Since a 1932 work from von Neumann, it has been considered that if two statistical mixtures are represented by the same density operator ρ, they should, in fact, be considered as the same mixture. In a 1970 paper, Zeh introduced a thought experiment with neutron spins, and suggested that, in that experiment, the density operator could not tell the whole story. Since then, no consensus has emerged yet, and controversies on the subject still presently develop. In his 1995 book, speaking of the use of the density operator, Peres spoke of a von Neumann postulate. In this paper, keeping the random variable used by von Neumann in his treatment of statistical mixtures, but also considering higher-order moments of this random variable, it is established that the two mixtures imagined by Zeh, with the same ρ, should however be distinguished. We show that the rejection of that postulate, installed on statistical mixtures for historical reasons, does not affect the general use of ρ, e.g., in quantum statistical mechanics, and the von Neumann entropy keeps its own interest and even helps clarifying that confusing consequence of the postulate identified by Peres.
Since a 1932 work from von Neumann, it has been considered that if two statistical mixtures are represented by the same density operator ρ, they should, in fact, be considered as the same mixture. In a 1970 paper, Zeh introduced a thought experiment with neutron spins, and suggested that, in that experiment, the density operator could not tell the whole story. Since then, no consensus has emerged yet, and controversies on the subject still presently develop. In his 1995 book, speaking of the use of the density operator, Peres spoke of a von Neumann postulate. In this paper, keeping the random variable used by von Neumann in his treatment of statistical mixtures, but also considering higher-order moments of this random variable, it is established that the two mixtures imagined by Zeh, with the same ρ, should however be distinguished. We show that the rejection of that postulate, installed on statistical mixtures for historical reasons, does not affect the general use of ρ, e.g., in quantum statistical mechanics, and the von Neumann entropy keeps its own interest and even helps clarifying that confusing consequence of the postulate identified by Peres.
Quantum information mobilizes the description of quantum systems, their states, and their behavior. Since a measurement postulate introduced by von Neumann in 1932, if a quantum system has been prepared in two different mixed states represented by the same density operator ρ, these preparations are said to have led to the same mixture. For more than 50 years, there has been a lack of consensus about this postulate. In a 2011 article, considering variances of spin components, Fratini and Hayrapetyan tried to show that this postulate is unjustified. The aim of the present paper is to discuss major points in this 2011 article and in their reply to a 2012 paper by Bodor and Diosi claiming that their analysis was irrelevant. Facing some ambiguities or inconsistencies in the 2011 paper and in the reply, we first try to guess their aim, establish results useful in this context, and finally discuss the use or misuse of several concepts implied in this debate.
Quantum electronics is significantly involved in the development of the field of quantum information processing. In this domain, the growth of Blind Quantum Source Separation and Blind Quantum Process Tomography has led, within the formalism of the Hilbert space, to the introduction of the concept of a Random-Coefficient Pure State, or RCPS: the coefficients of its development in the chosen basis are random variables. This paper first describes an experimental situation necessitating its introduction. While the von Neumann approach to a statistical mixture considers statistical properties of an observable, in the presence of an RCPS one has to manipulate statistical properties of probabilities of measurement outcomes, these probabilities then being themselves random variables. It is recalled that, in the presence of a von Neumann statistical mixture, the consistency of the density operator \r{ho} formalism is based on a postulate. The interest of the RCPS concept is presented in the simple case of a spin 1/2, through two instances. The most frequent use of the \r{ho} formalism by users of quantum mechanics is a motivation for establishing some links between a given RCPS and the language of the density operator formalism, while keeping in mind that the situation described by an RCPS is different from the one which has led to the introduction of \r{ho}. It is established that the Landau - Feynman use of \r{ho} is mobilized in a situation differing from both the von Neumann statistical mixture and the RCPS. It is shown that the use of the higher-order moments of a well-chosen random variable helps solving a problem already identified by Zeh in 1970.
Users of quantum mechanics are familiar with the concept of a statistical mixture as introduced by von Neumann, and with the use of a density operator in that context. A density operator may also be used in another situation, introduced by Landau, with a transient coupling between the two parts of a quantum bipartite system. But more than fifty years after a clarifying work by Feynman on the subject, a confusion still persists about what we call the Landau-Feynman situation. In this paper we establish that, when facing that situation, the right concept to be used is not the one of a mixed state - be it qualified as proper or improper -, but the one of entanglement.
Peres and Weinberg have debated the definition of entropy in nonlinear quantum mechanics. This issue can be resolved, it is contended, if one adopts an entropy measure consistent with the approach of Guiasu (following work of Wiener and Siegel) in which a probability distribution over the space of wavefunctions is estimated. This is tantamount to rejecting the von Neumann measure (S N ), -Tr rho ln rho , of the entropy of a density matrix ( rho ), which is at the basis of Peres' argument. As authority for such a rejection of S N , a detailed critique of this measure, given by Band and Park in the mid-1970s, can be cited.
After presenting the infinite operator-sum form solution to the Milburn equation dρ/dt = γ(UρU † – ρ) = γU[ρ,U † ], where U = e -iH/ℏγ , and verifying that this equation preserves the three necessary conditions of density operators during time evolution, we prove that the von Neumann entropy increases with time. We also point out that if A and B both obey the Milburn equation, then the product AB obeys (d/dt) (AB) = γU[AB, U†] – (1/γ) (dA/dt) (dB/dt), which violates the Milburn equation, this reflects that a pure state will evolve to a mixture in general.
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.