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A Short Note on Quantum Entropies

Amelia Carolina Sparavigna

Published Jan 11, 2016• 1 citation
OpenAlex

Abstract

In quantum statistical mechanics, the extension of the classical Gibbs entropy is the von Neumann entropy, obtained from a quantum-mechanical system described by means of its density matrix. Here we shortly discuss this entropy and the use of generalized entropies instead of it

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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.

•Mar 7, 2022•0 citations

On Gibbs distribution for quantum systems

Valery Vasil'evich Kozlov

Some problems from quantum statistical mechanics concerning the regular and chaotic behavior of quantum systems are discussed.

P-Adic Numbers Ultrametric Analysis and Applications•Jan 1, 2012•0 citations
Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.
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.
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.