Spontaneous disentanglement and thermalisation

The problem of quantum measurement can be partially resolved by incorporating a process of spontaneous disentanglement into quantum dynamics. We propose a modified master equation, which contains a nonlinear term giving rise to both spontaneous disentanglement and thermalisation. We find that the added nonlinear term enables limit cycle steady states, which are prohibited in standard quantum mechanics. This finding suggests that an experimental observation of such a limit cycle steady state can provide an important evidence supporting the spontaneous disentanglement hypothesis.

A modified Schrödinger equation having a nonlinear term that gives rise to suppression of entanglement (i.e.disentanglement) has been recently proposed [30].This nonlinear extension partially resolves the self-inconsistency associated with the measurement problem by making the collapse postulate of QM redundant.The proposed modified Schrödinger equation can be constructed for any physical system whose Hilbert space has finite dimensionality, and it does not violate norm conservation of the time evolution.The nonlinear term added to the Schrödinger equation has no effect on product (i.e.disentangled) states.The spontaneous disentanglement generated by the modified Schrödinger equation gives rise to a process similar to state vector collapse.
The nonlinear extension that was proposed in Ref. [30] is applicable only for bipartite systems, and only for pure states.To allow incorporating spontaneous disentanglement for more general cases, we propose here a modified master equation for the time evolution of the density operator ρ.The modified master equation [see Eq. ( 2) below] contains a nonlinear term that gives rise to spontaneous disentanglement.For a multipartite system, disentanglement between any pair of subsystems can be introduced by the added nonlinear term.Moreover, ther- * Electronic address: eyal@ee.technion.ac.il malisation can be incorporated by an additional nonlinear term added to the master equation (2).
In contrast to the modified master equation ( 2), which is nonlinear in ρ, in standard QM the time evolution of ρ is governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [18,31,32], which is linear in ρ.This linear dependency excludes any nonlinear dynamics in the time evolution of ρ (see appendix B of Ref. [33]), and, in particular, it excludes a limit cycle steady state for any quantum system having a Hilbert space of finite dimensionality and time independent Hamiltonian.On the other hand, as is demonstrated below, the modified master equation ( 2) yields rich nonlinear dynamics.In particular, both a Hopf bifurcation and a limit cycle steady state may occur.Note, however, that, nonlinearity in ρ does not necessarily imply that dynamical instabilities are possible, as was demonstrated in Ref. [34].
For the case H = 0, and for a fixed operator Θ, the modified master equation ( 2) yields an equation of motion for Θ given by The above result (3) implies for this case that the expectation value Θ monotonically decreases with time.
Hence, the nonlinear term in the modified master equation (2) can be employed to suppress a given physical property, provided that Θ quantifies that property.
Here the operator Θ is assumed to be given by Θ = γ H Q (H) + γ D Q (D) , where both rates γ H and γ D are positive, and both operators Q (H) and Q (D) are Hermitian.The first term γ H Q (H) , which gives rise to thermalisation [37,38], is discussed below in section III, whereas section IV is devoted to the second term γ D Q (D) , which gives rise to disentanglement.

III. THERMALISATION
Consider the master equation (2) for the case where H is time independent, γ D = 0 (i.e.no disentanglement), and Q (H) = βU H , where U H = H + β −1 log ρ is the Helmholtz free energy operator, β = 1/ (k B T ) is the thermal energy inverse, k B is the Boltzmann's constant, and T is the temperature.For this case, the thermal equilibrium density matrix ρ 0 , which is given by is a steady state solution of the master equation ( 2), for which the Helmholtz free energy U H is minimized [34,[37][38][39].The rate γ H represents the thermalisation inverse time.Note that γ H needs not be a constant.

IV. DISENTANGLEMENT
Consider the case where γ H = 0 (i.e.no thermalisation).As can be seen from Eq. (3), disentanglement can be generated by the term proportional to γ D in the Schrödinger equation (1), and by the term proportional to γ D in the master equation (2), provided that the operator Q (D) is chosen such that Q (D) quantifies entanglement [40][41][42][43][44][45][46][47][48][49][50].In Ref. [30] the operator Q (D) was chosen to be equal to Q (S) , where Q (S) is constructed using the Schmidt decomposition.This allowed deriving a modified Schrödinger equation having the form given by Eq. ( 1), which contains a nonlinear term that gives rise to pure state bipartite disentanglement.However, the Schmidt decomposition is inapplicable for both mixed states and for multipartite systems.Here we employ an alternative operator (henceforth denoted as Q (D) ), which can be used to derive a modified master equation having the form given by Eq. ( 2), and which is applicable for a general multipartite case [35], and for a general mixed state.Consider a multipartite system composed of three subsystems labeled as 'a', 'b' and 'c'.The Hilbert space of the system H = H a ⊗ H b ⊗ H c is a tensor product of subsystem Hilbert spaces H a , H b and H c .The dimensionality of the Hilbert space H L of subsystem L, which is denoted by d L , where L ∈ {a, b, c}, is assumed to be finite.A general observable of subsystem L can be expanded using the set of generalized Gell-Mann matrices λ Entanglement between subsystems a and b can be characterized by the matrix D ab ≡ ρ ab − ρ a ⊗ ρ b , where ρ ab is the reduced density matrix of the combined a and b subsystems, and ρ a (ρ b ) is the reduced density matrix of subsystem a (b).The following holds [see Eq. (A4) of appendix A] where for any given observable O a = O † a of subsystem a, and a given observable where I L is the d L × d L identity matrix, and where L ∈ {a, b, c}.
The above result (5) suggests that entanglement between subsystems a and b can be quantified by the nonnegative variable τ ab , which is given by τ ab = Q and where η ab is a positive constant.The (a, b) entry of the , and the (a, b) entry of the , and thus τ ab can be expressed as [compare to Eq. ( 31) of Ref. [40]] In a similar way, the entanglement between subsystems b and c, which is denoted by τ bc , and the entanglement between subsystems c and a, which is denoted by τ ca , can be defined.Deterministic disentanglement between subsystems L ′ and L ′′ can be generated by the modified master equation ( 2), provided that the operator Q (D) in Eq. ( 2) is replaced by the operator L ′ ,L ′′ .The entanglement variable τ ab is invariant under any single subsystem unitary transformation [40].Under such a transformation, the matrix C is transformed according to C → C ′ = T a CT T b .A completeness relation, which is satisfied by the generalized Gell-Mann matrices [see Eq. (8.177) or Ref. [51]], can be used to show that both the , and thus, as can be seen from Eq. ( 7), τ ab is invariant [see Eq. (8.881) of Ref. [51]].The invariance of τ bc and τ ca can be shown in a similar way.

V. BIPARTITE PURE STATE DISENTANGLEMENT
To gain some insight into the disentanglement process, the relatively simple case of a bipartite system in a pure state |ψ is considered.Subsystems are labeled as 'a' and 'b'.With the help of the Schmidt decomposition, |ψ can be expressed as where d m = min (d a , d b ), the coefficients q l are nonnegative real numbers, the tensor product |l a ⊗ |l b is denoted by |l, l , and {|l a } ({|l b }) is an orthonormal basis spanning the Hilbert space H a (H b ) of subsystem a (b).The normalization condition reads ψ |ψ = L 2 = 1, where the n'th moment L n is defined by For a product state, for which q l = δ l,l0 , where l 0 ∈ {1, 2, • • • , d m }, τ ab obtains its minimum value of τ ab = 0 [see Eqs. ( 6) and ( 8)].The maximum value of τ ab , which is given by [maximum entropy state, see Eq. ( 8)].The constant η ab is chosen to be given by For this choice τ ab is bounded between zero and unity.Consider for simplicity the case where the Hamiltonian vanishes, i.e.H = 0.For that case the modified Schrödinger equation (1) where the so-called capitalistic function For d m = 2 the factor 4 in Eq. ( 12) is replaced by 12.The identity K (m) l = ∂H (m) /∂q l , where the potential function H (m) is given by FIG. 1: Pure bipartite disentanglement.The time evolution of the coefficients q l is calculated using Eq. ( 12) for the case dm = 10 and γη ab = 1.The coefficients q l 0 , which initially, at time t = 0, is the largest one, i.e. q l 0 = max {q l }, is represented by the red curve.
implies that L 2m (i.e.L 6 for m = 3) monotonically increases in time (recall the normalization condition L 2 = 1).A similar derivation, based on the operator Q (S) , leads to a set of equations of motion similar to ( 12), but with m = 2 [30].For that case L 4 monotonically increases in time [see Eq. ( 13)].
The following holds [see Eq. ( 12)] For both cases Q (S) (for which m = 2) and Q (D) (for which m = 3), time evolution governed by Eq. ( 14) gives rise to disentanglement.Consider the case where initially, at time t = 0, q l0 = max {q l } for a unique positive integer As can be seen from Eq. ( 14), for this case |ψ evolves into the product state |l 0 , l 0 in the long time limit, i.e. q l → δ l,l0 for t → ∞.This behavior is demonstrated by the plot shown in Fig. 1.

VI. TWO SPIN 1/2
While thermalisation increases entropy, disentanglement decreases it (as is demonstrated by the plot shown in Fig. 1) [52].The interplay between thermalisation and disentanglement is explored below using a relatively simple system composed of two spins 1/2.For this case d a = d b = 2, and η ab = 1/3 [see Eq. (11)].The angular momentum vector operator of spin L is denoted by S L = (S Lx , S Ly , S Lz ), where L ∈ {a, b}.
An example for time evolution, which is obtained by numerically integrating the modified master equation (2), is shown in Fig. 2. Assumed parameters' values are listed in the figure caption.

VII. TRUNCATION APPROXIMATION
The simplest physical system suitable for the exploration of disentanglement is the above-discussed two spin 1/2 system.For some cases, further simplification can be achieved by implementing a truncation approximation.
Let H be the Hamiltonian of a two spin 1/2 system.The matrix representation of H in the basis {|00 , |01 , |10 , |11 } is assumed to be given by where the central 2 × 2 block of Ω is given by both the scalar ω s and the vector ω E = (ω Ex , ω Ey , ω Ez ) are real, and σ = (σ x , σ y , σ z ) is the Pauli matrix vector.
Consider the case where ω s β ≫ 1.For this case, for which the population of the states |00 and |11 is low, a truncation approximation can be employed.In this approximation it is assumed that the system's Hilbert space is spanned by the vector states |01 and |10 .Note that in the truncation approximation, τ ab for pure states is given by Eq. ( 15), with D = q 01 q 10 .The truncated density matrix is expressed as where the central 2 × 2 block of ρ is given by where µ ∈ [0, 1] is real, and n = (n x , n y , n z ) is a unit vector (i.e.n • n = 1).Note that Tr As can be seen from Eq. ( 21), in the truncation approximation the system's state is describe by a single real 3-dimensionla Bloch vector k = µn.With the help of the truncated density matrix ρ (20), one finds that the matrix Θ where the central 2 × 2 block of Θ (D) is given by The following holds thus for a given µ, the expectation value The expectation value Q (D) obtains its minimum value Q (D) = 0 for n 2 z = 1 and µ = 1.These two points (north and south poles of the Bloch sphere) represent fully disentangled states.
The entropy matrix S = − log ρ is given by where the central 2 × 2 block of S is given by (26) Note that the expectation value S , which is given by is bounded by S ∈ [0, log 2].With the help of Eq. ( 26) one finds that the central 2 × 2 block of (Sρ + ρS − 2 S ρ) is given by k 19), and recall the identity (σ The expectation value Θ is given by Θ = γ H Q (H) + γ D Q (D) , where and Q (D) is given by Eq. (24).The expectation value Q (H) is minimized at thermal equilibrium, for which The modified master equation ( 2) yields an equation of motion for the 3-dimensional Bloch vector k = µn given by An example of numerical integration of Eq. ( 30) is shown in Fig. 3.This example demonstrates a disentanglementinduced shift of a steady state fixed point away from thermal equilibrium (which is represented by a cyan cross symbol).Assumed parameters are listed in the figure caption. and The plots shown in Fig. 4 exhibit the time evolution of the single spin Bloch vectors k a and k b , where k L = ((1/2) S L+ + S L− , (−i/2) S L+ − S L− , S Lz ), and where L ∈ {a, b}.The case g = 0 (i.e.no dipolar coupling) is represented by the plots (a1) and (b1).For this case the steady state is a fixed point, which is labeled by a red cross symbol in Fig. 4(a1) and (b1).On the other hand, in the presence of sufficiently strong dipolar coupling, the steady state becomes a limit cycle, as is demonstrated by the plots shown in Fig. 4(a2) and (b2), for which g/ω a = 0.1.The limit cycle angular frequency is close to ω a .

IX. DISCUSSION AND SUMMARY
The above-discussed MFA greatly simplifies the analysis.For the two spin 1/2 system, the modified master equation (2) leads to a set [see Eq. (A6)] of 4 2 − 1 = 15 real equations of motion (generalized Bloch equation) for 15 real variables (generalized Bloch vector).On the other hand, for the same system, the MFA leads to a set of 2 2 − 1 + 2 2 − 1 = 6 real equations of motion for 6 real variables [see Eqs.(34) and (35)].
As is demonstrated by the plots shown in Fig. 4, the nonlinear terms in Eqs. ( 34) and ( 35) can give rise to a limit cycle steady state.Such limit cycle steady states cannot be obtained from the GKSL master equation, which is linear in ρ (see appendix B of Ref. [33]).On the other hand, in spite of the linear dependency of the GKSL equation on ρ, some theoretical studies have revealed nonlinear dynamics that is derived from this GKSL master equation [57][58][59][60][61].However, the origin of such nonlinearity is the assumption that entanglement between subsystems can be disregarded.It has remained unclear how such an assumption can be justified in the framework of standard QM.On the other hand, this assumption represents a limiting case for the modified master equation ( 2), for which disentanglement is sufficiently efficient.
The limit cycle steady state shown in Fig. 4(a2) and (b2) can occur only when the driving detuning ∆ is positive (i.e.driving is blue-detuned).This behavior demonstrates that disentanglement can give rise to detuning asymmetry.On the other hand, in the absence of disentanglement, the system's response is theoretically expected to be an even function of the detuning ∆ [e.g.see Eq. (4) of Ref. [62]].Many examples can be found in the published literature for a profound detuning asymmetry observed in spin systems under nutation driving or dynamical decoupling.In most papers the presented asymmetry is not discussed, however, a paper from 1955 [63], and another one from 2005 [64], explicitly state that the observed asymmetry is theoretically unexpected.Moreover, limit cycle steady states are experimentally observed in systems of correlated spins [65].Further study is needed to explore possible connections between experimentally observed nonlinear dynamics in spin systems [66] and disentanglement.
In summary, the spontaneous disentanglement hypothesis is inherently falsifiable, because it yields predictions, which are experimentally distinguishable from predictions obtained from standard QM.In particular, as was discussed above, the experimental observation of a limit cycle steady state in a system having a Hilbert space of finite dimensionality (i.e. a spin system) may provide a supporting evidence for the spontaneous disentanglement hypothesis.Moreover, such an experimental observation may yield some insight related to the question 'what determines the disentanglement rate γ D ?', which has remained entirely open.The set G (ab) can be used to expand the entire system density matrix ρ (which is assume to be normalized, i.e.Tr ρ = 1) as ρ =

FIG. 2 :
FIG. 2: Bell singlet state.The time evolution of the single spin Bloch vectors ka and k b is shown in (a) and (b), respectively.Initial values for ka and k b are denoted by green cross symbols.(c) The purity Tr ρ 2 .(d) The entanglement variable τ .(e) The expectation value PB of the projection PB = |ψB ψB|.(f) The expectation value UH of the Helmholtz free energy.Assumed parameters' values are γH/ωB = 0.005, γD/ωB = 0.05 and ωBβ = 10.