Abstract
We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices, and is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant measure in the Wasserstein metric. Standard techniques from the theory of products of random matrices cannot be applied under our assumptions, and new techniques are developed, such as maximum likelihood-type estimations.
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Notes
Complete positivity is stronger than positivity; namely by definition \(\phi \) is completely positive iff \(\phi \otimes \mathrm{Id}_{M_n({{\mathbb {C}}})}\) is positive for all \(n\in {{\mathbb {N}}}\).
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Acknowledgements
T.B. and C.P. would like to thank Y. Guivarc’h for his useful comments at an early stage of this work. Y.P. and C.P. would like to thank P. Bougerol for enlightening discussions about random products of matrices. Y.P. and C.P. would like to thank L. Miclo for relevant discussions regarding Markov chains. The research of T.B. has been supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. The research of T.B., Y.P. and C.P. has been supported by the ANR project StoQ ANR-14-CE25-0003-01 and CNRS InFIniTi project MISTEQ.
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Appendices
Appendix A: Equivalence of (Pur) and contractivity
We assume \({\text {supp}}\mu \subset \mathrm {GL}_k({{\mathbb {C}}})\). Recall that \(T_\mu \) is the smallest closed sub-semigroup of \(\mathrm {GL}_k({{\mathbb {C}}})\) that contains \({\text {supp}}\mu \). It is said to be contracting if there exists a sequence \((a_n)_{n\in {\mathbb {N}}}\subset T_\mu \) such that \(\lim _{n\rightarrow \infty } a_n/\Vert a_n\Vert \) exists and is a rank one matrix.
Proposition A.1
Assume \({\text {supp}}\mu \subset \mathrm {GL}_k({\mathbb {C}})\) and \(T_\mu \) is strongly irreducible. Then \(\mu \) verifies (Pur) if and only if \(T_\mu \) is contracting.
Proof
By Proposition 2.2 the implication (Pur)\(\Rightarrow \) contractivity follows by taking for \((a_n)\) a convergent subsequence of \((W_n(\omega ))\) for \(\omega \in {\text {supp}}{{\mathbb {P}}}^{\mathrm {ch}}\).
We prove the opposite implication by contradiction. Following [12, Lemma 3], under the assumptions of the proposition, \(T_\mu \) is contracting if and only if, for any two \(\hat{x}, \hat{y}\in {{\mathrm P}({{\mathbb {C}}}^k)}\) there exists a sequence of matrices \((a_n)\subset T_\mu \) such that
Now, assume that contractivity holds but (Pur) does not. Namely, that \(T_\mu \) is contracting but there exists an orthogonal projector \(\pi \) of rank \(\ge 2\), such that for any \(a\in T_\mu \),
Let x, y in the range of \(\pi \) be orthonormal vectors. Then \(\langle ax,ay\rangle =\langle x,y\rangle =0\), and \(\Vert ax\Vert , \Vert ay\Vert \) are nonzero, so that \(d(a\cdot \hat{x},a\cdot \hat{y})=1\). As this is true for any a in \(T_{\mu }\), contractivity cannot hold. This contradiction yields the proposition. \(\square \)
Appendix B: Set of invariant measures under assumption (Pur)
A quantum channel is a map \(\phi \) on \(\mathrm {M}_k({{\mathbb {C}}})\) of a form
where \(\mu \) is a measure satisfying the normalization condition (1). The decomposition of quantum channels to irreducible components was derived in [2, 5, 22]. The space \(\mathbb {C}^k\) is decomposed into orthogonal subspaces, one subspace is transient and in all other subspaces the map has a canonical tensor product structure. We recall these results.
There exists a decomposition
with the following properties. We denote by \(v^{(j)}\) the restriction of v to \(\mathbb {C}^{n_j}\).
-
(e1)
All invariant states are supported in the subspace \(L = \mathbb {C}^{n_1} \oplus \dots \oplus \mathbb {C}^{n_d} \oplus 0\),
-
(e2)
The restriction of v to this subspace is block diagonal,
$$\begin{aligned} v|_L = v^{(1)} \oplus \cdots \oplus v^{(d)}\oplus 0 \quad \mu {\text {-}}\mathrm {a.e.}\end{aligned}$$(38) -
(e3)
For each \(j=1, \dots ,d\) there is a decomposition \(\mathbb {C}^{n_j} = \mathbb {C}^{k_j} \otimes \mathbb {C}^{m_j}, \, n_j = k_j m_j\), a unitary matrix \(U_j\) on \(\mathbb {C}^{n_j}\) and a matrix \(\tilde{v}^{(j)}\) on \(\mathbb {C}^{k_j}\) such that
$$\begin{aligned} v^{(j)} = U_j \left( \tilde{v}^{(j)} \otimes \mathrm{Id}_{{{\mathbb {C}}}^{m_j}}\right) U_j^* \quad \mu -a.s. \end{aligned}$$(39) -
(e4)
There exists a full rank positive matrix \(\rho _j\) on \(\mathbb {C}^{k_j}\) such that
$$\begin{aligned} 0 \oplus \cdots \oplus U_j \left( \rho _j \otimes \mathrm{Id}_{{{\mathbb {C}}}^{m_j}}\right) U_j^* \oplus \cdots \oplus 0 \end{aligned}$$(40)is a fixed point of \(\phi \).
It follows from (e3) and (e4) that the set of fixed points for \(\phi \) is
The decomposition simplifies under the purification assumption.
Proposition B.1
Assume (Pur) holds. Then there exists a set \(\{\rho _j\}_{j=1}^d\) of positive definite matrices and an integer D such that the set of \(\phi \) fixed points is
Proof
The statement follows from the discussion preceding the proposition if we show that (Pur) implies \(m_1 = \dots = m_d =1\). Assume that one of the \(m_j\), e.g. \(m_1\), is greater than 1. Let x be a norm one vector in \(\mathbb {C}^{k_1}\). Then \(\pi = U_1\pi _{\hat{x}} \otimes \mathrm{Id}_{\mathbb {C}^{m_1}} U_1^*\oplus 0 \oplus \dots \oplus 0\) is a projection with rank bigger than 1, and by Eq. (39) we have, in the notation of (38) and (39),
for \(\mu ^{\otimes n}\)-almost all \(v_1,\ldots ,v_n\). This contradicts (Pur). \(\square \)
It is clear from Eq. (38) that to each extremal fixed point \(0 \oplus \dots \oplus \rho _j \oplus \dots \oplus 0\) corresponds a unique invariant measure \(\nu _j\) supported on its range \(F_j\). The converse is the subject of the next proposition.
Proposition B.2
Assume (Pur) holds. Then any \(\Pi \)-invariant probability measure is a convex combination of the measures \(\nu _j\), \(j=1,\ldots ,d\).
Proof
Let \(\nu \) be a \(\Pi \)-invariant probability measure. Let f be a continuous function. From Lemma 2.3,
Proposition 2.1 implies
with \(\rho _\nu \in {\mathcal {D}}_k\) a fixed point of \(\phi \). By Proposition B.1, (Pur) implies that there exist non negative numbers \(t_1,\ldots ,t_d\) summing up to one such that \(\rho _\nu =t_1\rho _1\oplus \cdots \oplus t_d\rho _d\oplus 0_{M_D({{\mathbb {C}}})}\). From the definition of \({{\mathbb {P}}}^{\rho _\nu }\),
where we used the abuse of notation \(\rho _j\equiv 0\oplus \cdots \oplus \rho _j\oplus \cdots \oplus 0\). Using Proposition 2.1, it follows that
Then Lemma 2.3 and the \(\Pi \)-invariance of each measure \(\nu _j\) yield the proposition. \(\square \)
Appendix C: Products of special unitary matrices
Proposition C.1
Assume \({\text {supp}}\mu \subset \mathrm {SU}(k)\). Let G be the smallest closed subgroup of \(\mathrm {SU}(k)\) such that \({\text {supp}}\mu \subset G\). For any \(\hat{x}\in {{\mathrm P}({{\mathbb {C}}}^k)}\), let \([\hat{x}]_G\) be the orbit of \(\hat{x}\) with respect to G and the action \(G\times {{\mathrm P}({{\mathbb {C}}}^k)}\ni (v,\hat{x})\mapsto v\cdot \hat{x}\). Namely, \([\hat{x}]_G:=\{\hat{y}\in {{\mathrm P}({{\mathbb {C}}}^k)}\ |\ \exists v\in G \text{ s.t. } \hat{y}=v\cdot \hat{x}\}\). Then, for any \(\hat{x}\), there exists a unique \(\Pi \)-invariant probability measure supported on \([\hat{x}]_G\), and this unique invariant measure is uniform in the sense that for any \(v\in G\) it is invariant by the map \(\hat{x}\mapsto v\cdot \hat{x}\).
Corollary C.2
With the assumption and definitions of the last proposition, if \(G=\mathrm {SU}(k)\), \(\Pi \) has a unique invariant probability measure and this probability is the uniform one on \({{\mathrm P}({{\mathbb {C}}}^k)}\).
Proof
The corollary being a trivial consequence of \(G=\mathrm {SU}(k)\Rightarrow [\hat{x}]_G={{\mathrm P}({{\mathbb {C}}}^k)}\ \forall \hat{x}\in {{\mathrm P}({{\mathbb {C}}}^k)}\), we are left with proving the proposition.
Let \(P_\mu \) be the Markov kernel on G defined by the left multiplication: \(P_\mu f(v)=\int _G f(uv)d\mu (u)\). Since G is compact as a closed subset of \(\mathrm {SU}(k)\), following [1, Proposition 4.8.1, Theorem 4.8.2], the unique \(P_\mu \)-invariant probability measure \(\mu _G\) on G is the normalized Haar measure on G. Since G is compact, Prokhorov’s theorem implies that for any \(u\in G\),
Let \(\hat{x}\in {{\mathrm P}({{\mathbb {C}}}^k)}\). Since \({\text {supp}}\mu \subset G\), for any \(\hat{y}\in [\hat{x}]_G\), \(\Pi (\hat{y}, [\hat{x}]_G)=1\). Then, \([\hat{x}]_G\) being compact, there exists a \(\Pi \)-invariant measure \(\nu \) supported on \([\hat{x}]_G\).
Let f be a continuous function on \([\hat{x}]_G\). Then,
For each \(\hat{y}\in [\hat{x}]_G\) let \(u_y\in G\) be such that \(\hat{y}=u_y\cdot \hat{x}\). The map \(v\mapsto vu_y\cdot \hat{x}\) being continuous, setting \(u=u_y\), the weak convergence (41) and Lebesgue’s dominated convergence theorem imply,
It follows that \(\nu \) is the image measure of \(\mu _G\) by the application \(v\mapsto v\cdot \hat{x}\). The left multiplication invariance of the Haar measure \(\mu _G\) yields the invariance of \(\nu \) by the map \(\hat{x}\mapsto v\cdot \hat{x}\) for any \(v\in G\). \(\square \)
Example C.3
Let \(\mu =\frac{1}{2}(\delta _{v_1}+\delta _{v_2})\) with,
Then \(G=\mathrm {SU}(2)\) and the uniform measure on \({\mathrm P}({{\mathbb {C}}}^2)\) is the unique \(\Pi \)-invariant probability measure.
Proof
Following Proposition C.1, it is sufficient to prove that any element of \(\mathrm {SU}(2)\) is the limit of a sequence of products of \(v_1\) and \(v_2\).
Let \(\sigma _1,\sigma _2,\sigma _3\) be the usual Pauli matrices:
The Pauli matrices being generators of \(\mathrm {SU}(2)\) in its fundamental representation, for any \(u\in \mathrm {SU}(2)\), there exist three reals \(\theta _1,\theta _2,\theta _3\in {{\mathbb {R}}}\) s.t.,
Especially, \(v_1=\exp (i\sigma _3)\) and \(v_2=\exp (i\sigma _1)\). Since for any \(j=1,2,3\), \(\exp (i\theta _j\sigma _j)=\exp (i(\theta _j+2\pi )\sigma _j)\), taking limits of sequences of powers of \(v_1\) or \(v_2\), for any \(\theta \in {{\mathbb {R}}}\), both
are elements of G. It remains to show that any \(u\in \mathrm {SU}(2)\) is a product of elements equal to \(\exp (i\theta \sigma _1)\) or \(\exp (i\theta \sigma _3)\) with \(\theta \) real.
Fix \((\theta _1,\theta _2,\theta _3)\in {{\mathbb {R}}}^3\). Then using spherical coordinates in \({{\mathbb {R}}}^3\), there exist \(r\in {{\mathbb {R}}}_+\), \(\theta \in [0,\pi ]\) and \(\varphi \in [0,2\pi [\) such that \(\theta _1=r\cos \theta \), \(\theta _2=r\sin \theta \cos \varphi \) and \(\theta _3=r\sin \theta \sin \varphi \). Then by direct computation,
It follows that as a product of elements of G, \(e^{i(\theta _1\sigma _1+\theta _2\sigma _2+\theta _3\sigma _3)}\in G\), hence \(G=\mathrm {SU}(2)\) and the example holds. \(\square \)
Example C.4
Let \(\mu =\frac{1}{2}(\delta _{v_1}+\delta _{v_2})\) with,
Then \(G=\{\pm \mathrm{Id}_{{{\mathbb {C}}}^2}, \pm v_1, \pm v_2, \pm v_1v_2\}\). For \(z\in {{\mathbb {C}}}\), let \(e_z=(1,z)^\mathsf {T}\) and \(e_\infty =(0,1)^\mathsf {T}\). With the conventions \(\infty ^{-1}=0\), \(0^{-1}=\infty \) and \(-\infty =\infty \), for any \(z\in {{\mathbb {C}}}\cup \{\infty \}\), \([\hat{e}_z]_G=\{\hat{e}_z, \hat{e}_{z^{-1}}, \hat{e}_{-z}, \hat{e}_{-z^{-1}}\}\) and the measure \(\frac{1}{4}(\delta _{\hat{e}_z}+\delta _{\hat{e}_{-z}}+\delta _{\hat{e}_{z^{-1}}}+\delta _{\hat{e}_{-z^{-1}}})\) is a \(\Pi \)-invariant probability measure.
The proof of this example is obtained by an explicit computation.
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Benoist, T., Fraas, M., Pautrat, Y. et al. Invariant measure for quantum trajectories. Probab. Theory Relat. Fields 174, 307–334 (2019). https://doi.org/10.1007/s00440-018-0862-9
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DOI: https://doi.org/10.1007/s00440-018-0862-9