency dependent Laplaee transform solution. For two particular classes of projection superoperators projected density operators arc shown to be equivalent to projected wave functions. Except for these two special eases, no projected wave function analogs of projected density operators exist. Along with the decoupled-motions approximation, projected interaction picture density operators arc applied to inelastic scattering events. Simple illustrations arc provided of how this formalism is related to previously established results for two-state processes, namely, the theory of resonant transfer events, the first order Magnus approximation, and the Landau–Zener theory.

Entanglement in composite systems due to external influences

2018 ◽  
Vol 33 (21) ◽  
pp. 1850128
Author(s):  
D. M. Gitman ◽  
M. S. Meireles ◽  
A. D. Levin ◽  
A. A. Shishmarev ◽  
R. A. Castro
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Density Operator ◽  
Time Dependent ◽  
Von Neumann Entropy ◽  
Entanglement Measure ◽  
Photon Beams ◽  
Von Neumann ◽  
Electron Positron ◽  
In The Beginning

In this paper, we consider two examples of an entanglement in two-qubit systems and an example of entanglement in quantum field theory (QFT). In the beginning, we study the entanglement of two spin states by a magnetic field. A nonzero entanglement appears for interacting spins. When the coupling between the spins is constant, we study the entanglement by several types of time-dependent magnetic fields. In the case of a constant difference between [Formula: see text] components of magnetic fields acting on each spin, we find several time-dependent coupling functions [Formula: see text] that also allow us to analyze analytically and numerically the entanglement measure. Considering two photons moving in an electron medium, we demonstrate that they can be entangled in a controlled way by applying an external magnetic field. The magnetic field affecting electrons of the medium affects photons and, thus, causes an entanglement of the photon beams. The third example is related to the effect of production of electron–positron pairs from the vacuum by a strong external electric field. Here, we have used a general nonperturbative expression for the density operator of the system under consideration. Applying a reduction procedure to this density operator, we construct mixed states of electron and positron subsystems. Calculating the von Neumann entropy of such states, we obtain the loss of information due to the reduction and, at the same time, the entanglement measure of electron and positron subsystems. This entanglement can be considered as an example of an entanglement in QFT.

scholarly journals Entropic Lower Bound for Distinguishability of Quantum States

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Seungho Yang ◽  
Jinhyoung Lee ◽  
Hyunseok Jeong
Keyword(s):  
Lower Bound ◽  
Density Operator ◽  
Success Probability ◽  
Quantum States ◽  
Von Neumann Entropy ◽  
Von Neumann

For a system randomly prepared in a number of quantum states, we present a lower bound for the distinguishability of the quantum states, that is, the success probability of determining the states in the form of entropy. When the states are all pure, acquiring the entropic lower bound requires only the density operator and the number of the possible states. This entropic bound shows a relation between the von Neumann entropy and the distinguishability.

ASYMPTOTIC ENTANGLEMENT IN THE PARAMETRIZED HADAMARD WALK

2012 ◽  
Vol 10 (06) ◽  
pp. 1250066 ◽  
Author(s):  
CLEMENT AMPADU
Keyword(s):  
Density Operator ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Von Neumann Entropy ◽  
Von Neumann ◽  
Fourier Representation ◽  
Asymptotic Entropy ◽  
Reduced Density

We study asymptotic entanglement properties of the Hadamard walk with phase parameters on the line using the Fourier representation. We use the von Neumann entropy of the reduced density operator to quantify entanglement between the coin and position degrees of freedom. We investigate obtaining exact expressions for the asymptotic entropy of entanglement, for different classes of initial conditions. We also determine under which conditions the asymptotic entropy of entanglement can be characterized as full, intermediate, or minimum.

scholarly journals Upper bounds on mixing rates

2013 ◽  
Vol 13 (11&12) ◽  
pp. 986-994
Author(s):  
Elliott H. Lieb ◽  
Anna Vershynina
Keyword(s):  
Density Operator ◽  
Random Variable ◽  
Upper Bounds ◽  
Von Neumann Entropy ◽  
Mixing Rate ◽  
Unitary Evolution ◽  
Von Neumann ◽  
Mixing Rates ◽  
Non Local ◽  
Expected Density

We prove upper bounds on the rate, called "mixing rate", at which the von Neumann entropy of the expected density operator of a given ensemble of states changes under non-local unitary evolution. For an ensemble consisting of two states, with probabilities of p and 1-p, we prove that the mixing rate is bounded above by 4\sqrt{p(1-p)} for any Hamiltonian of norm 1. For a general ensemble of states with probabilities distributed according to a random variable X and individually evolving according to any set of bounded Hamiltonians, we conjecture that the mixing rate is bounded above by a Shannon entropy of a random variable $X$. For this general case we prove an upper bound that is independent of the dimension of the Hilbert space on which states in the ensemble act.

Derivation of a Master Equation for Intramolecular Relaxation Using Projection Operator Techniques in Liouville Space

1972 ◽  
Vol 27 (3) ◽  
pp. 526-533
Author(s):  
Notker Rösch
Keyword(s):  
Master Equation ◽  
Coarse Graining ◽  
Relevant Information ◽  
Coarse Grained ◽  
Inhomogeneous Equation ◽  
Intramolecular Rearrangement ◽  
Von Neumann ◽  
Liouville Space ◽  
Space Formalism ◽  
Rearrangement Processes

Abstract A generalized master equation is derived to describe intramolecular rearrangement processes. It is an inhomogeneous equation, including memory effects. The derivation is based on the Liouville space formalism. Because chemically relevant information is contained in the off-diagonal elements of the density matrix, a non-diagonal coarse-graining projector is used. All necessary assumptions are stated explicitly. By making further approximations, the master equation can be reduced to an inhomogeneous von Neumann equation with an effective Liouville operator the imaginary part of which is responsible for relaxation-like coarse-grained solutions. All neglected terms are given in closed form. The character of the solutions of the master equation is discussed in "coordinate-free" manner, i.e. without referring to the underlying Hilbert space.

On Lower and Semi-Lower Density Operators

2007 ◽  
Vol 14 (4) ◽  
pp. 661-671
Author(s):  
Jacek Hejduk ◽  
Anna Loranty
Keyword(s):  
Density Operator ◽  
Baire Property ◽  
Density Operators ◽  
Dense Sets ◽  
Meager Sets ◽  
Algebras Of Sets

Abstract This paper contains some results connected with topologies generated by lower and semi-lower density operators. We show that in some measurable spaces (𝑋, 𝑆, 𝐽) there exists a semi-lower density operator which does not generate a topology. We investigate some properties of nowhere dense sets, meager sets and σ-algebras of sets having the Baire property, associated with the topology generated by a semi-lower density operator.

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