Spontaneous Collapse by Entanglement Suppression

A modified Schrödinger equation having an added nonlinear term, which gives rise to disentanglement, has been recently proposed. The process of quantum measurement is explored for the case of a pair of coupled spins. It is found that the deterministic time evolution generated by the modified Schrödinger equation mimics the process of wavefunction collapse. Added noise gives rise to stochasticity in the measurement process. Conflict with both principles of causality and separability can be avoided by postulating that the nonlinear term is active only during the time when subsystems interact. Moreover, in the absence of entanglement, all predictions of standard quantum mechanics are unaffected by the added nonlinear term.

Introduction -In standard quantum mechanics a measurement is described by a two-step process.The first step is governed by the standard Schrödinger equation.To avoid a possible paradoxical outcome of a description based only on the first step (undefined cat state [1]), a second step is postulated, in which the state vector collapses.However, it has remained unknown how such a second step can be self-consistently added [2][3][4].This difficulty has became known as the problem of quantum measurement.
In this work we explore an alternative to the collapse postulate, which is based on a modified Schrödinger equation that has an added nonlinear term giving rise to disentanglement [5,6].The proposed 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.We explore the dynamics of a system made of two coupled spins, and find that disentanglement gives rise to a process similar to state vector collapse.
Other types of nonlinear extensions of quantum mechanics [7] have been previously proposed and studied [8][9][10][11][12][13][14].Most previously proposed extensions give rise to a spontaneous collapse [15][16][17][18][19].In some cases, however, the proposed nonlinear models are inconsistent with wellestablished physical principles.Moreover, many predictions of standard quantum mechanics, that have been experimentally verified to very high precision, are significantly altered by some of the proposed nonlinear extensions.Such difficulties are discussed below in the final part of this paper for the case of our proposed modified Schrödinger equation.We find that possible conflicts with the principles of causality and separability, and with many experimentally confirmed predictions of standard quantum mechanics, can be avoided by postulating that disentanglement is active only when subsystems interact.
Disentanglement -Consider a system composed of two subsystems labeled as '1' and '2', respectively.The dimensionality of the Hilbert spaces of both subsystems, which is denoted by N 1 and N 2 , respectively, is assumed * Electronic address: eyal@ee.technion.ac.il to be finite.The system is in a normalized pure state vector |ψ given by where ) is an orthonormal basis spanning the Hilbert space of subsystem '1' ('2').
The purity P 1 (P 2 ) is defined by 2 ), where ρ 1 = Tr 2 ρ (ρ 2 = Tr 1 ρ) is the reduced density operator of the first (second) subsystem.By employing the Schmidt decomposition one finds that P 1 = P 2 ≡ P , where P = 1 − Q = 1 − ψ| Q |ψ , the operator Q is given by [see Eq. (A15) of appendix A, and Ref. [20]] and the state 2 , which depends on the matrix C corresponding to a given state |ψ , is given by (note that where Note that Q = 0 for a product state.In standard quantum mechanics Q is time independent when the subsystems are decoupled (i.e.their mutual interaction vanishes).
Consider a modified Schrödinger equation for the ket vector |ψ having the form where is the Planck's constant, H = H † is the Hamiltonian, the rate γ is positive, and the operator Q is given by Eq. ( 2).The added nonlinear term proportional to γ gives rise to disentanglement, however, it has no effect when |ψ represents a product state.Note that the norm conservation condition 0 = (d/dt) ψ |ψ is satisfied by the modified Schrödinger equation ( 4).Dipolar interaction -As an example, the dynamics generated by the modified Schrödinger equation ( 4) is explored for the case of dipolar interaction between two spins having spin quantum numbers S 1 and S 2 , respectively.The dipolar interaction is represented by the operator , where the rate ω d is positive, S n is the spin angular momentum vector operator of the n'th spin (n ∈ {1, 2}), and ûd = (sin θ cos ϕ, sin θ sin ϕ, cos θ) is a unit vector.
Time evolution examples for the case S 1 = 1/2 and S 2 = 21/2 are shown by the plots in Fig. 1.The initial state at time t = 0 is a product state, for which the spin 1/2 is pointing in the direction of the unit vector n1 (labeled by a red star symbol), and the spin 21/2 is pointing in the direction of the unit vector n2 = −ẑ (labeled by a cyan star symbol).The overlaid blue solid (dashed) lines connect the origin and the dipolar coupling unit vectors ûd (−û d ).The spin 1/2 Bloch vector k = ( /2) −1 S 1 is numerically calculated by integrating the modified Schrödinger equation ( 4) for the case H = V d .The black solid lines in Fig. 1(a1), (a2), (a3) and (a4) represent the spin 1/2 Bloch vector k evolving from its initial value n1 at time t = 0.The single-spin purity P = 1 − Q as a function of time t is shown in Fig. 1(b1), (b2), (b3) and (b4).
For the plots in Fig. 1 labeled by the numbers 1, 2 and 3, the dipolar unit vector ûd is given by ûd = x (i.e.ûd is perpendicular to n2 = −ẑ).These plots, which differ by the initial direction n1 of the spin 1/2 (labeled by red star symbols), demonstrate that the Bloch sphere is divided into two basins of attraction.The first (second) basin is the hemisphere n1 • ûd > 0 (n 1 • ûd < 0), and the corresponding attractor is ûd (−û d ).
While n2 • ûd = 0 for the plots in Fig. 1 labeled by the numbers 1, 2 and 3, the behavior when the initial spin S 2 direction n2 is not perpendicular to the dipolar coupling unit vector ûd is demonstrated by the plots labeled by the number 4. The plot in Fig. 1(a4) shows that the Bloch vector trajectory, from the initial value n1 (labeled by the red star symbol) towards the attractor at ûd becomes spiral-like when n2 • ûd = 0.The basins of attraction for this case (i.e.plots in Fig. 1 labeled by the number 4) are shown in Fig. 2.This example demonstrates that the dipolar unit vector ûd determines the spin 1/2 component that is being measured.The measurement process is deterministic however the outcome, which is either +1 (when n1 • ûd > 0) or −1 (when n1 • ûd < 0) is quantized.This behavior is demonstrated by the green dashdotted line in Fig. 3, in which the probability p + that the measurement outcome is +1 is plotted as a function of the angle θ 1 = cos −1 (n 1 • ûd ).For comparison, the red solid line represents the Born rule of standard quantum mechanics, for which p + (θ 1 ) = cos 2 (θ 1 /2).A simplified model is employed below to explore noise-induced stochasticity.
Noise -The effect of external noise is taken into account by applying a random rotation to the initial spin 1/2 Block vector n1 .The random rotation is characterized by an axis normal to n1 , and by a rotation angle φ r .As an example, consider the case where the rotation angle φ r has a wrapped Cauchy probability distribution f (φ r ) given by where φ 0 > 0 is a scale factor.Consider a rotated frame, in which the dipolar unit vector ûd is parallel to the unit vector ẑ.The unit vector n1 in this frame is denoted by n1R .The probability p + that the measurement outcome is +1 is calculated by spherical integration over the hemisphere z ′ ≥ 0 where θ 1R = cos −1 (n 1R • n′ ), and where n′ = (sin θ ′ cos ϕ ′ , sin θ ′ sin ϕ ′ , cos θ ′ ).As can be seen from the blue dashed line in Fig. 3, which is calculated using Eq. ( 6) with a scale factor of φ 0 = 0.5, noise-induced stochasticity mimics the behavior predicted by the Born rule (red solid line).
The measurement time -For the examples shown in Fig. 1, initially at time t = 0, the ket vector |ψ represents a product state having single-spin purity P = 1.The time dependency of P is shown in Fig. 1(b1), (b2), (b3) and (b4).In the short time limit of ω d t ≪ 1 the effect of the disentanglement term in the modified Schrödinger equation ( 4) is relatively weak (since Q is initially small), and consequently P rapidly drops due to entanglement generated by the dipolar interaction V d .At latter times, when disentanglement becomes sufficiently efficient, the single-spin purity P starts increasing.Interaction-induced generation of entanglement becomes inefficient when the spin 1/2 becomes nearly parallel or nearly anti-parallel to the dipolar unit vector ûd , and consequently the single-spin purity P approaches unity in the long time limit.
For sufficiently short times after turning on the interaction (i.e. after t = 0), time evolution is dominated by the effect of the dipolar interaction.When the effect of the disentanglement term is disregarded, one finds that in the short time limit the following holds d S n /dt ≃ ω n ûd × S n , where n ∈ {1, 2}, ω 1 = ω d −1 S 2 • ûd and ω 2 = ω d −1 S 1 • ûd .Thus, in the short time limit, the purity P is roughly given by P (6.192) and (8.701) of Ref. [20], and note that it is assumed that in the short time limit the spin states are nearly spin coherent states [22]].The above-derived expression for the purity time evolution P (t) reveals the dependence of short-time dynamics on the macroscopicity of the measuring apparatus (i.e. the second spin), which is represented by the spin number S 2 .
Vanishing Hamiltonian -To gain further insight into the disentanglement process generated by the nonlinear term −γ (Q − Q ) added to the Schrödinger equation (4), consider for simplicity the case where the Hamiltonian vanishes, i.e.H = 0.The Schmidt decomposition of a general state vector |ψ is expressed as where q l are non-negative real numbers, the tensor product |l 1 ⊗|l 2 is denoted by |l, l , and {|l 1 } ({|l 2 }) is an orthonormal basis spanning the Hilbert space of subsystem '1' ('2').Note that for a product state q l = δ l,l0 , where l 0 ∈ {1, 2, • • • , min (N 1 , N 2 )}.The normalization condition reads ψ |ψ = L 2 = 1, where the n'th moment L n is defined by Note that for a product state L n = 1 for any positive integer n (provided that |ψ is normalized).
In the Schmidt basis, the following holds [see Eqs. ( 2) and ( 3)] and Q = 1 − L 4 , and thus [see Eq. ( 4)] An example solution of the set of equations (10) for the case min (N 1 , N 2 ) = 10 and γ = 1 is shown in Fig. 4. The time evolution of the n'th moment L n is governed by [see Eqs. ( 8) and (10)] FIG. 4: Vanishing Hamiltonian.The plot shows an example solution of the set of equations ( 10) for the case min (N1, N2) = 10 and γ = 1.The solution for q l 0 (t) is represented by the red line, whereas the blue lines represent the solutions for q l (t) with l = l0.For this example, q l (t = 0) ≃ (min (N1, N2)) −1/2 , i.e. the initial value of the purity P is close to its smallest possible value of 1/ min (N1, N2).The corresponding initial entropy σ is close to its largest possible value of log (min (N1, N2)).In the limit t → ∞ the purity P → 1 (largest possible value) and the entropy σ → 0 (smallest possible value).
For any two integers l ′ , l ′′ ∈ {1, 2, • • • , min (N 1 , N 2 )} the following holds [see Eq. ( 10)] The above relation (12) implies that the ratio q l ′ /q l ′′ monotonically increases with time, provided that q l ′ > q l ′′ (recall that γ > 0).This behavior gives rise to disentanglement.Consider the case where initially, at time t = 0, q l0 = max {q l } for a unique positive integer l 0 ∈ {1, 2, • • • , min (N 1 , N 2 )}.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 in the limit t → ∞ (see Fig. 4).Note, however, that in the long time limit the state can be strongly affected by noise when initially the set {q l } doesn't have a unique member significantly larger than all others.
Discussion -As was already mentioned above, several types of nonlinear extensions of quantum mechanics have been proposed and explored [15,[23][24][25][26].However, it was found that for some cases, the proposed nonlinear extension gives rise to the violation of the causality principle by enabling superluminal signaling [27][28][29][30].More recently, it was shown that when a condition called 'convex quasilinearity' is satisfied by a given nonlinear master equation, the violation of the causality principle becomes impossible [31,32].Some of the proposed nonlinear extensions are inconsistent with the principle of separability [28,33,34].Moreover, any proposed extension must be ruled out if it alters predictions of standard quantum mechanics that have been experimentally confirmed.
The modified Schrödinger equation given by Eq. ( 4) has an important advantage compared to other proposals: the added nonlinear term −γ (Q − Q ) has no effect on product states.This implies that in the absence of entanglement, the added term does not vary any prediction of standard quantum mechanics.Moreover, possible conflicts with both principles of causality and separability can be avoided by postulating that γ ≃ −1 ψ| V † V |ψ 1/2 , where V is the coupling term in the Hamiltonian giving rise to the interaction between subsystems [γ is the disentanglement rate in Eq. ( 4)].This postulate implies that the added nonlinear term is active only when subsystems interact, and that time evolution is governed by the standard Schrödinger equation when subsystems are remote (i.e.decoupled).Note that for the examples shown in Fig. 1, the calculations are performed for the case γ = ω d .This demonstrates that a disentanglement rate γ having the order of −1 ψ| V † V |ψ 1/2 is sufficiently large to allow full suppression of entanglement.
Summary -Further theoretical study is needed to check whether quantum mechanics can be selfconsistently reformulated based on the proposed modified Schrödinger equation (4).We find that conflict with some well-established physical principles, as well as many experimental observations, can be avoided by postulating that γ ≃ −1 ψ| V † V |ψ 1/2 .
The expression given by Eq. ( 2) for the operator Q is applicable for the bipartite case, for which the entire system is divided into two subsystems.The multipartite case, however, for which the entire system is divided into more than two subsystems, requires a generalization of Eq. ( 2).Such generalization is discussed in Ref. [35].The generalization of the above discussed postulate (regarding the disentanglement rate γ) for the multipartite case states that disentanglement between two given subsystems is active only during the time when they interact.
Further insight can be gained from experimental study of entanglement in the region where environmental decoherence is negligible [36].Upper bounds imposed upon the disentanglement rate γ in Eq. ( 4) can be derived from lifetime measurements of entangled states.Experimental observations of deviation from the Born rule may provide supporting evidence for nonlinearity (see Fig. 3).