Stochastic Schrödinger Equations

A diffusive stochastic Schrödinger equation evolves a conditional pure state driven by Wiener increments. The noise scales as $\sqrt{\Delta t}$, so terms quadratic in noise survive at order $\Delta t$. Treating it as an ordinary differential equation with a noisy coefficient usually gives the wrong drift.

Helpful background. Quantum trajectories explains ensemble averaging; measurement feedback treats conditional density matrices.

A normalized Itô equation

Consider ideal monitoring of a Hermitian operator $A$, with collapse operator $L=\sqrt\kappa A$ and real Wiener process $W$. The normalized Itô equation is

$$d|\psi\rangle= \left[-\frac{i}{\hbar}H-\frac{\kappa}{2} (A-\langle A\rangle)^2\right]|\psi\rangle\,dt +\sqrt\kappa(A-\langle A\rangle)|\psi\rangle\,dW.$$

Here $\mathbb E[dW]=0$, $(dW)^2=dt$, and expectations are evaluated in the conditional state at the start of the increment. The Itô product rule gives

$$d\langle\psi|\psi\rangle =\langle d\psi|\psi\rangle+\langle\psi|d\psi\rangle +\langle d\psi|d\psi\rangle=0.$$

The quadratic noise term cancels drift-induced norm loss. Averaging projectors gives $\dot\rho=-i[H,\rho]/\hbar+\kappa\mathcal D[A]\rho$, where $\mathcal D[A]\rho=A\rho A-\{A^2,\rho\}/2$. The same written coefficients interpreted in the Stratonovich sense describe a different process. Conversion requires a drift correction, not merely renormalization.

Euler–Maruyama and refinement

Generate $\Delta W=\sqrt{\Delta t}\,\xi$ with independent standard normal $\xi$ and update $\psi_{n+1}=\psi_n+a(\psi_n)\Delta t+b(\psi_n)\Delta W$. Finite steps do not preserve norm exactly. Record the pre-normalization defect, normalize before the next conditional expectation, and verify convergence. Projection does not raise the method’s order.

For a normalized input, the drift and diffusion above satisfy

$$2\operatorname{Re}(\psi^\dagger a)+\|b\|^2=0, \qquad 2\operatorname{Re}(\psi^\dagger b)=0.$$

Consequently the Euler step’s leading norm defect is $\|b\|^2[(\Delta W)^2-\Delta t]$, with additional terms of orders $\Delta t\,\Delta W$ and $\Delta t^2$. The leading defect fluctuates in sign and has zero mean, but is not zero along each path. This is a useful local test of the Itô drift before any normalization hides it. Enforcing unit norm alone cannot establish that the drift, measurement statistics, or ensemble generator is correct.

Under appropriate regularity, Euler–Maruyama has strong order $1/2$ and weak order $1$. Strong error compares individual paths driven by the same Wiener realization. Couple refinements by setting each coarse increment equal to the sum of two fine increments. Comparing independent paths measures physical stochastic variation instead of discretization error.

Weak error concerns ensemble averages. Use enough trajectories to distinguish time-step bias from Monte Carlo error. Higher-order stochastic integrators require stochastic correction terms, not merely additional deterministic Runge–Kutta stages.

Monitored qubit benchmark

Take $H=0$, $A=Z=|0\rangle\langle0|-|1\rangle\langle1|$, and $\psi(0)=|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$. Then

$$\rho_{01}(t)=\frac12e^{-2\kappa t}, \qquad \mathbb E[\langle Z\rangle]=0.$$

Individual ideal trajectories remain pure and tend toward $Z$ eigenstates, while the ensemble becomes mixed. For $z=\langle Z\rangle$,

$$dz=2\sqrt\kappa(1-z^2)\,dW.$$

This tests the factor of two in dephasing, symmetric outcome probabilities, and trajectory purity. Values $|z|>1$ reveal numerical breakdown; clipping can mask an overly large step.

For $|z|<1$, Itô’s rule applied to $s=\operatorname{atanh}z$ gives

$$ds=2\sqrt\kappa\,dW+4\kappa z\,dt =2\sqrt\kappa\,dY, \qquad dY=2\sqrt\kappa z\,dt+dW.$$

Thus the integrated physical record gives $z(t)=\tanh[\operatorname{atanh}z(0)+2\sqrt\kappa\,Y(t)]$. Replacing $Y$ by an unshifted Wiener process would omit the drift in $s$ and change the measurement law. Eigenstates $z=\pm1$ are absorbing limits; do not evaluate their infinite inverse hyperbolic tangent numerically. The finite-bin measurement update supplies a positive update and the correct record distribution for this commuting benchmark.

Monitor norm defects before repair, ensemble trace and positivity, convergence of $\mathbb E[z^2]$, and agreement with a deterministic master equation. Check empirical noise variance and independence as well.

An inefficient detector generally leaves a mixed conditional state and requires a stochastic master equation. Additional unobserved unravelings can provide a pure-state sampling representation, but an individual such state is not the observer’s full conditional state.

Exercise

Derive the short-time slope of $\mathbb E[z^2]$ at $z(0)=0$.

Solution

Itô’s rule gives $d(z^2)=2z\,dz+(dz)^2$, so $d\mathbb E[z^2]/dt=4\kappa\mathbb E[(1-z^2)^2]$. Initially the slope is $4\kappa$. Omitting the quadratic term predicts zero information gain, directly exposing incorrect stochastic calculus.

References

• Jacobs, K., and Steck, D. A., “A straightforward introduction to continuous quantum measurement,” Contemporary Physics 47, 279–303 (2006), arXiv:quant-ph/0611067.
• Kloeden, P. E., and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer (1992), doi:10.1007/978-3-662-12616-5.