The Schrödinger problem on the non-commutative Fisher-Rao space

Abstract

We present a self-contained and comprehensive study of the Fisher-Rao space of matrix-valued non-commutative probability measures, and of the related Hellinger space. Our non-commutative Fisher-Rao space is a natural generalization of the classical commutative Fisher-Rao space of probability measures and of the Bures-Wasserstein space of Hermitian positive-definite matrices. We introduce and justify a canonical entropy on the non-commutative Fisher-Rao space, which differs from the von Neumann entropy. We consequently derive the analogues of the heat flow, of the Fisher information, and of the dynamical Schrödinger problem. We show the $\Gamma$-convergence of the $\varepsilon$-Schrödinger problem towards the geodesic problem for the Fisher-Rao space, and, as a byproduct, the strict geodesic convexity of the entropy.

Appendix A: Some technical lemmas and postponed proofs

Lemma A.1

(Constant-speed reparametrization) Let $(G_t,U_t)$ be a curve connecting $G_0,G_1$ with finite energy

$$\begin{aligned} E[G,U]=\int _0^1\int _\Omega \mathrm {d}G_t U_t:U_t\mathrm {d}t<+\infty . \end{aligned}$$

Then there exists a curve $({\check{G}}_t,{\check{U}}_t)_{t\in [0,1]}$ connecting ${\check{G}}_0=G_0$ to ${\check{G}}_1=G_1$ with

$$\begin{aligned} \Vert {\check{U}}_t\Vert _{L^2(\mathrm {d}{\check{G}}_t)}\equiv cst ~\text {and}~ E[{\check{G}},{\check{U}}] = \left( \int _0^1\Vert U_t\Vert _{L^2(\mathrm {d}G_t)}\mathrm {d}t\right) ^2 \le E[G,U] \end{aligned}$$

with strict inequality unless $\Vert U_t\Vert _{L^2(\mathrm {d}G_t)}$ is constant in time.

Proof

The argument is fairly standard (see e.g. [47, lemma 5.3] or [5, lemma 1.1.4]) hence we only sketch the idea. Consider the change of time variable

$$\begin{aligned} {\mathsf {s}}(t):=\int _0^t \Vert U_\tau \Vert _{L^2(\mathrm {d}G_\tau )}\mathrm {d}\tau , \text {and}~{\mathsf {t}}(s):= \min \{t\in [0,1]:\quad \mathsf s(t)=s\} \end{aligned}$$

(${\mathsf {t}}$ is the left inverse of ${\mathsf {s}}$). Setting

$$\begin{aligned} {\widetilde{G}}_s:=G_{{\mathsf {t}}(s)}, \qquad \widetilde{U}_s:=\frac{U_{{\mathsf {t}}(s)}}{\Vert U_{{\mathsf {t}}(s)}\Vert _{L^2(\widetilde{G}_s)}} \end{aligned}$$

gives an admissible path connecting ${\widetilde{G}}_0=G_0$ to ${\widetilde{G}}_L=G_1$ in time $s\in [0,L]$ with

$$\begin{aligned} L={\mathsf {s}}(1)=\int _0^1 \Vert U_\tau \Vert _{L^2(\mathrm {d}G_\tau )}\mathrm {d}\tau \end{aligned}$$

and unit speed $\Vert {\widetilde{U}}_s\Vert _{L^2(\mathrm {d}{\widetilde{G}}_s)}\equiv 1$. This is clear at least formally from the chain-rule

$$\begin{aligned}&\partial _s{\widetilde{G}}_s = \frac{\mathrm {d}{\mathsf {t}}}{\mathrm {d}s}(s) \partial _tG_t({\mathsf {t}}(s)) \\&\quad = \frac{1}{\frac{\mathrm {d}{\mathsf {s}}}{\mathrm {d}t}({\mathsf {t}}(s))}\left( G_{{\mathsf {t}}(s)} U_{\mathsf t(s)}\right) ^{Sym}\\&\quad = \frac{1}{ \Vert U_{\mathsf t(s)}\Vert _{L^2(\mathrm {d}G_{{\mathsf {t}}(s)})}}\left( G_{{\mathsf {t}}(s)} U_{{\mathsf {t}}(s)}\right) ^{Sym} \\&\quad =(\widetilde{G}_s {\widetilde{U}}_s)^{Sym} \end{aligned}$$

and can be made rigorous since the denominator $\Vert U_{\mathsf t(s)}\Vert _{L^2(\mathrm {d}G_{{\mathsf {t}}(s)})}$ only vanishes at the discontinuity points of ${\mathsf {t}}(s)$, which are countable (being ${\mathsf {t}}$ monotone nondecreasing) and therefore $\mathrm {d}s$-negligible. Scaling $t=Ls$

$$\begin{aligned} ({\check{G}}_t,\check{U}_t):=({\widetilde{G}}_{tL},L{\widetilde{U}}_{tL}) \qquad \text{ for } t\in [0,1] \end{aligned}$$

back to the unit interval and noticing that $\Vert \check{U}_t\Vert _{L^2(\mathrm {d}{\check{G}}_t)}\equiv L$ is constant, we get

$$\begin{aligned}&E[{\check{G}},{\check{U}}] = \int _0^1 \Vert \check{U}_t\Vert ^2_{L^2(\mathrm {d}{\check{G}}_t)} \,\mathrm {d}t = \int _0^1 L^2\mathrm {d}t=L^2 \\&\quad =\left( \int _0^1\Vert U_\tau \Vert _{L^2(\mathrm {d}G_\tau )}\mathrm {d}\tau \right) ^2 \le \int _0^1\Vert U_\tau \Vert _{L^2(\mathrm {d}G_\tau )}^2\mathrm {d}\tau = E[G,U] \end{aligned}$$

as desired. (The inequality is strict unless $(G,U)$ has constant speed as in our statement.) $\square$

Lemma A.2

(Refined Banach-Alaoglu [47]) Let $(X,\Vert \cdot \Vert )$ be a separable normed vector space. Assume that there exists a sequence of seminorms $\{\Vert \cdot \Vert _k\}$ ($k=0,1,2,\dots$) on X such that for every $x\in X$ one has

$$\begin{aligned} \Vert x\Vert _k\le C \Vert x\Vert \end{aligned}$$

with a constant C independent of k, x, and

$$\begin{aligned} \Vert x\Vert _k\underset{k\rightarrow \infty }{\rightarrow } \Vert x\Vert _0. \end{aligned}$$

Let $\varphi _k$ ($k=1,2,\dots$) be a uniformly bounded sequence of linear continuous functionals on $(X,\Vert \cdot \Vert _k)$, resp., in the sense that

$$\begin{aligned} c_k:=\Vert \varphi _k\Vert _{(X,\Vert \cdot \Vert _k)^*}\le C. \end{aligned}$$

Then the sequence $\{\varphi _k\}$ admits a converging subsequence $\varphi _{k_n}\rightarrow \varphi _0$ in the weak-$*$ topology of $X^*$, and

$$\begin{aligned} \Vert \varphi _0\Vert _{(X,\Vert \cdot \Vert _0)^*}\le c_0:=\liminf \limits _k c_{k}. \end{aligned}$$

(A.1)

Lemma A.3

(Refined Arzelà-Ascoli [5, 15]) Let $(X,\varrho )$ be a metric space. Assume that there exists a Hausdorff topology $\sigma$ on X such that $\varrho$ is sequentially lower semicontinuous with respect to $\sigma$. Let $(x^k)_t$, $t\in [0,1]$, be a sequence of curves lying in a common $\sigma$-sequentially compact set $K\subset X$. Let it be equicontinuous in the sense that there exists a symmetric continuous function $\omega :[0,1]\times [0,1]\rightarrow \mathbb {R}_+$, $\omega (t,t)=0$, such that

$$\begin{aligned} \varrho ((x^k)_t,(x^k)_{\bar{t}})\le \omega (t,{\bar{t}}). \end{aligned}$$

(A.2)

for all $t,{\bar{t}}\in [0,1]$. Then there exists a $\varrho$-continuous curve $x_t$ such that

$$\begin{aligned} \varrho (x_t,x_{{\bar{t}}})\le \omega (t,{\bar{t}}), \end{aligned}$$

(A.3)

and (up to a not relabelled subsequence)

$$\begin{aligned} (x^k)_t\rightarrow x_t \end{aligned}$$

(A.4)

for all $t\in [0,1]$ in the topology $\sigma$.

Proof of Proposition 3.4

First of all, let us show that the right-hand side is always finite and that the minimum is always attained. For any fixed $A_1\in {\mathcal {H}^+}$ it is easy to check that $(A_t,U_t):=(t^2A_1,\frac{2}{t} I)$ gives an admissible path connecting $A_0=0$ to $A_1$ with finite energy. In particular any two matrices $A_0,A_1\in {\mathcal {H}^+}$ can be connected through zero as $A_0\leadsto 0\leadsto A_1$ with finite cost, thus the problem is proper. For fixed $A_0,A_1$ consider now a minimizing sequence $(A^n_t,U^n_t)_{t\in [0,1]}$. Note that our Lemma 4.2 applies in particular when $\Omega =\{x\}$ is a one-point space, which gives here equicontinuity and pointwise relative compactness in the form

$$\begin{aligned} |A^n_t-A^n_s|_2\le C |t-s|^\frac{1}{2}, \qquad m^n_t=\hbox {tr}A^n_t\le M, \qquad s,t\in [0,1] \end{aligned}$$

uniformly in n. By the classical Arzelá-Ascoli theorem we get, up to extraction of a subsequence if needed,

$$\begin{aligned} A^n\rightarrow A \qquad \text{ uniformly } \text{ in } C([0,1];{\mathcal {H}^+}). \end{aligned}$$

This immediately shows that the matrix-valued measure $\mu ^n:=A^n_t\mathrm {d}t\rightarrow A_t dt=:\mu$ at least weakly-$*$ on $\mathbb {H}^+(0,1)$. Because $\Vert U^n\Vert _{L^2(\mathrm {d}\mu ^n)}^2=\int _0^1 A^n_tU^n_t:U^n_t\,\mathrm {d}t\le C$, an easy application of our Banach-Alaoglu variant (lemma A.2) in varying $L^2(\mathrm {d}\mu ^n)$ spaces gives a limit $U\in L^2(\mathrm {d}\mu )$ with

$$\begin{aligned} \int _0^1 A_t U_t:U_t\mathrm {d}t \le \liminf \limits _{n\rightarrow \infty } \int _0^1 A^n_tU^n_t:U^n_t\,\mathrm {d}t \end{aligned}$$

and such that $\int _0^1 A^n U^n:V\,\mathrm {d}t\rightarrow \int _0^1 A U:V\,\mathrm {d}t$ for any reasonably smooth test function V. This shows that this limit $(A_t,U_t)_{t\in [0,1]}$ is an admissible curve joining $A_0,A_1$ with energy $E[A,U]\le \liminf E[A^n,U^n]$ and this pair is therefore a minimizer.

In order to identify now the left-hand side and the right-hand side of (3.5) we proceed in two steps.

Step 1: assume that $A_0,A_1\in \mathcal {H}^{++}(d)$ are positive definite, and let ${\check{A}}_{0,1}:={\mathfrak {r}}(A_{0,1})$ be the corresponding real extensions as defined in (3.8). From Proposition 3.6 there holds $d_B^2(A_0,A_1)=\frac{1}{2} W^2_2({\mathcal {N}}(\check{A}_0),{\mathcal {N}}({\check{A}}_1))$. In this real setting it is known [68, Prop. A] that

$$\begin{aligned} W^2_2({\mathcal {N}}({\check{A}}_0),{\mathcal {N}}({\check{A}}_1)) = \frac{1}{4} \min \limits _{{\check{A}}\in {\mathcal {A}}({\check{A}}_0,{\check{A}}_1)} \int _0^1 {\check{A}}_t {\check{U}}_t:{\check{U}}_t\,\mathrm {d}t, \end{aligned}$$

where the infimum runs of course over real pairs $\frac{d\check{A}_t}{dt}=({\check{A}}_t {\check{U}}_t)^{Sym}$ with ${\check{U}}_t\in \mathcal S(2d)$. Complexifying back $({\check{A}},{\check{U}})\leadsto (A,U)$ gives the result.

Step 2: if now either $A_0$ or $A_1$ are only semi-definite we approximate $A^n_0:=A_0+\frac{1}{n} I\rightarrow A_0$ and $A^n_1:=A_1+\frac{1}{n} I\rightarrow A_1$. Clearly $A^n_{0,1}\in \mathcal {H}^{++}$ are positive-definite, hence step 1 applies.

Note from (3.4) that the left-hand side of (3.5) is of course continuous in $A_0,A_1$, hence it suffices to show that the optimal value in the right-hand side is continuous for this particular choice of $A^n_{0,1}\rightarrow A_{0,1}$. Observe that

$$\begin{aligned} {\mathcal {J}}^*(A_0,A_1):=\frac{1}{4} \min _{{\mathcal {A}}(A_0,A_1)}\int _0^1 A_t U_t :U_t \mathrm {d}t \end{aligned}$$

is a well-defined function of $A_0,A_1\in {\mathcal {H}^+}$, and we proved earlier that there always exists a minimizer. Arguing precisely as for the existence of the minimizers, cf. also Lemma , it is easy to prove that ${\mathcal {J}}^*$ is lower semi-continuous in both arguments. Indeed, up to a subsequence, any sequence $(A^n_t,U^n_t)_{t\in [0,1]}$ of minimizers in ${\mathcal {J}}^*(A_0^n,A_1^n)$ converges to an admissible candidate $(A_t,U_t)_{t\in [0,1]}$ connecting $A_0,A_1$, hence ${\mathcal {J}}^*(A_0,A_1)\le E[A,U]\le \liminf E[A^n,U^n]=\liminf {\mathcal {J}}^*(A^n_0,A^n_1)$ (regardless of the particular form of $A_0^n,A_1^n$).

In order to establish the upper continuity, let $A_0=R D_0 R^*$ be a spectral decomposition of $A_0$ and note that obviously $A^n_0=A_0+\frac{1}{n} I=R(D_0+\frac{1}{n} I)R^*$. Since $A^n_0$ and $A_0$ commute it is easy to check that

$$\begin{aligned} A^n_{0t}:=R\left[ (1-t)\sqrt{D_0}+t\sqrt{D_0+\frac{1}{n} I}\right] ^2R^*, \qquad U^n_{0t}:=2 R\left( \sqrt{D_0+\frac{1}{n} I}-\sqrt{D_0}\right) R^*\sqrt{A_{0t}^{-1}} \end{aligned}$$

defines an admissible path $(A^n_{0t})_{t\in [0,1]}$ between $A_0$ and $A^n_0$. Moreover, a straightforward computation shows that the corresponding energy is

$$\begin{aligned} \frac{1}{4} \int _0^1A^n_{0t} U^n_{0t}:U^n_{0t}\,\mathrm {d}t = \left| \sqrt{D_0}-\sqrt{D_0+I/n}\right| ^2_2 \xrightarrow [n\rightarrow \infty ]{} 0. \end{aligned}$$

(A.5)

(Actually this path is exactly the Bures geodesic between $A_0,A_0^n$.) A similar construction yields a path $(A^n_{1t})_{t\in [0,1]}$ connecting $A_1^n$ to $A_1$ with cost

$$\begin{aligned} \frac{1}{4} \int _0^1A^n_{1t} U^n_{1t}:U^n_{1t}\,\mathrm {d}t = \left| \sqrt{D_1}-\sqrt{D_1+I/n}\right| ^2_2 \xrightarrow [n\rightarrow \infty ]{} 0, \end{aligned}$$

(A.6)

where $D_1$ is the spectral decomposition of $A_1=QD_1Q^*$. Now pick a minimizer $({\tilde{A}}_t,{\tilde{U}}_t)_{t\in [0,1]}$ in the definition of ${\mathcal {J}}^*(A_0,A_1)$ and fix a small $\theta _n\in (0,1)$ to be determined shortly. Rescaling in time and concatenating the paths $A_0^n\leadsto A_0\leadsto A_1\leadsto A_1^n$ in the intervals $t\in [0,\theta _n]$, $t\in [\theta _n,1-\theta _n]$, and $t\in [1-\theta _n,1]$, respectively, we obtain an admissible path $({\hat{A}}^n,{\hat{U}}^n)$ from $A_0^n$ to $A_1^n$ whose energy is bounded as

$$\begin{aligned}&4 {\mathcal {J}}^*(A_0^n,A_1^n)\le \int _0^1{\hat{A}}_t^n{\hat{U}}^n_t:{\hat{U}}^n_t\,\mathrm {d}t \\&\quad = \frac{1}{\theta _n} \int _0^1A^n_{0t} U^n_{0t}:U^n_{0t}\,\mathrm {d}t +\frac{1}{1-2\theta _n} \int _0^1 {\tilde{A}}_t{\tilde{U}}_t:\tilde{U}_t\,\mathrm {d}t +\frac{1}{\theta _n} \int _0^1A^n_{1t} U^n_{1t}:U^n_{1t}\,\mathrm {d}t. \end{aligned}$$

By (A.5), (A.6) we see that the first and third integrals in the right-hand side tend to zero, while the second integral is exactly $4 \mathcal J^*(A_0,A_1)$ by definition of $({\tilde{A}},{\tilde{U}})$. Choosing $\theta _n\rightarrow 0$ sufficiently slowly and taking $\limsup$ in the previous inequality gives

$$\begin{aligned} \limsup _{n \rightarrow \infty } 4{\mathcal {J}}^*(A_0^n,A_1^n)\le 0+\lim _{n\rightarrow \infty }\frac{1}{1-2\theta _n}\int _0^1 {\tilde{A}}_t{\tilde{U}}_t:{\tilde{U}}_t\,\mathrm {d}t + 0 = 4\mathcal J^*(A_0,A_1) \end{aligned}$$

and the proof is complete. $\square$

Proof of Proposition 3.8

Let $\rho _0={\mathcal {N}}(A_0)$ and $\rho _1={\mathcal {N}}(A_1)$. By [51, Thm. 3.3 and Thm. 3.4], if one could write the (f, g) transform

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho _0=f_0g_0\\ \rho _1=f_1g_1 \end{array}\right. } \end{aligned}$$

(A.7)

for $(f_t)_{t\in [0,1]}$ a forward solution of the heat equation with initial datum $f_0$ and $(g_t)_{t\in [0,1]}$ a backward solution with terminal datum $g_1$, then the solution of (3.16) would be given by

$$\begin{aligned} \rho _t=f_tg_t. \end{aligned}$$

Since the product of Gaussian distributions is Gaussian, it is legitimate to try and solve for $f_0={\mathcal {N}}(B_0)$ and $g_1={\mathcal {N}}(C_1)$ as Gaussians. Since the heat flow for Gaussians is explicitly given by (3.14) we see that the corresponding forward and backward solutions read

$$\begin{aligned} {\left\{ \begin{array}{ll} f_t={\mathcal {N}}(B_t) &{} \qquad \text{ with } B_t=B_0+2t I \\ g_t={\mathcal {N}}(C_t) &{} \qquad \text{ with } C_t=C_1+2(1-t) I \end{array}\right. } \end{aligned}$$

Exploiting the algebraic product rule ${\mathcal {N}}(B)\times {\mathcal {N}}(C)=\mathcal N\left( [B^{-1}+C^{-1}]^{-1}\right)$, we see that, given $A_0,A_1$ the Schrödinger system (A.7) is equivalent to solving

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {N}}(A_0)={\mathcal {N}}(B_0){\mathcal {N}}(C_0)\\ {\mathcal {N}}(A_1)={\mathcal {N}}(B_1){\mathcal {N}}(C_1) \end{array}\right. } \qquad \Leftrightarrow \qquad {\left\{ \begin{array}{ll} A_0^{-1}= B_0^{-1}+[C_1+2I]^{-1} \\ A_1^{-1} = [B_0+2I]^{-1}+ C_1^{-1} \end{array}\right. } \end{aligned}$$

in $B_0,C_1\in \mathcal S^{++}$. It is then a simple exercise to check that this system has a unique solution, and in particular the covariance $A_t$ of $\rho _t=f_tg_t$ is fully determined by

$$\begin{aligned} A_t^{-1}=[B_0+2tI]^{-1} +[C_1+2(1-t)I]^{-1}. \end{aligned}$$

$\square$