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Entanglement production by statistical operators

In the problem of entanglement there exist two different notions. One is the entanglement of a quantum state, characterizing the state structure. The other is entanglement production by quantum operators, describing the action of operators in the given Hilbert space. Entanglement production by statistical operators, or density operators, is an important notion arising in quantum measurements and quantum information processing. The operational meaning of the entangling power of any operator, including statistical operators, is the property of the operators to entangle wave functions of the Hilbert space they are defined on. The measure of entanglement production by statistical operators is described and illustrated by entangled quantum states, equilibrium Gibbs states, as well as by the state of a complex multiparticle spinor system. It is shown that this measure is in intimate relation to other notions of quantum information theory, such as the purity of quantum states, linear entropy, or impurity, inverse participation ratio, quadratic R\'{e}nyi entropy, the correlation function of composite measurements, and decoherence phenomenon. This measure can be introduced for a set of statistical operators characterizing a system after quantum measurements. The explicit value of the measure depends on the type of the Hilbert space partitioning. For a general multiparticle spinor system, it is possible to accomplish the particle-particle partitioning or spin-spatial partitioning. Conditions are defined showing when entanglement production is maximal and when it is zero. The study on entanglement production by statistical operators is important because, depending on whether such an operator is entangling or not, it generates qualitatively different probability measures, which is principal for quantum measurements and quantum information processing.


Introduction
Entanglement is a principally important notion for several branches of quantum theory, such as quantum measurements, quantum information processing, quantum computing, and quantum decision theory (see books and reviews [1][2][3][4][5][6][7][8][9]). It is possible to distinguish three directions in studying entanglement for composite systems described in terms of tensor products of Hilbert spaces.
One is the entanglement of quantum states, characterized by wave functions in the case of pure states and by statistical operators, for mixed states. A wave function is entangled when it cannot be represented by a tensor product of wave functions pertaining to different Hilbert spaces. And the wave function is disentangled, when it can be represented as a product (1.1) A statistical operator is entangled if it cannot be represented as a linear combination of products of partial statistical operators acting in different Hilbert spaces [1][2][3][4][5][6][7][8]. And it is called separable, if it can be represented as a finite linear combination in which 0 ≤ p k ≤ 1 , andρ ik are statistical operators acting on partial Hilbert spaces [1][2][3][4][5][6][7]. The notion of states can be straightforwardly extended to a set of bounded operators, which, being complimented by the Hilbert-Schmidt scalar product, forms a Hilbert-Schmidt space, where the operators are isomorphic to states of this space. Then it is admissible to consider the entanglement of operators in a way similar to the entanglement of states, thus just lifting the notion of entanglement from the state level to the operator level [9][10][11][12][13][14].
A rather separate problem is the study of entangling properties of unitary operators acting on a set of given states. This can be characterized by considering the entanglement of states generated by these unitary operators acting on disentangled states. In the case of several states, one averages the appropriate measure of a unitary operator, say linear entropy, over a set of states with a given distribution [10,12,14,15]. One usually considers unitary operators, since information-processing gates are characterized by such operators. In that approach, the problem is reduced to the consideration of the entanglement of the states, obtained by the action of a unitary operator, under the given set of initial states. But this does not describe the entangling properties of an operator acting on the whole Hilbert space.
In the present paper, we consider the related problem of describing entangling properties of operators. An operator is called entangling, if there exists at least one separable pure state such that it becomes entangled under the action of the operator. Conversely, one says that an operator preserves separability if its action on any separable pure state yields again a separable pure state. It has been proved [16][17][18] that the only operators preserving separability are the operators having the form of tensor products of local operators and a swap operator permuting Hilbert spaces in the tensor product describing the total Hilbert space of a composite system. The action of the swap operator is trivial, in the sense that it merely permutes the indices labeling the spaces. This result of separability preservation by product operators has been proved for binary [16,19,20] as well as for multipartite systems [17,18,21]. The operators preserving separability can be called nonentangling [22,23]. While an operator transforming at least one disentangled state into an entangled state is termed entangling [24,25]. The strongest type of an entangling operator is a universal entangling gate that makes all disentangled pure states entangled [26].
The general problem is what could be a measure of entanglement production characterizing the entangling properties of an arbitrary operator defined on the whole Hilbert space of a composite system, but not only for some selected initial states from this space. Such a global measure of entanglement production by an arbitrary operator has been proposed in Refs. [27,28]. This measure is applicable to any system, whether bipartite or multipartite, and to any trace-class operator [29,30], which does not necessarily need to be unitary. The entanglement production has been investigated for several physical systems, such as multimode Bose-Einstein condensates of atoms in traps and in optical lattices [31][32][33] and radiating resonant atoms [34]. The entanglement production by evolution operators has also been studied [35].
As is mentioned above, the operator entanglement is usually considered for unitary operators, since the evolution operators as well as various information gates are unitary. However, it may happen important to study the entanglement production by nonunitary operators. For example, one may need to quantify the entanglement production by statistical operators. The entangling properties of the latter define the characteristic features of quantum measurements, as well as the structure of probability measure in quantum information processing and quantum decision theory. Also, it can be necessary to study thermal entanglement production characterized by the amount of entanglement produced by connecting an initially closed nonentangled quantum system to a thermal bath. There exists a variety of finite quantum systems that can be initially prepared in a desired pure state [36]. Then this system can be connected to a thermal bath in the standard sense of realizing a thermal contact that transfers heat but does not destroy the system itself, as a result of which the system acquires the thermal Gibbs distribution [37]. The immediate question is how much entanglement can be produced by this nonunitary procedure of connecting an initially closed quantum system to a thermal bath?
It is the aim of the present paper to study the entanglement production by statistical operators. In Sec. 2, we explain the operational meaning of entanglement production, introduce basic notations, concretize the difference between separable and nonentangling operators, and demonstrate how the problem of entanglement production by statistical operators naturally arises in the theory of quantum measurements, quantum information processing, and quantum decision theory. In Sec. 3, we define a general measure of entanglement production by arbitrary operators and specify the consideration for different types of statistical operators. The calculational procedure for this measure is demonstrated in Sec. 4 by several simple, but important, examples of entangled pure states. We explain in Sec. 5 how the introduced measure of entanglement production is connected with the other known quantities, such as the purity of quantum states, linear entropy, or impurity, inverse participation ratio, quadratic Rényi entropy, and the correlation function of composite measurements. This measure can be defined for a set of statistical operators characterizing a system after quantum measurements. Section 6 demonstrates that the decoherence phenomenon is connected with the increase of the entanglement-production measure. In Sec. 7, we study the entanglement production by an equilibrium Gibbs operator with the Ising type Hamiltonian, since such Hamiltonians are widely employed for representing qubit registers. The measure of entanglement production depends on the type of coupling between qubits, whether it is ferromagnetic or antiferromagnetic. In Sec. 8, we turn to complex multiparticle systems, for which it is admissible to consider different ways of partitioning the system degrees of freedom. In Sec. 9, we study a general case of a multiparticle spinor system, calculating the entanglement production measure for particle partitioning and for spin-spatial partitioning. Section 10 concludes.

Operational meaning of entanglement production
In order to avoid confusion, let us first of all concretize the difference between separable and nonentangling operators. We also stress the importance of the operator entanglement production in the process of quantum measurements [61]. Note that quantum measurements can be treated as decisions in decision theory [61][62][63], because of which the mathematical structure of quantum decision theory is the same as that of quantum measurement theory [13,64,65]. The difference is only in terminology, where a measurement is called a decision and the result of a measurement is termed an event.
An operator is defined on a Hilbert space and acts on wave functions (vectors) of this space. The property of the operator to produce entangled wave functions from disentangled ones is called entanglement production. The operational meaning of the entangling power of an operator is its ability of entangling the wave functions of the Hilbert space it acts on [16][17][18][19][20][21][22][23]26]. This notion is applicable to any operator acting on a Hilbert space, including statistical operators.

Separable versus nonentangling operators
One considers a system in a Hilbert space H characterized by a statistical operatorρ that is a semi-positive, trace-one operator. The pair {H,ρ} is called statistical ensemble. The considered system is composite, with the Hilbert space being a tensor product Each space H i possesses a basis {|n i }, so that An operator algebra {Â} is defined on the space H, consisting of trace-class operators, for which 0 = |Tr HÂ | < ∞ .
An operatorÂ, acting on a disentangled function of H can either result in another disentangled function or transform the disentangled function into an entangled function. The sole type of a nonentangling operator, except the trivial swap operator changing the labelling, has the factor form [17,18,21] which, as is evident, is defined up to a multiplication constant. The notion of separable states can be extended to operators [9][10][11][12][13][14]. Then a separable operator is such that can be represented as the finite linear combination where λ k are complex-valued numbers. A nonentangling operator is a particular case of a separable operator, when λ k is proportional to δ kko , i.e., it is a rank-1 separable operator. But the principal difference of a general separable operator from a nonentangling operator is that the former does entangle disentangled functions. This is evident from the action of a separable operator on a disentangled function, yieldinĝ which is an entangled function, if λ k is not proportional to δ kko . Observable quantities are represented by self-adjoint operatorsÂ. For a system characterized by a statistical operatorρ, the measurable quantities are given by the averages Â ≡ Tr HρÂ . (2.7) The peculiarity of measurements are essentially different for the systems with an entangling or nonentangling statistical operators. Even if one is measuring an observable corresponding to a nonentangling operator (2.4), but the statistical operator being entangling, the related average is not reducible to a product of partial averages, where Such a reduction is possible only when the statistical operator is also nonentangling.

Structure of probability measure
Similarly, in quantum decision theory, an event is represented by an operatorP that is either a projector or, more generally, an element of a positive operator-valued measure [1-4, 7, 9]. The event operatorP plays the role of an operator of observable. And the probability of the event is defined by the average p(P ) ≡ P = Tr HρP , (2.10) which takes the values in the interval 0 ≤ p(P ) ≤ 1. A composite event, describing the set of independent partial events, has the form of a nonentangling operator i . (2.11) If the system statistical operator is entangling, the probability of the composite event cannot be reduced to the product of the probabilities of partial events, The reduction is possible only if the statistical operator is also nonentangling. Thus the structure of the probability measure is principally different for the cases of either entangling or nonentangling statistical operators.

Measure of entanglement production
One usually considers the entangling properties of unitary operators describing gates acting on bipartite systems, but in general, the operator does not need to be unitary. If an operatorÛ acts on a bipartite state function |ϕ 12 , one gets a new functionÛ|ϕ 12 defining the corresponding bipartite stateρ 12 =Û|ϕ 12 ϕ 12 |Û + . Then the problem is reduced to studying the entangled structure of this stateρ 12 by means of the known entanglement measures of bipartite states, such as entangling power, linear entropy, and like that [10,12,14,66]. However, this does not describe the global entangling property of an operator on the whole Hilbert space where it is defined. A general measure of entanglement production, applicable to arbitrary (not necessarily unitary) operators acting on the whole Hilbert space, containing any number of factors, has been suggested in Refs. [27,28]. Here, we shall use this measure for quantifying the entangling properties of statistical operators.

Arbitrary operators
The definition of the measure is as follows. Let us be interested in the entangling properties of an operatorÂ acting on a composite Hilbert space (2.1). The idea is to compare the action of this operator on H with the action of its nonentangling product counterpart that is a product of the reduced operatorŝ

. Thereforê
By the theorem proved for binary products [16,19,20], as well as for an arbitrary number of factors [17,18,21], the product operator form (3.4) never entangles any functions. The entanglement production measure for the operatorÂ is defined as The logarithm can be taken with respect to any base. This quantity (3.5) satisfies all conditions required for being classified as a measure [27,28,67]. Thus it enjoys the properties: (i) it is semipositive and bounded for the finite number of factors N; (ii) it is continuous in the sense of norm convergence; (iii) it is zero for nonentangling operators; (iv) it is additive; (v) it is invariant under local unitary operations.
As the norm here, it is convenient to accept the Hilbert-Schmidt norm which does not depend on the chosen basis. Respectively, the norm of a partial reduced operator, acting on H i , is

Statistical operators
Our aim is to consider statistical operators, for which the nonentangling counterpart reads aŝ The normalization condition is Tr Hρ⊗ = Tr Hρ = 1 .
Therefore we need to study the measure in which Explicitly, the measure writes as

Pure states
In the case of pure states, statistical operators have the form where |ψ is a normalized wave function. This statistical operator is idempotent, so that ||ρ|| = Tr Hρ 2 = 1 (ρ 2 =ρ) . (3.14) Then measure (3.10) becomes Or, taking into account the above relations, for pure states, we get Notice that the defined measure is valid for arbitrary systems, with any statistical operators, and with any number of factors in the Hilbert space (2.1). Also, any operators, unitary or not, can be considered [28].

Separable states
As has been mentioned in Sec. II, separable operators are, generally, entangling. Now, we can demonstrate this by explicitly calculating the measure of entanglement production for a separable statistical operator. Let us consider a separable statê where normalized wave functions |n ik belong to H i and are orthogonal, n ik ′ |n ik = δ kk ′ . Taking into account the propertieŝ we get the norm The partial statistical operators arê with the propertiesρ Then for the norm of the nonentangling counterpart, we find The entanglement production measure (3.10) becomes This is evidently nonzero, provided that N > 1 and p k = δ kk 0 .

Gibbs states
For an equilibrium system, characterized by a Hamiltonian H, the Gibbs statistical operator iŝ where β is inverse temperature. With the partial operatorŝ the nonentangling counterpart isρ Introducing the notations we can represent the entanglement production measure as Thus, for a given Hamiltonian, we need to calculate the functions (3.27) and (3.28), and the partition function Z.

Entangled pure states
Before going to more complicated problems, it is useful to illustrate how the measure is calculated for simple cases of pure states. Generally, depending on the definition of the employed norm, the entanglement production measure can be slightly different [28]. Here we use the Hilbert-Schmidt norm. We shall see that for bipartite systems with entangled states, the entanglement production measure ε(ρ) coincides with the entanglement von Neumann entropy S(ρ i ) ≡ −Tr H iρ i lnρ i . The examples considered in this section illustrate how the measure is calculated, which will allow us to shorten the explanation of intermediate calculations in the following more complicated cases.

Einstein-Podolsky-Rosen states
The corresponding statistical operator iŝ The reduced operators areρ The corresponding norms are Then we find the entanglement production measure Note that in this case, the entanglement entropy S(ρ i ) ≡ −Tr H iρ i lnρ i coincides with measure (3.10).

Bell states
The statistical operator isρ Calculations are similar to the previous case, giving Again, this coincides with the entanglement entropy S(ρ i ) = log 2.

Greenberger-Horne-Zeilinger states
These states are a generalization of two-particle Bell states to N particles, so that with The entanglement entropy for N-particle states is not defined.

Multicat states
Such states are a generalization of the Schrödinger cat states to N objects, where with c i being complex numbers satisfying the normalization |c 1 | 2 + |c 2 | 2 = 1 . The reduced operators areρ Calculating the norms The maximal entanglement production is reached for |c i | 2 = 1/2, yielding  With the norms The maximal entanglement production happens for |c

Purity of quantum state
The purity of a quantum stateρ in the Hilbert space H is defined as and shows the closeness of the state to a pure state. For a pure state γ(ρ) = 1, while for a completely mixed state γ(ρ) = 1/d. Purity describes the spread of the state over the given basis.
Similarly to the purity of the total stateρ, it is possible to introduce the purity of the partial states The purity of the nonentangling stateρ ⊗ becomes Then measure (3.12) can be presented as The denominator of the fraction under the logarithm in equation (5.6) has the meaning of an effective purity of the nonentangling state of a system composed of partial subsystems. Hence measure (5.6) shows how much the purity of the given stateρ is larger than the effective purity of the nonentangling stateρ ⊗ corresponding to the system composed of partial subsystems.

Linear entropy or impurity
The linear entropy of a stateρ is given by the expression Because of its relation (5.8) to the state purity, the linear entropy is also called impurity. In the same way, the linear entropy of the nonentangling stateρ ⊗ is Therefore, measure (3.12) can be written as By partitioning the system, with a stateρ, into subsystems, with partial statesρ i , one gets the system composed of the subsystems, with the nonentangling stateρ ⊗ , whose purity is smaller than the purity of the initial stateρ. Consequently, the impurity, that is, the linear entropy, of the nonentangling stateρ ⊗ is larger than that of the stateρ. In that sense, measure (5.11) describes how much the impurity of the stateρ ⊗ increases, as compared to the stateρ, before the partitioning.

Inverse participation ratio
Sometimes purity is used as a measure of localization and linear entropy, as a measure of delocalization. This is because these concepts are closely connected with the notion of inverse participation ratio [68][69][70][71][72]. Inverse participation ratio can be introduced as a measure of localization in the real space for characterizing Anderson localization or in phase space for describing semiclassical localization. For the purpose of the present paper, it is more convenient to introduce the inverse participation ratio characterizing Hilbert-space localization [73], which can be defined as This definition shows that the inverse participation ratio is equivalent to the transition probability averaged over time. For the basis formed by the eigenvectors of the Hamiltonian from the eigenproblem in which ρ mn ≡ m|ρ(0)|n . This expression is valid for both nondegenerate or degenerate spectrum. For a nondegenerate spectrum, this simplifies to 14) The inverse participation ratio varies in the range which is the same as the variation range of state purity. Remembering definition (5.1) of state purity and introducing the notation for the second-order coherence function [75] we obtain the relation R −1 (ρ) = γ(ρ) − C 2 (ρ) (5.16) between the purity and inverse participation ratio. When the nondiagonal terms are less important than the diagonal ones, the inverse participation ratio is approximately equal to the state purity. This is why the latter can also serve as a measure characterizing localization in a Hilbert space. Therefore measure (3.12) shows to what extent the system stateρ is more localized in the Hilbert space than the stateρ ⊗ of the partitioned system.

Quantum Rényi entropy
In the quantum setting, the Rényi entropy of order α, for a stateρ acting on a Hilbert space H, is given by the form Here we are interested in the quadratic Rényi entropy which is connected with the state purity and varies in the range Since the state purity can characterize Hilbert-space localization, the quadratic Rényi entropy can serve as a measure of impurity and delocalization [74,76,77].
The quadratic Rényi entropy can also be defined for partial states, Then the quadratic Rényi entropy for the partitioned stateρ ⊗ reads as Therefore measure (3.12) can be represented as the difference quantifying how much the Rényi entropy of the partitioned stateρ ⊗ is larger than that of the initial stateρ. In other words, measure (5.23) is half of the difference, measured in terms of the Rényi entropy, of the stateρ from the product stateρ ⊗ . For a pure stateρ, the Rényi entropy is zero, Then for a pure state, measure (5.23) becomes one half of the sum of partial Rényi entropies In the case of a bipartite system, the partial Rényi entropies play the role of the system entanglement entropies. In such a case, measure (5.25) coincides with the entanglement entropy, Notice that for bipartite systems the Rényi entropy is claimed to be available for measuring [78].
Recall that in the general case, measure (5.23) quantifies how much the Rényi entropy of the partitioned stateρ ⊗ overweights the Rényi entropy of the initial stateρ. Since the Rényi entropy shows the degree of delocalization, the measure (5.23) defines to what extent the partitioned stateρ ⊗ is more delocalized than the initial stateρ.

Correlation in composite measurements
Entangling property of a statistical operator is of great importance for studying correlations in composite measurements. For simplicity, we consider here a bipartite system, with the Hilbert space although the generalization to larger composite systems is straightforward. Let us examine a composite measurement, represented by the operatorÂ B , formed by two measurements described by the operatorsÂ on H A andB on H B , respectively. The operatorsÂ andB correspond to the operators of local observables.
The correlation between these two measurements is characterized by the correlation function which explicitly reads as As is clear, the value of the correlation function depends on the entangling property of the system stateρ. If the system state is nonentangling, such that it can be represented as a product of the partial states, then the correlation function is zero, But if the system stateρ is entangling, the correlation function is not zero, which implies that the two measurements cannot be made independently of each other, since they are correlated with each other. The stronger the entangling ability ofρ, that is, the larger its entanglement production measure (3.10), the larger the absolute value |C AB | of the correlation function (5.28).

State reduction after measurements
For each system stateρ, we can define the measure of entanglement production ε(ρ). Moreover, if the system is subject to measurements, then there appear the whole set of possible states and, respectively, the set of the related measures.
Let the system be in a stateρ. And let us be interested in an observable represented by the operatorQ acting on the Hilbert space H. The basis of this space can be taken as defined by the eigenproblemQ | n = Q n | n , If the result of the measurement of this observable is Q n , then, according to the von Neumann -Lüders theory [61,79], the system state reduces tô ρ n =P nρPn Tr H (ρP n ) .
Generally, the operatorsP n here are the projectors on subspaces associated with the eigenvalues Q n . For a nondegenerate spectrum of Q n , which we assume for simplicity in what follows, P n = |n n|. where Tr H (ρP n ) 2 = ρ 2 nn , Tr HρPn = ρ nn , ρ mn ≡ m |ρ | n .
Hence Tr Hρ 2 n = 1. The corresponding product state iŝ Thus, the entanglement production measure of the new state is Altogether, we get a set of the measures for different multi-indices n.

Decoherence in nonequilibrium systems
In nonequilibrium systems, the stateρ(t) depends on time, which can lead to the temporal evolution of the measure ε(ρ(t)). This evolution is closely connected with such an important phenomenon as decoherence [80]. Below, we show that the phenomenon of decoherence is in intimate relation to the measure ε(ρ(t)). Generally, the observable quantity (6.3) can be written as the sum of a diagonal and nondiagonal terms The effect of decoherence implies [80] that the nondiagonal term tends to zero with time, so that lim t→∞ ρ mn (t) = 0 (m = n) . (6.6) This happens because of the interaction between the subsytem of interest and the surrounding. Decoherence appears even when the surrounding is represented by measuring devices realizing the so-called nondestructive, nondemolition, or minimally disturbing measurements [81][82][83][84].
These distributions are the densities of states normalized so that Then the matrix elements (6.17) can be written as with the notation Factors (6.21) enjoy the properties To proceed further, let us assume that the system is sufficiently large, so that the state distributions g mn and g αβ are measurable, similarly to the density of states of macroscopic systems [85]. And by definition (6.19) these functions are integrable. Then by Riemann-Lebesgue lemma [86], one has lim t→∞ D mn (t) = 0 (m = n) , Hence in the expressions the nondiadonal parts tend to zero with increasing time.
The increase of the entanglement production measure, as is explained in Sec. 5.4, means that the difference, measured by the Rényi entropy, of the system stateρ(t) from the nonentangling product stateρ ⊗ (t) increases under decoherence. In other words, the growing entanglement production measure implies that the system state becomes more entangling as a result of decoherence.

Two-qubit register in thermal bath
As an example of an equilibrium Gibbs state, let us consider the Gibbs state of a two-qubit register in thermal bath. Such states are often met in quantum information theory. The Gibbs state is defined in the usual way, as in Eq. (3.24), where the influence of the thermal bath is characterized by the bath temperature. Note that this description is equivalent to the method, when one models a system-bath interaction, after which one averages out the bath degrees of freedom, under the assumption of thermal contact between the system and the bath [87][88][89], which is effectively represented by the statistical operator of the Gibbs state defined in Eq. (3.24), depending on the bath inverse temperature β.

Calculating entanglement-production measure
The system Hamiltonian is a sum of a noninteracting part H 0 and an interaction term H int . The noninteracting part has the Zeeman form where S z i are spin 1/2 operators and B plays the role of an external field. The interaction term describes the qubit coupling. When J > 0, the coupling is called ferromagnetic, while if J < 0, it is named antiferromagnetic. The Hamiltonian acts on the Hilbert space H being the closed linear envelope over the basis formed by the Hamiltonian eigenfunctions.
Since H 0 and H int commute, one has The exponential operators can be reduced to non-exponential forms [90]. Noticing that where we find Respectively, taking into account the relations, we get Combining Eqs. (7.7) and (7.9) yields Thus we come to function (3.28) in the form The entanglement production measure (3.29) takes the form When the qubits are not coupled, so that J → 0, but B = 0, then And there is no entanglement production: In the opposite case, when B → 0, but J = 0, we have In what follows, we assume that B is positive, B > 0. A more detailed analysis of the entanglement production measure (7.16) should be done separately for the ferromagnetic and antiferromagnetic coupling.

Entanglement production under ferromagnetic coupling (J > 0)
According to Eq. (7.19), there is no entanglement production without qubit coupling. Nontrivial behavior of measure (7.16) exists only for J = 0. It is therefore convenient to introduce the dimensionless variables so that measure (7.16) becomes a function of these variables, ε(ρ) = ε(T, h) . (7.24) In the definition of measure (7.16), for concreteness, we take the natural logarithm. The asymptotic behavior of the measure is as follows.
At low temperature, but finite h, we have In the opposite regime of small h, but finite temperature, we get At high temperature, but finite h, the measure is which shows that lim And for large h, but finite temperature, we find The general behavior of the entanglement production measure, under ferromagnetic coupling, as a function of the dimensionless variables T and h, is demonstrated in Fig. 1. The maximal value of the measure max ε(T, h) = 1 2 ln 2 = 0.347 (7.35) is reached when, first, h → 0, under finite T , after which T → 0. At high temperature, but finite h, we find And when h → ∞, at finite temperature, we obtain with the same coefficients b 2 and b 3 as in the high-field limit (7.32). Therefore lim h→∞ ε(T, h) = 0 (T ≥ 0) . (7.46) Figure 2 shows the general behavior of the entanglement production measure, under antiferromagnetic coupling, as a function of the dimensionless variables T and h. The maximal value (7.37) is reached at low temperature and h < 1.

Hilbert space partitioning
For many systems, as studied in the previous sections, the partitioning of the Hilbert space has been uniquely fixed. For more complex systems, the type of partitioning may be not unique. Respectively, the entangling properties of the system statistical operator depend on which parts of the system are being entangled. To illustrate how different kinds of partitioning could arise, let us consider a system of N particles with spins. For brevity, we can combine the spatial, r i , and spin, σ i , degrees of freedom in the notation x i = {r i , σ i }. The system wave function |Ψ nl depends on the multi-indices n and l for the spatial and spin states, respectively. The function can be treated as a column with respect to all its variables, so that its normalization reads as As usual, summation with respect to discrete indices is assumed. The system statistical operator acts on the Hilbert space H.

Particle partitioning
The natural partitioning of the system Hilbert space is with respect to particles composing the system. Then we can define the real-space single-particle Hilbert space as a closed linear envelope over a single-particle basis depending on real-space coordinates. Similarly, a spin-dependent basis defines the Hilbert space Then a single-particle Hilbert space is The total system Hilbert space can be represented as a tensor product of single-particle spaces. Following the general scheme, we define the reduced statistical operatorŝ whose tensor product induces the nonentangling operator Then the entanglement production of the statistical operator (8.3), with respect to the Hilbert space partitioning (8.7), is quantified by the measure Keeping in mind indistinguishable particles, we get Therefore measure (8.10) becomes Note that we have no problems dealing with indistinguishable particles, while the definition of state entanglement for indistinguishable particles confronts some problems [91,92]. All we need is to correctly symmetrize the system wave function depending on whether bosons or fermions are considered.

Spin-spatial partitioning
It is also interesting to study the entanglement between spin and spatial degrees of freedom [93][94][95][96][97]. To consider such a spin-spatial entanglement production, it is necessary to partition the system Hilbert space onto spin and spatial degrees of freedom. For this purpose, we introduce the real-space Hilbert part (8.13) and the spin Hilbert space (8.14) Then the total Hilbert space is a tensor product of the spatial and spin parts 9 Multiparticle spinor quantum system

Permutation-invariant wavefunctions of spinor particles
This section formulates general properties of many-body wavefunctions of indistinguishable spinor particles with separable spin and spatial degrees of freedom. Such a system is described by the HamiltonianĤ spat +Ĥ spin , whereĤ spat is spin-independent,Ĥ spin is spatially-homogeneous, and each ofĤ spat andĤ spin is permutation-invariant. The wavefunctions are composed from the spin Ξ [λ] tl and spatial Φ [λ] tn functions, which form bases of irreducible representations of the symmetric group S N of N-symbol permutations (see [39-41, 54, 98]). This means that a permutation P of the particles transforms each basis function to a linear combination of the functions in the same representation, t ′ n . t ′ t (P) satisfy the following relations, where E is the identity permutation. These relations provide the proper bosonic or fermionic permutation symmetry of the total wavefunction Ψ [λ] nl . The representation dimension is given by For spin-1/2 particles, the Young diagrams have two rows and are unambiguously related to the total spin S, λ = [N/2 + S, N/2 − S]. The representation dimension can be expressed as An explicit expression for the spin wavefunction is obtained [99] in the case of commutativê H spin and the total spin projection operatorŜ z , (9.10) The wavefunction is unambiguously determined by the total spin S and its projection S z , which is the half of the difference of the occupations of the two spin states | ↑ and | ↓ . The normalization factor is expressed as [99] C SSz = 1

Spin-spatial partitioning
The spin-spatial entanglement production measure can be evaluated for a generic system of indistinguishable spinor particles with separable spin and spatial degrees of freedom. Due to the orthogonality of the spin and spatial functions, Ξ [λ] t ′ n = δ tt ′ , we have for the total wavefunction (9.7) and, similarly,ρ tn . (9.13) Then Tr Hspat ρ spat and Tr H spin ρ spin Therefore, according to Eq. (8.19), the entanglement production measure ε(ρ nl ) = ln f λ (9.16) depends only on the representation dimension (9.8).
The leading term of the asymptotic expansion in the limit N → ∞ can be evaluated using the Stirling formula in Eq. (9.8) as λ m ln λ m . (9.17) Its maximum, ε(ρ nl ) = N ln M, is attained for equal lengths of the Young diagram rows λ m = N/M. For spin-1/2 particles, the entanglement production measure decreases when S increases (see Fig. 3). The measure vanishes at S = N/2, when the total wavefunction is a single product of the spin and spatial functions. The plot for N → ∞ is obtained using the leading term in the asymptotic expansion, In the asymptotic limit, the entanglement production measure attains its maximum of ε(ρ nl ) = N ln 2 at S = 0, when the Young diagram rows have the equal length.

Particle partitioning
The particle entanglement production measure can be evaluated in the particular case of noninteracting particles with s = 1 2 . If there are several spatial orbitals |0 , . . . |M − 1 , there are multiple spatial wavefunctions for the given λ and the orbital occupations, even ifĤ spat commutes with the orbital occupations (see [40,54,98]). However, if there are only two spatial orbitals, |+ and |− , andĤ spat commutes with the "isotopic spin"Î z = N j=1 (|+(j) +(j)|−|−(j) −(j)|)/2, the spatial wavefunction is unambiguously determined by the total spin S and the eigenvalue I z ofÎ z (it is nothing but the half of the difference of the orbital occupations) and can be represented for bosons like (9.10), Here the normalization factor is defined by Eq. (9.11). Given S, the system state is specified by two independent spin projections, S z , and I z . Then the multi-indices n and l can be specifically chosen as I z and S z . Ground states of such systems were analyzed in Refs. [100,101] using SU(2) symmetry (SU(2) and symmetric groups are closely related, having a common set of basic functions of irreducible representations, see [40]).
In the particular basic, the reduced statistical operators (8.8) have the following explicit form, 20) where n can be + or −, σ can be ↑ or ↓, and the summation is performed over all n i ′ and σ i ′ with i ′ = i. Their matrix elements can be expressed as matrix elements of the projection operator IzSz |n(i) |σ(i) n(i)| σ(i)|Ψ

[λ]
IzSz δ nn ′ δ σσ ′ . (9.21) Due to permutation symmetry of the total wavefunction, the matrix element n, σ|ρ IzSz |P nσ |Ψ , as an operator in the spin space, is a component of an irreducible spherical vector (see [98]). Then its matrix elements between states with arbitrary S z can be related to ones for S z = S using the Wigner-Eckart theorem (see [98]). In the case of two spacial orbitals, the same can be done for I z too, providing IzSz is an eigenfunction ofŜ z andÎ z , the matrix element of the reduced statistical operators can be related to the one for S z = I z = S, n, σ|ρ  where Ξ was calculated in Ref. [98]. Using similar expressions for the spatial matrix elements, Eqs. Its maximal value N ln 2 is attained at S z = I z = 0 for any S and N (see Figs. 4 and 5). In the case of the spin-spatial partition, this value can be reached only in the limit N → ∞. The particle and spin-spatial entanglement production measures both vanish at S z = I z = S = N/2. However, given 0 < S z < N/2 or 0 < I z < N/2, the particle entanglement increases with S, being maximal at S = N/2, when the total wavefunction is a single product of the spin and spatial functions and the spin-spatial entanglement vanishes.

Conclusion
Dealing with statistical operators, one can consider two different notions. One is the state entanglement characterizing the structure of the given statistical operator. The other notion is the entanglement production by the statistical operator, describing the action of the statistical operator on the given Hilbert space and showing how this action creates entangled functions from disentangled ones. These two notions are principally different and should not be confused. The operational meaning of the entangling power of statistical operators is the same as for any other operator defined on a Hilbert space: it shows the ability of an operator to produce entangled wave functions of the given Hilbert space. Throughout the paper, the notion of entanglement production has been used in line the commonly accepted in mathematical literature [16][17][18][19][20][21][22][23]26].
Entangling properties of statistical operators play an important role in several branches of quantum theory, such as quantum measurements, quantum information processing, quantum computing, and quantum decision theory, where one deals with composite measurements and composite events, related to composite probability measures. Entangling properties of statistical operators influence the structure of probability measures they generate. Depending on whether the statistical operator is entangling or not, the resulting probability measure can be either not factorizable or factorizable, as is discussed in Sec. 2.
We have defined the measure of entanglement production by statistical operators and illustrated it by several examples of entangled pure states, equilibrium Gibbs states, and by the case of a multiparticle spinor system. The relation of the introduced measure to other known concepts, such as quantum state purity, linear entropy or impurity, inverse participation ratio, quadratic Rényi entropy, and correlators in composite measurements, is thoroughly discussed. The measure can be defined for a collection of quantum systems or for a set of operators characterizing a quantum system after measurements. The phenomenon of decoherence is also shown to be intimately related to entanglement production.
For complex spinor systems, the measure of entanglement production depends on the type of partitioning of the total Hilbert space. Thus, it is possible to realize particle partitioning or spin-spatial partitioning. Both these cases are analyzed. The analysis demonstrates when the entanglement production is maximal and when it tends to zero, which can be used in the applications of quantum theory mentioned above.