The law of large numbers for quantum stochastic filtering and control of many-particle systems

Abstract

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles in a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Schrödinger equations or Hartree equations, or Gross–Pitaevskii equations. In this paper, we extend some of these convergence results to a stochastic framework. Specifically, we work with the Belavkin stochastic filtering of many-particle quantum systems. The resulting limiting equation is an equation of a new type, which can be regarded as a complex-valued infinite-dimensional nonlinear diffusion of McKean–Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.

Appendix A. A technical estimate

Lemma 1.

Let $\gamma$ be a one-dimensional projector in a Hilbert space, $\Gamma$ be a density matrix (positive operator with unit trace ) and $L$ be a bounded operator in this Hilbert space. Then

$$|{-}4\operatorname{tr}(L\gamma L\Gamma) +2\operatorname{tr}(\Gamma(L\gamma+\gamma L)) \operatorname{tr}(\Gamma L+\gamma L) -4\operatorname{tr}(\Gamma\gamma)\operatorname{tr}(\Gamma L) \operatorname{tr}(\gamma L)|\leqslant 20\|L\|^2\operatorname{tr}((1-\gamma)\Gamma) $$

(43)

for a self-adjoint $L$, and

$$\begin{aligned} \, &|{-}\operatorname{tr}(\gamma L\Gamma L^*+\gamma L^*\Gamma L +\gamma L^*\Gamma L^*+\gamma L\Gamma L)+{} \nonumber \\ &\qquad{}+\operatorname{tr}(\gamma\Gamma L^*+\gamma L\Gamma) \operatorname{tr}(\gamma(L^*+L)) +\operatorname{tr}(\gamma\Gamma L+\gamma L^*\Gamma) \operatorname{tr}(\Gamma_N(L^*+L))-{} \nonumber \\ &\qquad{}-\operatorname{tr}(\Gamma \gamma) \operatorname{tr}(\Gamma(L^*+L))\operatorname{tr}(\gamma(L^*+L))| \leqslant 28\|L\|^2\operatorname{tr}((1-\gamma)\Gamma) \end{aligned}$$

(44)

for a general $L$.

Proof.

By the approximation argument, it suffices to prove the Lemma for a finite-dimensional Hilbert space $\mathbb C^n$.

Let $\alpha=\operatorname{tr}((1-\gamma)\Gamma)$. We choose an orthonormal basis such that $\gamma$ is the projection on the first basis vector. By the positivity of $\Gamma$, it follows that

$$|\Gamma_{jk}|\leqslant\alpha\quad\text{for}\quad j,k\ne 1,\qquad \text{and}\qquad \max(|\Gamma_{j1}|,|\Gamma_{1j}|) \leqslant\sqrt{\alpha}\quad \text{for}\quad j\ne 1. $$

(45)

Let $L$ be a self-adjoint matrix. Then the expression under the modulus sign in the left-hand side of (43) becomes

$$\begin{aligned} \, &{-}4(L\Gamma L)_{11}+2[(L\Gamma)_{11} +(\Gamma L)_{11}](\operatorname{tr}(\Gamma L)+L_{11}) -4\Gamma_{11}L_{11}\operatorname{tr}(\Gamma L)= \\ &\qquad=-4\sum_{j,k}L_{1j}\Gamma_{jk}L_{k1}+{} \\ &\qquad\hphantom{={}}+2\biggl[2L_{11}\Gamma_{11}+\sum_{j\ne 1} (\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1})\biggr] (\operatorname{tr}\,(\Gamma L)+L_{11}) -4\Gamma_{11}L_{11}\operatorname{tr}(\Gamma L)= \\ &\qquad=-4L_{11}\sum_{j\ne 1}(L_{1j}\Gamma_{j1} +L_{j1}\Gamma_{1j})-4\sum_{j\ne 1,\,k\ne 1}L_{1j}\Gamma_{jk}L_{k1}+{} \\ &\qquad\hphantom{={}}+2\sum_{j\ne 1}(\Gamma_{1j}L_{j1} +L_{1j}\Gamma_{j1})(\operatorname{tr}(\Gamma L)+L_{11})= \\ &\qquad=2\sum_{j\ne 1}(\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1})L_{11}(\Gamma_{11}-1) -4\sum_{j\ne 1,\,k\ne 1}L_{1j}\Gamma_{jk}L_{k1}+{} \\ &\qquad\hphantom{={}}+2\sum_{j\ne 1}(\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1}) (\operatorname{tr}\,(\Gamma L)-L_{11}\Gamma_{11})= \\ &\qquad=-2\sum_{j\ne 1}(\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1})L_{11}\alpha -4\sum_{j\ne 1,\,k\ne 1}L_{1j}\Gamma_{jk}L_{k1}+{} \\ &\qquad\hphantom{={}}+2\biggl(\,\sum_{j\ne 1} (\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1})\biggr)^2 +2\sum_{j\ne 1} (\Gamma_{1j}L_{j1}+L_{1j}\Gamma_{j1}) \sum_{j\ne 1,\,k\ne 1}L_{kj}\Gamma_{jk}. \end{aligned}$$

Here, all terms are of the order $\alpha$ because of (45).

More precisely,

$$\biggl|\,\sum_{j\ne 1,\,k\ne 1}L_{kj}\Gamma_{jk}\biggr| =|\operatorname{tr}[(1-\gamma)L(1-\gamma)\Gamma(1-\gamma)]| \leqslant\|L\|\operatorname{tr}[(1-\gamma)\Gamma]\leqslant\|L\|\alpha.$$

Moreover,

$$\sum_{j\ne 1}|\Gamma_{j1}|^2 =\sum_{j\neq 1}|\Gamma_{1j}|^2 \leqslant\Gamma_{11}\sum_{j\ne 1}\Gamma_{jj} \leqslant\Gamma_{11}\alpha\leqslant\alpha,$$

and therefore

$$\begin{aligned} \, \biggl|\,\sum_{j\ne 1}(\Gamma_{1j}L_{j1})\biggr|^2 &\leqslant\sum_{j\ne 1}|\Gamma_{1j}|^2\sum_{j\ne 1}|L_{j1}|^2 \leqslant\|L\|^2\alpha, \\ \biggl|\,\sum_{j\ne 1}(\Gamma_{j1}L_{1j})\biggr|^2 &\leqslant\|L^T\|^2\alpha=\|L\|^2\alpha, \end{aligned}$$

whence

$$\biggl|\,\sum_{j\ne 1}(\Gamma_{1j}L_{j1} \pm L_{1j}\Gamma_{j1})\biggr| \leqslant 2\|L\|\sqrt{\alpha}. $$

(46)

Finally,

$$\biggl|\,\sum_{j\ne 1,\,k\ne 1}L_{1j}\Gamma_{jk}L_{k1}\biggr|^2 \leqslant\sum_{j,k} |L_{1j}|^2 |L_{k1}|^2 \sum_{j\ne 1,k\ne 1}|\Gamma_{jk}|^2 \leqslant\|L\|^4\biggl(\,\sum_{j\ne 1}|\Gamma_{jj}|\biggr)^{\!2} \leqslant\|L\|^4\alpha^2,$$

where the estimate

$$|\Gamma_{jk}|^2 \leqslant \Gamma_{jj}\Gamma_{kk}$$

for all $j$ and $k$ was used (arising from the positivity of $\Gamma$).

Putting the estimates together, we obtain (43).

For a general $L$, we can write $L=L^s+L^a$, where $L^s=(L+L^*)/2$ is self-adjoint and $L^a=(L-L^*)/2$ is anti-Hermitian. Substituting this in the left-hand side of (44) leads to several cancelations, such that the expression under the modulus sign in the left-hand side becomes

$$\begin{aligned} \, &{-}4\operatorname{tr}(L^s\gamma L^s\Gamma) +2\operatorname{tr}(\Gamma(L^s\gamma+\gamma L^s)) \operatorname{tr}(\Gamma L^s+\gamma L^s) -4\operatorname{tr}(\Gamma\gamma)\operatorname{tr} (\Gamma L^s)\operatorname{tr}(\gamma L^s)+{} \\ &\qquad{}+2\operatorname{tr}(\gamma[\Gamma,L^a]) (\operatorname{tr}(\Gamma L_s)-\operatorname{tr}(\gamma L^s)). \end{aligned}$$

Everything apart from the last term is already estimated by (43).

By (46) (which is valid for arbitrary $L$),

$$|\operatorname{tr}(\gamma[\Gamma,L^a])| =|[\Gamma,L^a]_{11}| =\biggl|\,\sum_{j\ne 1}(\Gamma_{1j}L^a_{j1}-L^a_{1j}\Gamma_{j1})\biggr| \leqslant 2\|L\|\sqrt{\alpha}$$

and

$$\begin{aligned} \, &|\operatorname{tr}(\Gamma L_s)-\operatorname{tr}(\gamma L^s)| =\biggl|(\Gamma_{11}-1)L^s_{11} +\sum_{j\ne 1}(\Gamma_{1j}L^s_{j1}-L^s_{1j}\Gamma_{j1}) +\sum_{j,\,k\ne 1}(\Gamma_{jk}L^s_{kj})\biggr| \leqslant 4\|L\|\sqrt{\alpha}, \end{aligned}$$

which implies (44). $\blacksquare$

Remark 6.

Our proof of Lemma 1 is based on some remarkable cancelation of terms in concrete calculations via coordinate representations. We do not see any intuitive reasons for its validity. Nor is it clear whether this can be extended to arbitrary density matrices $\gamma$, not just one-dimensional projectors.

Appendix B. McKean–Vlasov diffusions in Hilbert spaces

The McKean–Vlasov nonlinear diffusions are well-studied processes, due to a large variety of applications. However, there seem to be only very few publications pertaining to the infinite-dimensional case (see [32] and the references therein). Here, for completeness, we therefore provide some basic results for a class of McKean–Vlasov diffusions in Hilbert spaces, stressing explicit bounds and errors. Namely, we are interested in the Cauchy problems

$$dX_t=AX_t\,dt+b_t(X_t,\mathbf EX_t^{\otimes K})\,dt +(\sigma(X_t),dB_t),\qquad X_0=Y, $$

(47)

in a complex Hilbert space $\mathcal H$ equipped with the scalar product ${(\,\cdot\,,\,\cdot\,)}$ and the corresponding norm ${\|\,\cdot\,\|}$, and (as an auxiliary tool) in more standard equations

$$dX_t=AX_t\,dt+b_t(X_t,\xi_t)\,dt+(\sigma(X_t),dB_t),\qquad X_0=Y. $$

(48)

Here, $K=1$ or $K=2$ (which are the most important cases for applications), $B_t$ is the standard $n$-dimensional Wiener process defined on some complete probability space $(\Omega,\mathcal F,\mathbf P)$, $\sigma(Y)=(\sigma_1(Y),\dots,\sigma_n(Y))$ with each $\sigma_j$ a continuous map $\mathcal H\to\mathcal H$, $b$ a continuous map $\mathbb R\times\mathcal H\times\mathcal H^{\otimes K}\to\mathcal H$, $A$ a generator of a strongly continuous operator semigroup $e^{At}$ of contractions in $\mathcal H$, and $\xi_t$ a given (deterministic) continuous curve in $\mathcal H^{\otimes K}$. With some abuse of notation we let ${(\,\cdot\,,\,\cdot\,)}$ and ${\|\,\cdot\,\|}$ also denote the scalar product and the norm in the tensor product Hilbert space $\mathcal H^{\otimes K}$. In (47), $\mathbf E$ denotes the expectation with respect to the Wiener process. A solution process of (47) is called a nonlinear diffusion of the McKean–Vlasov type, because of the dependence of the coefficient on this expectation.

To obtain effective bounds for growth and continuous dependence on the parameters of Eqs. (48), we follow the strategy developed systematically in [33] for deterministic Banach space-valued equations. Namely, we use the generalized fixed-point principle of Weissinger type in the following form (see, e.g., Propositions 9.1 and 9.3 in [33] for simple proofs). If $\Phi$ is a map from a complete metric space $(M,\rho)$ to itself such that

$$\rho(\Phi^k(x),\Phi^k(y))\leqslant\alpha_k\rho(x,y),$$

with $a=1+\sum_j\alpha_j<\infty$, then $\Phi$ has a unique fixed point $x^*$, and $\rho(x,x^*)\leqslant a\rho(x,\Phi(x))$ for any $x$. Moreover, for any two maps $\Phi_1$ and $\Phi_2$ satisfying these conditions and such that $\rho(\Phi_1(x),\Phi_2(x))\leqslant\epsilon$ for all $x$, the corresponding fixed points allow the estimate

$$\rho(x_1^*,x_2^*)\leqslant\epsilon a.$$

We start with (48) and work with the so-called mild form of this equation:

$$X_t=Y+\int_0^te^{A(t-s)}[b_s(X_s,\xi_s)\,ds+(\sigma(X_s),dB_s)]. $$

(49)

For a Hilbert space $\mathcal B$ that is either $\mathcal H$ or $\mathcal H^{\otimes 2}$, let $C_{\mathrm{ad}}([0,T],\mathcal B)$ denote the Banach space of adapted continuous $\mathcal B$-valued processes, equipped with the norm

$$\|X_.\|_{\mathrm{ad},T}=\sup_{t\in[0,T]}\sqrt{\mathbf E\|X_t\|^2}. $$

(50)

We let $C([0,T],\mathcal B)$ denote its subspace of deterministic curves. For elements $\xi_.$ of this subspace, the norm becomes the standard sup-norm

$$\|\xi_.\|_{\mathrm{ad},T}=\sup_{t\in[0,T]}\|\xi_t\|.$$

We let $C_{Y,\mathrm{ad}}([0,T],\mathcal B)$ and $C_Y([0,T],\mathcal B)$ denote the subsets of these spaces consisting of curves with ${X(0)=Y}$.

In our estimates, we encounter the so-called Le Roy function of index $1/2$

$$R(z)=\sum_{k=0}^\infty\frac{z^k}{\sqrt{k!}}, $$

(51)

which plays the same role for stochastic equations as the exponential and the Mittag-Leffler function play for deterministic equations.

Proposition 1.

Let

$$ \begin{aligned} \, &\|b_t(Z_1,\xi_1)-b_t(Z_2,\xi_2)\| \leqslant\varkappa_1\|Z_1-Z_2\|+\varkappa_2\|\xi_1-\xi_2\|, \\ &\|\sigma(Z_1)-\sigma(Z_2)\| \leqslant\varkappa_3\|Z_1-Z_2\|. \end{aligned}$$

(52)

Then for any $T>0$, $Y\in\mathcal H$, and $\xi=\xi_.\in C([0,T],\mathcal H^{\otimes K})$, Eq. (49) has the a unique global solution $X_.\in C_{\mathrm{ad}}([0,T],\mathcal H)$, and it satisfies the estimate

$$\|X_.-Y\|^2_{\mathrm{ad},T} \leqslant 2tM^2(t)\biggl[\,\int_0^t|b_s(Y,\xi_s)|^2\,ds+\sigma^2(Y)\biggr], $$

(53)

where

$$M(t)=R\Bigl(\sqrt{2(\varkappa_3^2+\varkappa_1^2)}\max(\sqrt t,t)\Bigr). $$

(54)

Moreover, for two initial conditions $Y_1$ and $Y_2$ and two curves $\xi^1$ and $\xi^2$, the corresponding solutions satisfy the estimate

$$\|X^1_.-X^2_.\|^2_{\mathrm{ad},T} \leqslant 2M^2(t)(\|Y^1-Y^2\|^2 +t^2\varkappa_2^2\|\xi^1_.-\xi^2_.\|^2_{\mathrm{ad},T}). $$

(55)

Proof.

A solution of (49) is a fixed point of the map

$$[\Phi_{Y,\xi}(X_.)](t) =Y+\int_0^te^{A(t-s)}[b_s(X_s,\xi_s)\,ds+(\sigma(X_s),dW_s)] $$

(56)

in $C_{Y,\mathrm{ad}}([0,T])(\mathcal H)$. For two curves $X^1_.,X^2_.\in C_{Y,\mathrm{ad}}([0,T])(\mathcal H^{\otimes K})$, by the contraction property of the semigroup $e^{At}$ and Ito’s isometry, we have

$$\begin{aligned} \, &\mathbf E\|[\Phi_{Y,\xi}(X^1_.)](t)-[\Phi_{Y,\xi}(X^2_.)](t)\|^2\leqslant \\ &\leqslant 2\mathbf E\biggl\|\int_0^te^{A(t-s)}(b_s(X^1_s,\xi_s) -b_s(X^2_s,\xi_s))\,ds\biggr\|^2+{} \\ &\hphantom{-{}}+2\mathbf E \biggl\|\int_0^te^{A(t-s)}(\sigma(X^1_s) -\sigma(X^2_s))\,dW_s\biggr\|^2\leqslant \\ &\leqslant 2\mathbf E \biggl(\int_0^t\varkappa_1\|X^1_s-X_s^2\|\,ds\biggr)^2 +2\varkappa_3^2\mathbf E\int_0^t\|X^1_s-X^2_s\|^2\,ds. \end{aligned}$$

Applying the Cauchy–Schwarz inequality to the first integral gives

$$\biggl(\int_0^t\varkappa_1\|X^1_s-X_s^2\|\,ds\biggr)^2 \leqslant\varkappa_1^2t\int_0^t\|X^1_s-X_s^2\|^2\,ds,$$

whence

$$\mathbf E\|[\Phi_{Y,\xi}(X^1_.)](t)-\Phi_{Y,\xi}(X^2_.)](t)\|^2 \leqslant 2(\varkappa_3^2+\varkappa_1^2t)\mathbf E \int_0^t\|X^1_s-X^2_s\|^2\,ds.$$

This estimate is easy to iterate. Namely, for $t\leqslant 1$, the $k$th power of $\Phi$ satisfies the estimate

$$\mathbf E\|[\Phi^k_{Y,\xi}(X^1_.)](t)-[\Phi^k_{Y,\xi}(X^2_.)](t)\|^2 \leqslant 2^k(\varkappa_3^2+\varkappa_1^2)^k \frac{t^k}{k!}\|X^1_.-X^2_.\|^2_{\mathrm{ad},T}$$

and therefore,

$$\|[\Phi^k_{Y,\xi}(X^1_.)-\Phi^k_{Y,\xi}(X^2_.)]\|_{\mathrm{ad},T} \leqslant\frac{1}{\sqrt{k!}}[2(\varkappa_3^2+\varkappa_1^2)t]^{k/2} \|X^1_.-X^2_.\|_{\mathrm{ad},T}.$$

For $t\geqslant 1$, we have the estimate for the $k$th power of $\Phi$,

$$\mathbf E\|[\Phi^k_{Y,\xi}(X^1_.)](t) -[\Phi^k_{Y,\xi}(X^2_.)](t)\|^2 \leqslant 2^k(\varkappa_3^2 +\varkappa_1^2)^k \frac{t^{2k}}{(2k-1)!!}\|X^1_.-X^2_.\|^2_{\mathrm{ad},T},$$

whence

$$\|[\Phi^k_{Y,\xi}(X^1_.)-\Phi^k_{Y,\xi}(X^2_.)]\|_{\mathrm{ad},T} \leqslant\frac{1}{\sqrt{k!}}[2(\varkappa_3^2+\varkappa_1^2)t^2]^{k/2} \|X^1_.-X^2_.\|_{\mathrm{ad},T}.$$

Therefore, the conditions of the generalized fixed-point theorem above is satisfied for $a=M(t)$, implying all statements of the proposition. $\blacksquare$

A solution of the mild form

$$X_t=Y+\int_0^te^{A(t-s)}[b_s(X_s,\mathbf EX_s^{\otimes K})\,ds +(\sigma(X_s),dB_s)] $$

(57)

of Eq. (47) is a fixed point of the map $\Gamma\colon C_Y([0,T],\mathcal H^{\otimes K}) \to C_Y([0,T],\mathcal H^{\otimes K})$ that sends $\xi_.$ to $\mathbf E X_.^{\otimes K}$, where $X_.$ is the solution of (49). By (55), for two curves $\xi^1$ and $\xi_2$, we have estimates for the corresponding solutions $X_.^1$ and $X_.^2$:

$$\|\mathbf E(X_t^1)-\mathbf E(X_t^2)\| \leqslant 2M(t)t\varkappa_2\|\xi^1_.-\xi^2_.\|_{\mathrm{ad},t}$$

for $K=1$ and

$$\begin{aligned} \, \|\mathbf E(X_t^1)^{\otimes 2}-\mathbf E (X_t^2)^{\otimes 2}\| &\leqslant\mathbf E\|(X_t^1-X_t^2)\otimes (X^1_t)\| +\mathbf E\|(X^2_t)\otimes(X_t^1-X_t^2)\|\leqslant \\ &\leqslant 2M(t)t\varkappa_2\|\xi^1_.-\xi^2_.\|_{\mathrm{ad},t} \max(\|X^1_.\|_{\mathrm{ad},t},\|X^2_.\|_{\mathrm{ad},t}) \end{aligned}$$

for $K=2$.

Therefore, for small times $t\leqslant t_0$, the map $\Gamma$ is a contraction and hence has a unique fixed point. For ${K=1}$, the time $t_0$ is independent of $Y$ and we can therefore build a unique global solution by iterations. We have thus proved the following statement.

Proposition 2.

Let the assumptions of Proposition 1 hold. If $K=1$, Eq. (57) has a unique global solution for any initial $Y$, and

$$\|\mathbf E X_.-Y\|_{\mathrm{ad},T}\leqslant C(T),$$

with a constant $C(T)$ depending on $\varkappa_1$, $\varkappa_2$, $\varkappa_3$, and $\|Y\|$. If $K=2$, Eq. (57) has a unique local solution for times of the order of $\|Y\|^{-1}$.

We finally note a situation where solutions of mild equations also solve the SDEs.

Proposition 3.

Let $D$ be an invariant core for the semigroup $e^{At}$, which is itself a Banach space with respect to some norm $\|\,\cdot\,\|_D$. Let $b$ and $\sigma$ be continuous maps $\mathbb R\times\mathcal H\times\mathcal H^{\otimes K}\to D$ and $\mathcal H\to D$. Then for any $Y\in D$, the solutions of mild equations (57) and (49) solve the corresponding SDEs.

Proof.

This follows from the direct application of Ito’s rule. The differentiability required here is a consequence of the assumptions made. $\blacksquare$