Abstract
Does a semiclassical particle remember the phase space topology? We discuss this question in the context of the Berezin–Toeplitz quantization and quantum measurement theory by using tools of topological data analysis. One of its facets involves a calculus of Toeplitz operators with piecewiseconstant symbols developed in an appendix by Laurent Charles.
This is a preview of subscription content, access via your institution.
Notes
 1.
We thank Laurent Charles for noticing that assumption (\(\diamondsuit \)) is not needed when \(m < 1\).
 2.
Warning: Here and below we work with persistence modules parameterized by positive real numbers \({\mathbb {R}}_+\). The group \({\mathbb {R}}_+\) acts by multiplication on the set of parameters. The notion of interleaving and the stability theorem are adjusted accordingly.
 3.
The Liouville measure of the previous sections is \(\mu (A) = \nu (A) / \nu (M)\).
References
Avin, C., Lando, Y., Lotker, Z.: Radio cover time in hypergraphs. In: Proceedings of the 6th International Workshop on Foundations of Mobile Computing. ACM, September, pp. 3–12 (2010)
Barron, T., Ma, X., Marinescu, G., Pinsonnault, M.: Semiclassical properties of Berezin–Toeplitz operators with \(C^k\)symbol. J. Math. Phys. 55(4), 042108 (2014)
Barron, T., Polterovich, L.: Private exchange (2015)
Bordemann, M., Meinrenken, E., Schlichenmaier, M.: Toeplitz quantization of Kähler manifolds and \({\rm gl}(N)\), \(N\rightarrow \infty \) limits. Commun. Math. Phys. 165, 281–296 (1994)
Bouche, T.: Convergence de la métrique de FubiniStudy d’un fibré linéaire positif. Ann. Inst. Fourier (Grenoble) 40(1), 117–130 (1990)
Busch, P., Lahti, P.J., Pellonpää, J. P., Ylinen, K.: Quantum Measurement. Springer, Berlin (2016)
Charles, L.: Quantization of compact symplectic manifold. J. Geom. Anal. 26, 2664–2710 (2016)
Charles, L.: Berezin–Toeplitz operators, a semiclassical approach. Commun. Math. Phys. 239, 1–28 (2003)
Charles, L., Polterovich, L.: Quantum speed limit versus classical displacement energy. Ann. Henri Poincaré 19, 1215–1257 (2018)
Chazal, F., de Silva, V., Glisse, M., Oudot, S.: The Structure and Stability of Persistence Modules. Springer, Berlin (2016)
Douçot, B., Estienne, B.: Private communication (2017)
Edelsbrunner, H.: A Short Course in Computational Geometry and Topology. Springer, Berlin (2014)
Latschev, J.: VietorisRips complexes of metric spaces near a closed Riemannian manifold. Arch. Math. (Basel) 77, 522–528 (2001)
Lloyd, S., Garnerone, S., Zanardi, P.: Quantum algorithms for topological and geometric analysis of data. Nat. Commun. 7, 10138 (2016)
Ma, X., Marinescu, G.: BerezinToeplitz quantization and its kernel expansion, geometry and quantization. Trav. Math. 19, 125–166 (2011)
Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)
Merzbacher, E.: The early history of quantum tunneling. Phys. Today 55, 44–50 (2002)
Niyogi, P., Smale, S., Weinberger, S.: Finding the homology of submanifolds with high confidence from random samples. Discrete Comput. Geom. 39, 419–441 (2008)
Oudot, S.Y.: Persistence Theory: From Quiver Representations to Data Analysis. American Mathematical Society, Providence (2015)
Polterovich, L.: Quantum unsharpness and symplectic rigidity. Lett. Math. Phys. 102, 245–264 (2012)
Polterovich, L.: Quantum footprints of symplectic rigidity. EMS Newsl. 12(102), 16–21 (2016)
Stöckmann, H.J.: Quantum Chaos. An Introduction. Cambridge University Press, Cambridge (1999)
Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)
Acknowledgements
The work on this paper started during Leonid Polterovich’s stay at University of Chicago in the Winter of 2015. It was completed during his visits as a Mercator Fellow to Universität zu Köln and RuhrUniversität Bochum in 2017. He is grateful to these institutions for their warm hospitality. He thanks Shmuel Weinberger for useful discussions, as well as Laurent Charles, Yohann Le Floch, Vukašin Stojisavljević and Jun Zhang for helpful comments on the manuscript. His special thanks go to Laurent Charles for encouraging him to add Sect. 6.2 and for writing the Appendix. He thanks the referee for helpful remarks, and Andrei Iacob for superb copyediting. Laurent Charles would like to thank Benoit Douçot and Benoit Estienne for discussions on related subjects, and Leonid Polterovich for giving him the opportunity to write the Appendix.
Author information
Affiliations
Corresponding author
Additional information
Leonid Polterovich: Partially supported by the European Research Council Advanced Grant 338809 and by SFB/Transregio 191 of the Deutsche Forschungsgemeinschaft.
Appendices
Appendix: Toeplitz operators with piecewise constant symbol (by Laurent Charles)
In Berezin–Toeplitz quantization, we consider a symplectic compact manifold M, a set \(\Lambda \subset {\mathbb {R}}_{>0}\) having 0 as a limit point, and for any \(\hbar \in \Lambda \), a Hermitian complex line bundle \(L_\hbar \) and a finitedimensional subspace \(\mathcal {H}_{\hbar } \) of \({\mathcal {C}}^{\infty }(M, L_{\hbar })\). The space \( \mathcal {H}_{\hbar } \) has a natural scalar product \(\langle \Psi , \Psi ' \rangle _{\mathcal {H}_{\hbar }} = \int _M ( \Psi , \Psi ') \; d\nu \), where \((\Psi , \Psi ')\) is the pointwise scalar product and \(\nu \) the Liouville measure^{Footnote 3} of M. To any function \(f \in {\mathcal {C}}^{\infty }(M)\), we associate an endomorphism \(T_{\hbar } (f): \mathcal {H}_{\hbar } \rightarrow \mathcal {H}_\hbar \) such that
When the spaces \(\mathcal {H}_{\hbar }\) are conveniently defined, the family \((T_{\hbar }, \hbar \in {\Lambda })\) enjoys usual semiclassical properties. Typically, for a Kähler manifold M equipped with a positive line bundle L, we choose \({\Lambda }:= \{\hbar = 1/k, \; k \in {\mathbb {N}}^* \}\), \(L_{\hbar } := L^{\otimes k}\) and define \(\mathcal {H}_{\hbar }\) as the space of holomorphic sections of \(L^k\). These definitions can be extended to any quantizable M, cf. Ma and Marinescu (2011) or Charles (2016) for instance. For the purpose of this appendix, we only need that the reproducing kernel of \( \mathcal {H}_{\hbar }\) satisfies the two estimates (22) and (23).
In the definition (18) of \(T_{\hbar } (f)\), instead of a smooth function f, we can more generally consider any distribution \(f \in \mathcal {C}^{\infty } (M)\). Indeed, in equation (18), the pointwise scalar product \((\Psi , \Psi ' )\) is a smooth function, so the integral of f against \((\Psi , \Psi ' ) d \nu \) still makes sense and defines an endomorphism \(T_{\hbar } (f) \). The map \(T_{\hbar } :\mathcal {C}^{\infty }(M) \rightarrow {\text {End}} ( \mathcal {H}_{\hbar })\) is linear and positive in the sense that \(T_\hbar (f) \geqslant 0\) when \(f \geqslant 0\). The question is whether the asymptotic properties of \(T_{\hbar } (f)\) still hold, in particular the estimates of the trace and the product.
Proposition A.1
For any \(f \in \mathcal {C}^{\infty } (M)\), we have
The proof is an immediate generalization of the smooth case and will be given later. For the multiplicative properties of \(T_{\hbar }\), the regularity is crucial. For instance, let us recall two estimates proved in Barron et al. (2014). Let \(R_{\hbar } (f,g) = T_\hbar ( f) T_{\hbar } (g)  T_{\hbar } (fg)\). When f and g are both of class \(\mathcal {C}^\ell \) with \(\ell =1\) or 2, \( \Vert R_{\hbar } (f,g) \Vert = \mathcal {O}( \hbar ^{\ell /2})\). When f and g are only assumed to be continuous, \( \Vert R_\hbar ( f,g)\Vert \) tends to 0 in the semiclassical limit \(\hbar \rightarrow 0\). It is not proved that these estimates are sharp, but we believe they are.
Our goal is to extend these multiplicative properties to a subalgebra of \(L^{\infty } ( M)\) containing the characteristic functions of smooth domains. By a smooth domain, we mean a 0codimensional smooth submanifold with boundary. For any endomorphism T of \(\mathcal {H}_\hbar \), we introduce its Schatten norm normalized by the dimension \(d(\hbar ) = \dim \mathcal {H}_{\hbar }\),
If \((T_\hbar )\) is a family of endomorphisms depending on \(\hbar \), we write \(T_{\hbar } = \mathcal {O}_p (\hbar ^m) \) for \(\Vert T_{\hbar } \Vert _p = \mathcal {O}(\hbar ^m)\). For any measurable set A of M, denote by \(\chi _A \in L^{\infty } (M)\) its characteristic function. We say that A is a good set if
We say that a function \(f \in L^{\infty } (M)\) is a simple function if it has the form
where \(m\in {\mathbb {N}}\), \({\lambda }_1\),...,\({\lambda }_m\) are real numbers and \(A_1\),...,\(A_m\) are good sets.
Theorem A.2

1.
Any smooth domain of M is a good set.

2.
The good sets form an algebra, that is, they are closed under taking complement, finite intersection and finite union.

3.
If \(f, g \in L^{\infty }(M)\) are simple functions, then fg is simple and
$$\begin{aligned} T_\hbar (f) T_\hbar (g) = T_{\hbar } (fg) + \mathcal {O}_2 ( \hbar ^{1/4}). \end{aligned}$$ 
4.
If f is simple and takes only nonnegative values, then \(f^{1/2}\) is simple and
$$\begin{aligned} T_\hbar (f)^{1/2} = T_{\hbar } ( f^{1/2} ) + \mathcal {O}_4 ( \hbar ^{1/8}) . \end{aligned}$$
Interestingly, only the first assertion relies on the estimates (22) and (23) of the Bergman kernel. The proof of the other assertions is independent and does not use any difficult result.
Corollary A.3
Let \(f_1\),...,\(f_m\) be m simple nonnegative functions. Let \(P_\hbar = T_\hbar (f_1)^{1/2}\cdots T_\hbar (f_m)^{1/2}\). Then
Remark
It is essential that we use a Schatten norm in the definition (19) of a good set. Indeed, for any measurable set A, \(0 \leqslant T_\hbar ( \chi _A) \leqslant 1 \), so \( 0 \leqslant T_{\hbar } ( \chi _A )  T_{\hbar } ( \chi _A ) ^2 \leqslant 1/4\). When A is a good domain such that A and its complement are nonempty, we will prove in a forthcoming paper that \(T_{\hbar } ( \chi _A)\) has an eigenvalue \({\lambda }( \hbar )\) converging to 1 / 2 when \(\hbar \rightarrow 0\). Therefore,
The curious reader can think about the case where M is the twosphere and A a hemisphere. In this case, we can explicitly compute the spectrum of \(T_{\hbar } (\chi _A)\), as we learned from Douçot and Estienne (2017), cf. also Barron and Polterovich (2015).
\(\square \)
Proofs
Let \(( \Psi _i)\) be an orthonormal basis of \(\mathcal {H}_{\hbar }\) and define the Bergman kernel
where \(\overline{L}\) is the conjugate line bundle of L. We will need the diagonal estimate
where we identify \(L_x^k \otimes \overline{L}_x^k\) with \({\mathbb {C}}\) by using the metric of L. The second estimate we need is
for any \(m \in {\mathbb {N}}\), with some positive constants C and \(C_m\) independent of x, y. Here d is any distance on M obtained by embedding M into an Euclidean space and restricting the Euclidean distance. In the Kähler case, (22) was first proved in Bouche (1990) and was subsequently extended in Zelditch (1998) to convergence in the \({\mathcal {C}}^{\infty }\)topology. Estimate (23) follows from Corollary 1 of Charles (2003).
Proof of Proposition A.1
We have
by (22), which holds in the \({\mathcal {C}}^{\infty }\)topology. \(\square \)
In the sequel, to lighten the notation, we set \(T_{A} := T_{\hbar } ( \chi _A)\) for any measurable set A of M. We denote by \(A^c\) the complement of A.
Lemma A.4
A measurable set A of M is good if and only if
Proof
Since \(0 \leqslant T_A \leqslant 1\), \( T_A  T_A^2 \geqslant 0\). Hence
Using that \(d(\hbar ) = (2 \pi \hbar )^{n} \nu (M) ( 1+ \mathcal {O}( \hbar ))\), we see that A is good if and only if \({\text {tr}} ( T_A ( 1  T_A)) = \mathcal {O}( \hbar ^{n +1/2})\). To conclude the proof, observe that for any measurable subsets A and B
Indeed, computing the trace in the orthogonal basis \((\Psi _i)\), we have
By the definition (21) of the Bergman kernel,
which proves (25). \(\square \)
Proof of assertion 1 of Theorem A.2
We will deduce from estimate (23) that any smooth domain A in M satisfies (24). Consider a finite cover \((U_{\alpha }, \; 1 \leqslant {\alpha }\leqslant N)\) of M such that each \(U_{{\alpha }}\) is the domain of a coordinate system \((x_i)\) in which \(A \cap U_{{\alpha }} = \{ x \in U_{\alpha }; \; x_1(x) \geqslant 0 \}\). Denote by \(\Delta \) the diagonal of \(M^2\). Let \((f_{\alpha }, \; 0\leqslant {\alpha }\leqslant N)\) be a partition of unity of \(M^2\) subordinated to the cover \( ( \Delta ^c, U_1^2, \ldots , U_m^2 )\). It suffices to show that for each \({\alpha }\)
For \({\alpha }=0\), this follows from the fact that \({\text {supp}} f_0 \cap \Delta = \emptyset \), so that \(K_{\hbar } (x,y)  = \mathcal {O}( \hbar ^{\infty })\) uniformly on \({\text {supp}} f_0\). Let \({\alpha }\geqslant 1\) and choose a coordinate system \((x_i)\) on \(U_{\alpha }\) as above. Introduce on \(U_{\alpha }^2\) the coordinate system
Then by (23) there exists a constant C such that for any \((x,y) \in {\text {supp}} f_{\alpha }\), we have
where \(t^2 = \sum t_i^2\). Furthermore \((A \times \overline{A^c}) \cap U_{\alpha }^2 = \{ 0 \leqslant s_1 \leqslant t_1 \}\). So the integral in (26) is bounded above by
where \(s'_{\infty } = \max ( s_2, \ldots ,  s_{2n 1})\) and M is chosen so that the support of \(f_{\alpha }\) is contained in \(\{ s'_{\infty } \leqslant M \}\). Integrating with respect to the \(s_i\)’s and making the change of variable \(t= t \hbar ^{1/2}\), we obtain
Hence, the estimate (26) holds. \(\square \)
Proof of assertion 2 of Theorem A.2
Since \(T_{A^c} = 1  T_A\), \(T_A  T_A^2 = T_{A^c}  T_{A^c}^2\), so A is a good set if and only if \(A^c\) is a good set. The intersection of two good sets A, B is good because
and thanks to Lemma A.4. \(\square \)
Proof of assertion 3 of Theorem A.2
It suffices to prove that for any good sets A and B,
We first show that
with \(\mu (A \cap B) = \nu ( A \cap B) /\nu (M)\). Introduce the good sets \(a = A {\setminus } B\), \(b = B {\setminus } A\) and \(C = A \cap B\). Then A is the disjoint union of a and C, so \(T_A = T_a + T_C\). In the same way, \(T_B = T_b + T_C\). Therefore,
Since a and b are disjoint, we deduce from (25) that
Further, since a is good, we obtain that \(d(\hbar )^{1} {\text {tr}} (T_a T_b) = \mathcal {O}( \hbar ^{1/2})\). By the same argument, \(d(\hbar )^{1} {\text {tr}} ( T_a T_C) = \mathcal {O}( \hbar ^{1/2})\) and \(d(\hbar )^{1} {\text {tr}} (T_C T_b) = \mathcal {O}( \hbar ^{1/2})\). Finally, since C is good,
by Proposition A.1. Summing the various estimates, we get (28).
Second, we compute the HilbertSchmidt norm of \(T_A T_B\). We will use the following consequence of the Hölder inequality: for any \(\hbar \)dependent endomorphisms S, \(S'\), T of \(\mathcal {H}_{\hbar }\),
We have:
Now we come to the proof of (27). With \(C = A \cap B\), we have
Next, by (31),
To estimate the trace of \(T_A T_B T_C\), we use as above the sets \(a = A {\setminus } B\) and \(b = B {\setminus } A\). By (29),
Using that C is good and (30), we have
by (28). Similarly, using that C is good, (30) and (28), we have
because \(a\cap C = \emptyset \). By the same argument, \( d(\hbar )^{1} {\text {tr}} ( T_b T_C^2 ) = \mathcal {O}( \hbar ^{1/2})\). Finally, C being good, \( d(\hbar )^{1} {\text {tr}} (T_a T_b T_C) = d(\hbar )^{1} {\text {tr}} (T_a T_b T_C^2) + \mathcal {O}( \hbar ^{1/2})\), and by the Hölder inequality,
where we have applied (31) to C, a and b, C and used the fact that these sets are pairwise disjoint. Gathering these estimates we conclude that
Exchanging A and B, we get the same estimate for \(d(\hbar )^{1} {\text {tr}} ( T_B T_a T_C)\). Now (33) and (32) imply that \(\Vert T_A T_B  T_C \Vert _2^2 = \mathcal {O}( \hbar ^{1/2})\). \(\square \)
Proof of assertion 4 of Theorem A.2
Since the good sets are closed under taking the complement and finite intersections, we see that any simple function f can be written as a sum (20) with the \(A_i\) being pairwise disjoint good sets. We can furthermore assume that these sets are nonempty. Assume that f is nonnegative, then all the coefficients \({\lambda }_i\) are nonnegative. Then \(f^{1/2}\) is simple. Set \(S_{\hbar } (f) = T_{\hbar }(f^{1/2})^2\). By the third assertion of Theorem A.2, we have
Using that the square root is an operator monotone function, we have
and the righthand side is a \(\mathcal {O}( \hbar ^{1/8})\) by (34). In other words, \( T_\hbar ( f^{1/2}) = T_{\hbar } ( f) ^{1/2} + \mathcal {O}_4 ( \hbar ^{1/8}).\) \(\square \)
Proof of Corollary A.3
By assertion 4 of Theorem A.2 and (30), we have
with \(Q_{\hbar } = T_{\hbar } (f_1) \cdots T_{\hbar } ( f_m)\). By assertion 3 of Theorem A.2,
Hence, by (30) and Proposition A.1,
and the result follows. \(\square \)
Rights and permissions
About this article
Cite this article
Polterovich, L. Inferring topology of quantum phase space. J Appl. and Comput. Topology 2, 61–82 (2018). https://doi.org/10.1007/s4146801800180
Received:
Accepted:
Published:
Issue Date:
Keywords
 Persistent homology
 Barcode
 BerezinToeplitz quantization
 Quantum measurement
Mathematics Subject Classification
 81Sxx
 55U99