The Landau-Feynman Transiently Open Quantum System: Entanglement and Density Operators

Abstract

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.

1. Introduction

2025 has been declared the International Year of Quantum Science and Technology by the General Assembly of the United Nations. The field of Quantum Information has been developing for several decades, and keeps stimulating a reflection about major concepts introduced in the very beginning of Quantum Mechanics. In this Journal, we recently published a paper [1] in order to answer a question already identified by Zeh in 1970 [2]. The aim of the present paper is to treat a situation which is quite distinct from the one considered by Zeh and by our previous paper, but which is still a source of much confusion. Considerations about the situation discussed by Zeh in [2] and about the situation treated in this paper are more specifically important in the context of Quantum State Tomography (QST) [3] or in that of standard Quantum Process Tomography (QPT) [3], two major topics of Quantum Information Processing.

• We first recall a few definitions:

• In the Dirac formalism, a pure state of a quantum system $\Sigma$ is described by a ket, denoted for instance as $\mid \psi >.$
• It may happen that the state of $\Sigma$ cannot be described by a ket, but only by a collection of normalized (but not necessarily orthogonal) kets, {$\mid {\psi }_i, >,$$p_i$}, with $0\le p_i\le 1$ and ${\Sigma }_i{p}_i=1.$$\Sigma$ is then said to be in the mixed state or statistical mixture {$\mid {\psi }_i>,$$p_i$}, a concept introduced by von Neumann, who also associated its density operator $\rho ={\Sigma }_i{p}_i\mid {\psi }_i><{\psi }_i\mid$ with it. This density operator is also, possibly improperly, called the density matrix.
• If a quantum system $\Sigma$ is composed of two (distinguishable) parts ${\Sigma }_1$ (state space ${\mathcal{E}}_1$) and ${\Sigma }_2$ (state space ${\mathcal{E}}_2$), the state space of $\Sigma$ is the tensor product ${\mathcal{E}}_1\otimes {\mathcal{E}}_2$ of ${\mathcal{E}}_1$ and ${\mathcal{E}}_2.$ States represented by kets of the form $\mid {\Psi }_{1i}>\otimes \mid {\Psi }_{2j}>,$ where $\mid {\Psi }_{1i}>$ is a ket of ${\mathcal{E}}_1$ and $\mid {\Psi }_{2j}>$ a ket of ${\mathcal{E}}_2$, are called product states. Most kets of ${\mathcal{E}}_1\otimes {\mathcal{E}}_2$ are not product states, but can only be written as a weighted sum of product states. They are called entangled states.

Deliberately, here we do not introduce the expressions “proper mixed state” and “improper mixed state”, because these two expressions do not belong to the general vocabulary of quantum mechanics, and because, in this paper, we will show that these expressions were introduced in order to avoid a confusion, but failed.

In [2], Zeh considered the spin of two neutrons (or of two neutron beams), first submitted to well-chosen and different preparations. The first preparation mobilized the first neutron and the second one mobilized the second neutron. Although their preparations were different, the two neutrons then possessed the same density operator $\rho$. Zeh claimed that the density operator $\rho$ could not tell the whole story of (the spin of) these neutrons, since it was unable to establish any distinction between their preparation processes.

In our paper [1], we considered an isolated quantum system $\Sigma .$ The word isolated may be ambiguous, as an isolated system may be submitted to forces deriving from a scalar and/or a vector potential. In the absence of any force, the system is said to be free. An isolated system obeys the Schrödinger equation, which is postulated by Quantum Mechanics (QM), and mobilizes the Hamiltonian of the system, a concept introduced from classical mechanics. An isolated system is also sometimes said to be closed, and then, if not closed, the system is said to be open (existence of a coupling Hamiltonian between that system and its environment).

If the system of interest ${\Sigma }_1$ is not isolated, a simple situation is the following one:

• That system ${\Sigma }_1$ is coupled to another system ${\Sigma }_2$, and the whole system $\Sigma$, composed of these two parts (bipartite system), is itself isolated.
• Moreover, one is able to write explicitly the Hamiltonian of the whole system, as the sum of the Hamiltonian of each subsystem and of a coupling Hamiltonian.

• In contrast, if e.g., the system of interest is a spin $1/2$, and if this spin is coupled to a macroscopic system, one will generally not try to write the whole Hamiltonian. For instance, if that spin is coupled to a bath, which imposes it its temperature, and in a situation identified as thermal equilibrium, the effect of the bath is described with a scalar parameter, namely the bath temperature, imposed to the spin.

If the global system $\Sigma$ is first prepared in a pure state described with a ket $\mid \Psi \left(t_0\right)>$ in the Dirac formalism and is thereafter isolated from its environment (One should understand that, unless specifically supposed, forces acting upon it and deriving from a scalar and/or vector potential are still present.), its future behaviour is still described with a pure state, $\mid \Psi \left(t\right)>,$ which obeys the Schrödinger equation. It may happen that, at $t_0$, the state of that isolated system $\Sigma$ is not pure, but can only be described with a so called (von Neumann) mixed state or statistical mixture. Such a mixed state consists of a collection of (normed) kets {$\mid {\psi }_i>,$$p_i$}, where the randomly drawn ket $\mid {\psi }_i>$ has the statistical weight $p_i$ in the mixture. Moreover, for any $p_i,$$0\le p_i$$\le 1$ and ${\Sigma }_i{p}_i$ $=1$. The mean value of an observable $\widehat{O}$ in the mixture, i.e., the quantity ${\Sigma }_i{p}_i<{\psi }_i\mid \widehat{O}\mid {\psi }_i>$, can be written as $Tr\rho \widehat{O},$ where $\rho$ = ${\Sigma }_i{p}_i\mid {\psi }_i><{\psi }_i\mid$ is the density operator. As stressed e.g., by Peres [4], considering that $\rho$ contains the whole information present in the definition of the statistical mixture through the {$\mid {\psi }_i>,$ $p_i$} collection is not the result of some established property, as sometimes thought, but the result of a postulate made by von Neumann [5]. We discussed the relevance of that postulate in our recent paper [1]. The present paper is focused on another situation in which a density operator is also introduced, which we call the Landau-Feynman situation for reasons to be shortly developed.

Thinking that discussions about the use of the density operator or about the pure state concept are presently out of date and should be moved into the domain of the history of Science is not consistent with the present reality. For instance, a 2022 paper by Castellani [6] is entitled All quantum mixtures are proper, and, in fact, its content is precisely related to the Landau-Feynman situation. The concept of an improper mixture was already present in a 1966 paper by Bernard D’Espagnat [7], and the existence of the Castellani paper does testify that, some sixty years later, the thinking about that subject is not stabilized yet. And, specifically in the context of QST, which aims at estimating quantum states, and in that of standard QPT, with its use of QST, it would be useful to fully appreciate the content of that Castellani paper. In Section 2 of the present paper, the idea behind the improper mixture concept introduced by d’Espagnat, which recently led to the 2022 paper by Castellani, is presented and discussed.

When speaking of a von Neumann mixed state, one generally refers to the already cited 1932 canonical book by von Neumann [5]. In fact, a density operator $\rho$ had already been introduced in two different physical contexts:

• By von Neumann [8], in the situation he studied in more detail in his 1932 book.
• By Landau [9], in the case of permanently coupled systems.

• None of these contexts corresponds to what we call the Landau-Feynman situation, which appeared in volume III of the course by Landau and Lifschitz [10], and which is presented in Section 3. The analysis made by Feynman in his 1972 book focused on quantum statistical mechanics [11] is detailed in Section 4. It allows one to appreciate the role of entanglement, and the conditions to be fulfilled when a density operator is introduced in that situation. These results, discussed in Section 5, show that both the claim by Castellani and the concept of an improper mixture should be rejected. A conclusion is given in Section 6. A simple instance of the Landau-Feynman situation, implying two distinguishable spins 1/2 transiently coupled with a Heisenberg cylindrical exchange coupling, is examined in Appendix A. In Appendix B, considering the Landau-Feynman situation, it is shown that, whereas it it possible to introduce two density operators ${\rho }_1$ and ${\rho }_2$ through partial tracing, it would generally be wrong to claim that the state of the whole system $\Sigma$ is ${\rho }_1$⊗${\rho }_2$.

2. From d’Espagnat to Castellani

In a 1966 paper [7], D’Espagnat considered an implicitly isolated system composed of two parts, each one being initially in a pure state. The two subsystems then interacted, and were then again uncoupled. This situation is precisely the one which we call the Landau-Feynman situation. D’Espagnat observed that, after the disappearance of that interaction, the whole system was still in a pure state, and that one could be interested in only one of the subsystems. He then wrote: “after the interaction has taken place, one immediately sees that, in general, neither of them is a pure case. One then usually says that they are a mixture”. D’Espagnat then proposed to say that they are in “an improper mixture”, and that a von Neumann mixture is “a proper mixture”.

In this paper, we stress that D’Espagnat did identify a problem, but that introducing the expression “improper mixture” kept an ambiguity, and therefore did not solve that problem. Its persistence is testified by the existence of the 2022 paper [6] by Castellani. From the beginning of his paper, Castellani:

• assumes “that the state of a system is completely described by its density operator”, which in fact is just a postulate (cf. [1] and its references),
• considers “the state of a subsystem in a composite system to be completely described by its reduced density matrix”. As detailed in Section 4 and Section 5, this claim is wrong.

The Castellani paper, which expresses a disagreement with an article published by D’Espagnat in 2001 [12] (itself a reply to a 2001 paper from K.A. Kirkpatrick [13]), ignores the content of the 1970 Zeh paper [2]. That paper by Castellani refers to the 1932 book by von Neumann [5], in which anyway what we call the Landau-Feynman situation was not discussed. The approach taken by Castellani in [6] moreover ignores the content of the 1972 book [11] by Feynman on the subject. The whole approach followed by Castellani in his paper [6] imposes us to detail the concise treatment given by Feynman in [11], which will be done in Section 4.

3. Landau and the Use of a Density Operator

Texts briefly considering the historical introduction of the density operator generally claim that both Landau and von Neumann introduced the concept in 1927. It is true that, in a 1927 paper [9], Landau (born in 1908), in the first section, entitled “Coupled systems in wave mechanics”, wrote: “A system cannot be uniquely defined in wave mechanics; we always have a probability ensemble (statistical treatment). If the system is coupled with another, there is a double uncertainty in its behaviour”. Landau then considered an observable attached to the first subsystem. In the presence of the above coupling, and using the formalism of the wave function, he introduced an operator through an integration over the variables of the second subsystem. This corresponds to a Partial Trace procedure. And, unambiguously, the title of his paper contains the word damping (in its English version [9]) associated with an irreversible process if, in the instance treated by Landau, one focuses on the electrons. Such a permanent coupling (as opposed to a transient one) is not what, in this paper, we call the Landau-Feynman situation.

Roughly thirty years later (1956, Russian version, and page 38 of the English 1965 version [10]), in volume III of their course, devoted to non-relativistic quantum mechanics, Landau and Lifshitz provided the following investigation. They first supposed that a “closed system as a whole is in some state described by a wave function $\Psi (q,x),$ where x denotes the set of coordinates of the system of interest, and q the remaining coordinates of the whole system”. Integrating over the q variables, which corresponds to introducing a partial trace, they introduced an operator which they again called a density matrix (this should today correspond to the concepts of a master equation and Lindblad operator, or generalized density operator, cf. our brief comment in Section 5). Then, in a second step only, they “suppose that the system” (of interest, corresponding to x) “is closed, or became so at some time”. That second situation is the one which we call the Landau-Feynman situation. And it is the situation considered by d’Espagnat (and by K.A. Kirkpatrick or Castellani, cf. Section 3).

As explained in Chapter 2 of his Statistical Mechanics [11], and developed in Section 4, Feynman suppressed the possible confusion resulting from the use of the expression density or statistical operator by both von Neumann and Landau under different assumptions, and more specifically with the possible existence of a transient coupling of the system of interest with a second system in the Landau and Lifschitz course.

4. The Treatment of the Landau Case by Feynman

Chapter 2 of the quite synthetic 1972 book by Feynman, Statistical Mechanics [11], is entitled Density matrices. That book is rarely cited, and, instead of just mentioning its existence, we hereafter detail the argumentation developed by Feynman on the situation introduced by Landau (Section 3).

Feynman divides the universe into two parts: the system of interest (we call it ${\Sigma }_1$) and the rest of the universe. We will consider that the rest of the universe is just the environment of the system of interest at our chosen time scale, and call it ${\Sigma }_2$. Feynman then introduces a complete set of orthonormal kets, {$\mid {\phi }_i>$}, for ${\Sigma }_1,$ and a complete set of orthonormal kets, {$\mid {\theta }_j>$}, for ${\Sigma }_2.$ The most general (normed) pure state for the whole system can be written as:

$$\mid \psi >={\Sigma }_{i,j}{c}_{ij}\mid {\phi }_i>\mid {\theta }_j>.$$

(1)

Feynman introduces an observable acting on the system of interest only, i.e., of the form $A\otimes I_2$, where $I_2$ is the unit operator in the state space of ${\Sigma }_2$. He considers the mean value of this observable in state $\mid \psi >,$ i.e., the quantity:

$$\begin{array}{ccc}\hfill <\psi \mid A\otimes I_2\mid \psi >& =& {\Sigma }_{ij{i}^{\prime }{j}^{\prime }}{c}_{ij}^{*}{c}_{i^{\prime }{j}^{\prime }}<{\theta }_j\mid <{\phi }_i\mid A\otimes I_2\mid {\phi }_{i^{\prime }}>\mid {\theta }_{j^{\prime }}>\hfill \\ & =& {\Sigma }_{ij{i}^{\prime }}{c}_{ij}^{*}{c}_{i^{\prime }j}<{\phi }_i\mid A\mid {\phi }_{i^{\prime }}>,\hfill \end{array}$$

(2)

which can be written as

$$<\psi \mid A\otimes I_2\mid \psi >={\Sigma }_{i{i}^{\prime }}<{\phi }_i\mid A\mid {\phi }_{i^{\prime }}>{\rho }_{1,i^{\prime }i}$$

(3)

with

$${\rho }_{1,i^{\prime }i}={\Sigma }_j{c}_{ij}^{*}{c}_{i^{\prime }j}.$$

(4)

Feynman then defines an operator acting in the Hilbert space of ${\Sigma }_1$, which we will note ${\rho }_1$ (Feynman notes it as $\rho$), and such that, for any $i^{\prime },$ i: ${\rho }_{1,i^{\prime }i}=<{\phi }_{i^{\prime }}\mid {\rho }_1\mid {\phi }_i>.$

Manifestly, for any $i^{\prime },i$: ${\Sigma }_j{c}_{ij}^{*}{c}_{i^{\prime }j}={\left({\Sigma }_j{c}_{i^{\prime }j}^{*}{c}_{ij}\right)}^{*}$ i.e., ${\rho }_{1,i^{\prime }i}={\rho }_{1,i{i}^{\prime }}^{*}$, which means that ${\rho }_1$ is Hermitian. Therefore ${\rho }_1$ does possess a diagonal form, and its eigenvalues are real. Moreover, since a Trace does not depend on the chosen basis, and since $\mid \psi >$ is normed, this trace is equal to one:

$$Tr{\rho }_1={\Sigma }_i{\rho }_{1,ii}={\Sigma }_{ij}\mid c_{ij}{\mid }^2=1.$$

(5)

Feynman then shows that all the eigenvalues of ${\rho }_1$ are non-negative. Detailing his approach, we introduce the operator $A\otimes I_2,$ with $A=\mid i^{\prime }><i^{\prime }\mid$, $\mid i^{\prime }>$ being an eigenstate of ${\rho }_1,$ with eigenvalue $w_{i^{\prime }}$: ${\rho }_1={\Sigma }_i{w}_i\mid i><i\mid .$ We then calculate the mean value of $A\otimes I_2$ in state $\mid \psi >:$

$$\begin{array}{ccc}\hfill <\Psi \mid \mid i^{\prime }><i^{\prime }\mid \otimes I_2\mid \Psi >& =& <\Psi \mid i^{\prime }><i^{\prime }\mid \otimes {\Sigma }_j\mid {\theta }_j><{\theta }_j\mid \Psi >\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& =& {\Sigma }_j<\Psi \mid i^{\prime }>\mid {\theta }_j><i^{\prime }\mid <{\theta }_j\mid \Psi >\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& =& {\Sigma }_j\mid <i^{\prime }\mid <{\theta }_j\mid \Psi >{\mid }^2.\hfill \end{array}$$

(8)

This mean value $<\Psi \mid \mid i^{\prime }><i^{\prime }\mid \otimes I_2\mid \Psi >$ is also equal to:

$$\begin{array}{ccc}\hfill Tr\{{\rho }_1\mid i^{\prime }><i^{\prime }\mid \}& =& Tr\{{\Sigma }_i{w}_i\mid i><i\mid \mid i^{\prime }><i^{\prime }\mid \}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}& =& Tr\{w_{i^{\prime }}\mid i^{\prime }><i^{\prime }\mid \}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& =& w_{i^{\prime }}.\hfill \end{array}$$

(11)

Therefore: $w_{i^{\prime }}={\Sigma }_j\mid <i^{\prime }\mid <{\theta }_j\mid \Psi >{\mid }^2:$ any eigenvalue of ${\rho }_1$ is non-negative.

In the situation analyzed by von Neumann, with a statistical mixture $\{\mid {\Psi }_i>,$$p_i\}$ of normed kets $\mid {\Psi }_i>,$ examined in our paper [1], the density operator $\rho ={\Sigma }_i{p}_i\mid {\Psi }_i><{\Psi }_i\mid$ was a formal tool. It allowed one to get an expresssion of the mean value of an observable $\widehat{O}$ in the presence of that statistical mixture as a Trace: $Tr\rho \widehat{O}$. In the present situation, a density operator ${\rho }_1$ acting in the state space of ${\Sigma }_1$ has been introduced, allowing one to write that the mean value of A, an observable of ${\Sigma }_1$, is equal to $Tr{\rho }_{1}A.$ One may now introduce the following density operator: ${\rho }_w$ (w: whole):

$${\rho }_w=\mid \psi ><\psi \mid ,$$

(12)

describing the whole system in state $\mid \psi >$. The introduction of both ${\rho }_w$ and ${\rho }_1$ allows one to describe the contribution of the collection of the normed states $\mid {\phi }_{i^{\prime }}>\mid {\theta }_{j^{\prime }}>$ for a given $i^{\prime }$ in a compact way: one can verify that ${\rho }_1$ can be interpreted as the following partial Trace (over the state space of ${\Sigma }_2$):

$${\rho }_1=T{r}_{{\Sigma }_2}{\rho }_w.$$

(13)

If, at some time $t_0,$ the system and its environment are both prepared in a pure state, and if, then, during a time interval $\tau ,$ a coupling exists between the system and its environment, at the end of this time interval, here taken as the origin of time ($t_0$ being therefore negative), the whole system is still in a pure state $\mid \psi \left(0\right)>$, obeying the Schrödinger equation, but, except accidently, $\mid \psi \left(0\right)>$ is not a product state. If, at $t=0,$ one is interested in A acting in the state space of ${\Sigma }_1$ only, one can introduce the density operator ${\rho }_w=\mid \psi \left(0\right)><\psi (0\mid$ describing the state of the whole system $\Sigma$, and then, using a partial Trace, obtain the mean value $<\psi \left(0\right)\mid A\mid \psi \left(0\right)>.$ One may be interested in the mean value of A at a time $t>0$, i.e., after the disappearance of the transient coupling of ${\Sigma }_1$ with ${\Sigma }_2$ (but within, of course, our time scale). Feynman introduces the eigenstates $\mid E_n>$ of $H_1,$ the Hamiltonian of ${\Sigma }_1:$$H_1\mid E_n>=E_n\mid E_n>$. Then, if $f\left(H_1\right)$ is a function of $H_1,$$f\left(H_1\right)\mid E_n>=f\left(E_n\right)\mid E_n>$ (definition of a function of an operator). He then first writes ${\rho }_1$ (our own notations are kept here) as ${\rho }_1={\Sigma }_i{w}_i\mid i\left(0\right)><i\left(0\right)\mid$, ($\mid i\left(0\right)>$ was previously written as $\mid i>$), and then ${\rho }_1\left(t\right)$ as ${\rho }_1\left(t\right)={\Sigma }_i{w}_i\mid i\left(t\right)><i\left(t\right)\mid$. Detailing the argumentation from Feynman [11], one first observes that, for $t\ge 0:$

$$\begin{array}{ccc}\hfill \mid i\left(t\right)>& =& {\Sigma }_n\mid E_n><E_n\mid \mid i\left(t\right)>\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& =& {\Sigma }_n\mid E_n>e^{-i{E}_{n}t/\hslash }<E_n\mid \mid i\left(0\right)>\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& =& e^{-i{H}_{1}t/\hslash }\mid i\left(0\right)>.\hfill \end{array}$$

(16)

Since ${\rho }_1\left(t\right)={\Sigma }_i{w}_i\mid i\left(t\right)><i\left(t\right)\mid$, one gets:

$$\begin{array}{ccc}\hfill {\rho }_1\left(t\right)& =& {\Sigma }_i{w}_i{e}^{-i{H}_{1}t/\hslash }\mid i\left(0\right)><i\left(0\right)\mid e^{i{H}_{1}t/\hslash }\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& =& e^{-i{H}_{1}t/\hslash }{\rho }_1\left(0\right)e^{i{H}_{1}t/\hslash }.\hfill \end{array}$$

(18)

Then deriving with respect to time, one gets, for $t\ge 0:$

$$i\hslash {\displaystyle \frac{d{\rho }_1}{dt}}=[H_1,{\rho }_1]$$

(19)

(Feynman used units with $\hslash =1$). This equation is also obeyed in the presence of a von Neumann mixture, and usually called the Liouville - von Neumann equation.

5. Discussion

In Section 4 it was explained that if a system $\Sigma$ composed of two parts ${\Sigma }_1$ and ${\Sigma }_2$ is isolated and described by a density operator ${\rho }_w$ (w: whole), one may introduce either ${\rho }_1=T{r}_{{\Sigma }_2}{\rho }_w$ or ${\rho }_2=T{r}_{{\Sigma }_1}{\rho }_w,$ a formal tool helpful in the calculation of the mean value of an observable attached to one subsystem only, once a possible coupling between the two subsystems has disappeared. As detailed in Appendix B, it would generally be quite wrong to think that the whole system is then described by ${\rho }_1\otimes {\rho }_2,$ generally describing a statistical mixture, whereas one knows that the whole system is in a pure state $\mid \Psi \left(t\right)>.$ Already in 1939, F. London and Bauer wrote (English translation, page 248 in [14]): “While the combined system I+II, which we suppose isolated from the rest of the world, is and remains in a pure state, we see that during the interaction systems I and II individually transform themselves from pure cases into mixtures. This is a rather strange result”. In fact, the concept to be used in that situation is not the one of a mixture, but the concept of entanglement: with the notations used by F. London and Bauer, I and II are initially in a product state, and, as a consequence of their interaction, the state of the whole system becomes an entangled (pure) state, and then it is meaningless to try and speak of the state of either subsystem I or subsystem II. The word entanglement had been introduced four years earlier by Schrödinger in the context of the EPR discussion. That problem mentioned by F. London and Bauer was not treated in the 1955 canonical book Quantum Mechanics from Schiff [15]. Since the appearance of the 1972 book on statistical mechanics from Feynman, the problem was presented neither in the book Quantum Theory: Concepts and Methods by Peres [4] nor in the 2013 Lectures on Quantum Mechanics by Steven Weinberg [16]. It was discussed by Le Bellac in his 2006 book [17]. But Le Bellac adopted the point of view introduced by D’Espagnat, with a distinction between so-called proper and improper mixtures, instead of identifying the presence of a pure, entangled, state.

We already stressed that D’Espagnat did identify a problem, but that his distinction did not solve it. Before ending this discussion, we briefly comment on some of the arguments introduced in the controversy about that proper mixed state/improper mixed state distinction proposed by D’Espagnat:

• We first examine the approach adopted by K.A. Kirkpatrick. After his 2001 paper [13], he published another ArXiv paper [18] on the subject in 2004. But we must immediately add that trying to adopt, even momentarily, the approach taken by K.A. Kirkpatrick would mean introducing confusion again, for the following unquestionable reason: D’Espagnat introduced a physical problem, namely an isolated bipartite quantum system, with a transient internal coupling between its two parts. K.A. Kirkpatrick systematically ignores that reality. One is then forced to conclude that the content of the cited papers from K.A. Kirkpatrick has nothing to do with the problem discussed by D’Espagnat.
• We then come to the 2022 paper [6] by Castellani. We first mention the fact that his reference 2 to one of the 1927 papers by Landau is surprising. As we observed in our present paper, the title chosen by Landau immediately indicates that in that paper Landau considers a so-called open system. We then recall that, from the beginning, Castellani considers “the state of a subsystem in a composite system to be completely described by its reduced density matrix”. Accepting, even momentarily, such an assumption would just be reintroducing the confusion which the present paper is trying to eliminate. One prefers to insist upon the idea that the concept of entanglement should be taken seriously.

The reader interested in the situation found when a coupling Hamiltonian does persist between the two parts of an isolated bipartite sytem, i.e., the situation when each subsystem is said to be open, may consult the book The Theory of open quantum systems, by Breuer and Petruccione [19], and identify the importance of the so-called master (or Lindblad) equation in that situation.

6. Conclusions

When the two parts of an isolated bipartite system are initially in a pure state, and are then momentarily coupled to each other, one may focus on one of these subsystems once the internal coupling has disappeared. If one wants to calculate the mean value of some observable attached to that subsystem, one may introduce a density operator, through partial tracing over the second subsystem. A 2022 paper and a 2006 book, both cited in this paper, show that the meaning of that density operator is still under debate. The aim of the present paper was to stress once for all that it would be wrong to try and interpret the introduction of this so-called reduced density operator through a reference to some mixed state, be it either proper or improper. The right concept in this context is the one of entanglement: once the transient internal coupling has disappeared, the bipartite system keeps in an entangled pure state (except when, either it exceptionally keeps in a pure product state, or, at some specific times, it accidentally becomes unentangled).