Asymptotic statistical theory of irreducible quantum Markov chains

Abstract

In this paper we investigate the asymptotic statistical theory of irreducible quantum Markov chains, focusing on identifiability properties and asymptotic convergence of associated quantum statistical models. We show that the space of identifiable parameters for the stationary output is a stratified space called an orbifold, which is obtained as the quotient of the manifold of irreducible dynamics by a compact group of state preserving symmetries. We analyse the orbifold's geometric properties, the connection between periodicity and strata, and provide orbifold charts as the starting point for the local asymptotic theory. The quantum Fisher information rate of the system and output state is expressed in terms of a canonical inner product on the identifiable tangent space. We then show that the joint system and output model satisfies quantum local asymptotic normality while the stationary output model converges to a product between a quantum Gaussian shift model and a mixture of quantum Gaussian shift models, reflecting the underlying periodicity. These strong convergence results provide the basis for constructing asymptotically optimal estimators of dynamical parameters. We provide an in-depth analysis of the model with smallest dimensions, consisting of two-dimensional system and environment units.

Figures (6)

• Figure 1: Conceptual representation of the local geometry of stationary QMCs with two dimensional system and ancillas, and limit quantum statistical models for 3 one-parameter models. Panel a) depicts the (12 dimensional) manifold of isometries $\mathscr{V}^{\rm irr}$ and the action of the symmetry group $\mathscr{G}$. Locally, the action consists of translations along the $Z$ axis together with the $\mathbb{Z}_2$ action of reflections around the $XZ$ plane that represents periodic QMCs. The blue arrows correspond to the transformations generating the group, while the two vertical red lines constitute a single arbitrarily chosen orbit. At every point, the non-identifiable tangent space $\mathcal{T}^{\rm nonid}$ consists of vectors in the $Z$ direction while $\mathcal{T}^{\rm id}$ is parallel to the $XY$ plane. The latter carries the action of the stabiliser group $\mathbb{Z}_2$ and decomposes into the direct sum of eigenspaces ${\cal V}_0$ and ${\cal V}_1$ consisting of vectors along the $X$ and respectively the $Y$ axis. Panel b) represents the orbifold obtained as the quotient of $\mathcal{V}^{\rm irr}$ under the action of $\mathscr{G}$, which can be identified with the $XY$ plane folded along the $X$ axis, represented as a half plane including the 'edge' axis $X$; the axis corresponds to the (4 dimensional) manifold $\mathscr{P}_{2,4}$ of equivalence classes of periodic isometries, while the plane represents the (8 dimensional) manifold $\mathscr{P}^{\rm prim}$ of equivalence classes of primitive QMCs. The red dot corresponds to the orbit represented by red lines in $a)$. We analyse the asymptotics of three one-dimensional models with distinct geometries and limit models. In the parameter space these are the following lines in the $XY$ plane: a line (I) along the $X$ axis, a line (II) which is non-orthogonal to either $X$ or $Y$ axis, and a line (III) parallel to the $Y$ axis. For clarity, the lines are not represented in panel a) but their orbifold projections are represented in panel b) by purple arrows: (I) lies inside the singular manifold, (II) is the 'broken' line that crosses the singular manifold in one point, and (III) is a half line in the $Y$ direction, exhibiting non-identifiability. Panel $c)$ reports a representation in the phase space of the limit quantum statistical models corresponding to the three one-parameter models, locally around a point at the intersection with the $X$ axis (periodic QMC). Orange circles represent quantum coherent states with standard vacuum covariance. (I) is a pure quantum Gaussian shift model where the mean moves along the $Q$ axis. (II) is the tensor product of a pure Gaussian shift model as the one considered in (I) and a mixture of two coherent states with opposite means: the means move together when the local parameter changes. Finally, the third model is described by a mixture of coherent states lying on the $Q$ axis with opposite means. We remark that the local parameter is identifiable in (I) and (II), but not in (III).
• Figure 2: Quantum channel as a black box transformation mapping the initial system state $\rho$ to final state $\mathcal{T}_*(\rho)$ through the unitary interaction with an environment system in a fixed initial state $\tau$.
• Figure 3: Panel a) Bloch ball representation of qubit states. Pure states are represented as vectors on the unit sphere with basis vectors $0\rangle$ and $|1\rangle$ as north and south poles respectively. The pure state statistical model $|\psi_\theta\rangle$ covers a two dimensional neighbourhood of the north pole. Each state is a rotation of $|0\rangle$ with rotation parameter $\theta = (\theta_1, \theta_2) \in \Theta$. Panel b) Illustration of quantum local asymptotic normality for pure qubit states. The i.i.d. model consisting of an ensemble of $n$ identically prepared qubits in state $|\psi^{(n)}_{u,v}\rangle:= |\psi_{u/\sqrt{n},v/\sqrt{n}}\rangle^{\otimes n}$ where $u,v$ are local rotation parameters. For large $n$ the local i.i.d. model converges to quantum Gaussian model consisting of a single sample from the coherent state $|u,v\rangle$ whose mean is given by the local paramters.
• Figure 4: A discrete-time quantum Markov chain. A sequence of identically prepared input units interact successively with a system via the unitary $U$. After the interaction, the output units are correlated and carry information about the unitary $U$.
• Figure 5: In Figure a) the axis represents a choice of coordinates for $\mathbb{R}^2$, while the red sector in Figure b) is a graphic representation of $\mathbb{R}^2/\langle \gamma,\sigma \rangle$. Arrows in figure a) represent the five transformations in the dihedral group minus the identity: red arrows correspond to rotations and blue arrows to symmetries. Disks in Figure a) and b) represent charts and their image: the blue is a chart for $\pi((0,0))$, the green for $\pi(x)$ for some point $x$ laying on the $X$ axis and the red one for a point in $A_2$. The blue dot in Figure b) represents $\Sigma_0$, the red dotted half-lines (without the origin) represent $\Sigma_1$ and all the other points are regular points.

Theorems & Definitions (66)

• Theorem 1: Kraus Theorem
• Theorem 2: Quantum Cramér-Rao bound
• Lemma 1
• Proposition 1
• Definition 1: Equivalent models
• Definition 2: Statistical deficiency and Le Cam distance
• Theorem 3: weak QLAN
• Theorem 4: strong QLAN
• Lemma 2: Weak to strong convergence for pure state models
• Lemma 3: Strong convergence of mixed statistical models