For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
For a quantum state undergoing unitary Schrodinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system comprised of a set of highly-interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
For a quantum state undergoing unitary Schrödinger time evolution, the von Neumann entropy is constant. Yet the second law of thermodynamics, and our experience, show that entropy increases with time. Ingarden introduced the quantum operator entropy, which is the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator [R. S. Ingarden, Quantum information theory, Rep. Math. Phys. 10, 43 (1976)RMHPBE0034-487710.1016/0034-4877(76)90005-7]. These entropies characterize the missing information about a particular observable inherent in the quantum state itself. The von Neumann entropy is the quantum operator entropy for the case when the operator is the density matrix. We examine pure state unitary evolution in a simple model system composed of a set of highly interconnected topologically disordered states and a time-independent Hamiltonian. An initially confined state is subject to free expansion into available states. The time development is completely reversible with no loss of quantum information and no course graining is applied. The positional entropy increases in time in a way that is consistent with both the classical statistical mechanical entropy and the second law.
The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: ? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.
The von Neumann entropy definition is -Tr ( d ln(d)) and is linked to: = Tr (d A) ((1)). Here d is a matrix as is A i.e Aij = . As a result d is often not diagonal. It seems further that the idea of entropy is linked to an operator (2). In other words, one might have certain probabilities linked to the operator P momentum and a different set to X. For a diagonal d, von Neumann’s form is the same as Shannon’s entropy, but uses the eigenvalues of d. For a pure quantum bound state Wn(x), it seems that one considers the two operators X and P and writes two entropies using Shannon’s entropy form: - Integral dx W(x)W(x) ln(WW) and -Integral dp a(p)a(p) ln(a(p)a(p)) where W(x)=wavefunction= Sum over p a(p)exp(ipx). Here P(x)=W(x)W(x) and P(p)=a(p)a(p). For example in (3), one may see calculations of these “distinct” entropies for a particle in a box with infinite potential walls. X and P are linked to measurables and for a quantum bound system one cannot know exactly x and p at the same time. One may write < KE > for a bound state in two ways: Integral dp a(p)a(p) pp/2m or Integral dx W(x) (-1/2m d/dx dW/dx). Using Shannon’s entropy approach one obtains two different entropy values Sx and Sp. With respect to von Neumann’s entropy, it is zero for a pure state. Although one cannot know x and p at the same time (in terms of a measurement) that does not prevent one from writing x and p together (i.e. both known) in exp(ipx) for the wavefunction of a free particle. In this note we suggest that one might consider product probabilities linked to more than one operator when considering entropy. For example, for <KE> one may write Integral dx P(x) Integral dp pp/2m P(p/x) thus acknowledging that both x and p play a role. In this case, P(x)P(p/x) is the probability to use in Shannon’s entropy as we have shown in (4). In (3) it is suggested that P(x)P(p/x) be used in order to incorporate both variables x and p. Using P(x)P(p) in Shannon’s entropy yields the same overall entropy when integrated over x and p i.e. S=Sx+Sp. One reason that the sum is useful is that it eliminates parameters such as L (box length) for a particle in a box with infinite walls (or k the spring constant for an oscillator. This allows for a quantum adiabatic transformation of a pure state to be isentropic. This follows from a Fourier transformation i.e. W(x)=Sum over p a(p)exp(ipx). The two variables p and x scale with L in the opposite manner i.e. x/L and pL (5). Thus we argue that entropy is not necessarily linked to only one operator, but rather for a pure state is linked to variables which appear in a Fourier transform. Usually a Fourier transform is simply a mathematical procedure, but not in the case of the quantum pure state wavefunction where the variable p is actually momentum. In this note we try to investigate how the quantum relationship between p and x emerges. We argue that it exists at the level of the classical action for a free particle (relativistic or nonrelativistic) A=-Et+px where dA/dx partial =p and dA/dt partial = -E. Thus p and x are variable pairs in classical dynamics and this carries over to classical statistical phase space linked to p and x. We argue that a quantum dynamical link between p and x is also present in A i.e. -id/dx exp(iA) = p exp(iA). In other words p and x are linked dynamically in quantum mechanics as well (for a free particle) with exp(-iEt+ipx) being part of this link. The Fourier series form is already present in this solution as one may create a momentum distribution i.e. Sum over p a(p)exp(ipx). This depends on p and x and both affect any operator based on p and x (or one of these). Thus we expect that probabilities linked to both p and x should appear in an expression of entropy for a quantum bound state.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
We consider a closed macroscopic quantum system in a pure state ψ evolving unitarily with Hamiltonian H in a finite-dimensional Hilbert space H. Following von Neumann, we assume a decomposition of H into mutually orthogonal subspaces Hν, where each ν represents a macro state, and Hν contains the pure states compatible with ν. We define the quantum Boltzmann entropy for a macro state ν as SqB(ν)=kBlogdimHν, where kB is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqB predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of H and Hν.
Note (March 8, 2022) von Neumann's entropy may be written in Shannon's form for the eigenvalues of the density operator. If the density operator is written as Sum over j mj |j><j| with the set |j> not being orthonormal, then von Neumann's entropy would not be Shannon's entropy -Sum over j mj ln(mj) because the set mj are not the eigenvalues of the density matrix. In the literature, Shannon's entropy, however, is sometimes used with the weights of the projection states i.e. mj weights of |j>. For example, see: Kak, S. The Transactional Nature of Quantum Informationhttps://arxiv.org/pdf/0907.2465.pdf Note: (March 10, 2022) As an example of a state linked to the "fictitious" P(x)P(p) consider a single particle in a box with infinite potential walls. Outside the box W(x)=0 so P(x)=0, but P(p)=a(p)a(p) is nonzero for all p. Thus, one needs to consider P(x)P(p) together to make sense of the region outside of the box. Using the argument that one cannot know p and x exactly and so should consider P(x) and P(p) separately seems to lead to considerations of P(p) being nonzero i.e changes of measuring particle momenta outside the box which constradicts P(x)=0 outside. In (1) it is argued that von Neumann’s density operator d = Sum over i bi |i><i| = Sum over j mj |j><j> where in one case |i> represents an orthonormal basis and in the other of |j> a different nonorthogonal basis is used. The problem is that von Neumann entropy -Sum over i bi ln(bi) and -Sum over j mjln(mj) are not the same, they depend on the projection operator used. The author of (1) goes on to argue that historically it was suggested this might mean that a quantum state in quantum statistical mechanics is not defined clearly. We wish to consider the case of a pure bound state for which we have defined S=entropy=Sp+Sx where Sp=-Integral dp P(p) ln(P(p)) and Sx = - Integral dx P(x) ln(P(x)). According to quantum mechanics, one may not know p and x exactly, but S=Sp+Sx is equivalent to using P(p)P(x) in Shannon’s entropy formula. (Alternatively, one may use P(p/x)P(x).) As a result, the so-called state (x,p) associated with P(p)P(x) assumes knowledge of p and x exactly and so is not a real quantum state perhaps in the manner suggested in (1). In order to consider this issue, we argue that a quantum bound problem may be thought of in two ways: the classical mechanics type picture, and the statistical mechanical type one. The first approach is consistent with using the wavefuntion W(x) in the time-independent Schrodinger equation. P(p) and P(x) do not appear, but this is still a statistical picture because KE(x) = {Sum over p pp/2m a(p) exp(ipx)}/W(x). KE(x)+V(x)=E, nevertheless, has the form of a classical mechanical particle. In other words, it has one total E at any x and a potential value V(x) at x. In fact, KE(x) may be considered to represent a vrms(x) which is the velocity of the classical particle, although this is a purely mathematical construct. One may define P(p/x) = a(p)exp(ipx)/W(x) which is complex and allows for interference among exp(ipx) so this picture is very different from that of usual classical probabilities. The second picture of quantum mechanics involves P(x) or P(p) which behave as classical probabilities (i.e. are real, positive and sum to 1). For example: Operator ave = < W | B | W> is associated with this picture (as is von Neumann’s density matrix). One may write energy as: E= Sum over p P(p)pp/2m + Sum over x V(x) P(x), but this equation does not lead to solutions. One must use the Schrodinger to equation to solve for W(x) and find a(p) from W(x)=Sum over p a(p)exp(ipx) and then form P(x)=W(x)W(x) and P(p)=a(p)a(p). Nevertheless quantum statistics, in order to be linked with classical statistical mechanics, seems to be formulated in terms of P(p) and P(x). For example the diagonal elements of von Neumann’s density matrix contain this type of probability. Given that quantum mechanics may be viewed in two different statistical manners, (A) in terms of W(x), the Schrodinger equation and P(p/x) or (B) in terms of P(p) and P(x), a formulation of quantum statistical mechanics based on P(p) and P(x) may lead to unusual results. In particular, P(p) and P(x) represent exact p and x values, but according to quantum theory one cannot know these at the same time. Thus talking of P(p) and P(x) (in a classical way) seems to mean referring to a quantum state that is not really defined, perhaps one of the states noted in (1). This may not be a problem, however, if the underlying statistics follows from W(x) and a(p) and P(p/x) for which p and x are defined, but are linked to complex probabilities.