A quantum central limit theorem for non-equilibrium systems: Exact local relaxation of correlated states

We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states - pure or mixed - which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly relate our findings to ideas of typicality and concentration of measure.


Introduction
In what sense can closed local many-body systems in a non-equilibrium situation relax to an apparent equilibrium? Instances of that question have a long tradition in the literature. This apparent contradiction of having entropy preserved in any closed system and at the same time arrive at an equilibrated situation for long times can yet be resolved by acknowledging that merely some observables, and in particular entire subsystems, can well appear as if they had truly relaxed, even in closed systems. In local observations, such a composite system would then look entirely equilibrated.
Early work formulating a quite similar intuition, albeit not in quantum lattice systems, was concerned with fermions freely moving in space [1]. The seminal work in Ref. [2,3] rigorously developed such an intuition for classical harmonic systems on cubic lattices. Refs. [4,5,6] consider that question in instances of a bosonic or fermionic fully Gaussian setting, where initial states are Gaussian, and the evolution is governed by a quadratic Hamiltonian. Ref. [7] then arrives at a true local relaxation theorem for subsystems where initial states are not taken to be Gaussian: Yet, in the course of the non-equilibrium dynamics of the system, locally, the states become Gaussian, so maximum entropy states for given second moments. This is true without having to invoke a time average. In such lattice models, the intuition is that non-equilibrium generates local excitations at each site that then travel ballistically through the lattice, resulting in a mixing at each site with excitations from further and further separated sites coming in [7,8].
Some of the revived significant recent interest in equilibration and relaxation in closed quantum many-body systems, needless to say, has been sparked off by novel experiments with cold atoms in optical lattices or on low-dimensional structures. For theory work relating to those experiments, see, e.g., Refs. [11,12,13,14,10,9] and references therein. This development is quite intriguing, as questions of an apparent local relaxation in non-equilibrium dynamics can be measured and probed under very precisely tunable laboratory conditions.
In this work, we present a fully rigorous relaxation theorem, generalizing the findings of Ref. [7], for one-dimensional quantum lattice systems. Physically speaking, the results presented here can be viewed as a relaxation theorem forming a "theoretical laboratory", describing an idealized situation of non-equilibrium dynamics in the Bose-Hubbard setting: The system being initially held in some ground or thermal state of the full Bose-Hubbard-Hamiltonian, and then suddenly switched or "quenched" to a parameter regime of strong hopping.
It is shown that for general pure or mixed initial states satisfying quite weak assumptions concerning the clustering of correlations-properties that are expected to hold in particular for ground states of local many-body models-, locally, the system will relax to a Gaussian state under the dynamics generated by a quadratic Hamiltonian. That is to say, we prove the "local relaxation conjecture" for one-dimensional clustering states under quadratic Hamiltonians on a ring of L sites. To wit, for an initial state̺ 0 that is evolved in time under a quadratic Hamiltonian and a given finite subset S of the chain, we show that for every ǫ > 0 and sufficiently large L there is a relaxation time t relax , and a recurrence time t rec (which grows unboundedly with L), such that the reduction of the time-evolved state to S,̺ S (t), fulfils in particular, lim The state̺ G (t) is a Gaussian state with the same second moments as̺ S (t). The proof features the explicit forms of t relax and t rec in terms of the coupling parameters in the Hamiltonian and the decay behaviour of correlations of the initial state.
The key ingredients in the proof are a quantum Lindeberg central limit theorem, a notion of locality of dynamics as is manifest by a Lieb-Robinson bound, and a Bernstein-Spohn-blocking argument reminiscent of the classical situation. Essentially, we will show that for a given time excitations arising from far away sites will hardly influence the dynamics at a subset of sites, whereas the close sites give rise to a mixing such that a Lindeberg central limit theorem can be invoked. In several ways, our quantum treatment reminds of the classical situation presented in Ref. [2,3], with the blocking argument being essentially identical, with significant differences in many other aspects. In this work, we focus on the important case of a one-dimensional setting and allow for assumptions on the initial state that can be relaxed, in order to render the argument simpler than a fully general argument [15].
After rigorously proving convergence of characteristic functions to Gaussian functions in phase space, we prove closeness in trace-norm of the respective reduced states.
In this way, we derive in the trace-norm topology a closeness to maximum entropy states under the second moments. In some ways, the arguments reminds of the situation of discrete time in convergence to stationary states in quantum cellular automata [16]. We also finally briefly discuss the findings in the light of recent results concerning concentration of measure and notions of typicality [17,18,19].

Preliminaries
In this section, we are collecting preliminaries that are being used in the proof. We also specify the model under consideration here and will specify the assumptions on the initial states in the subsequent section. Note that the assumptions on such initial states are very mild and do by no means only include pure or Gaussian states; merely the model Hamiltonian as such is taken to be a specific nearest-neighbor Hamiltonian.

System and dynamical setting
We consider a chain L = {1, . . . , L} equipped with periodic boundary conditions. Each lattice site is associated with a bosonic canonical degree of freedom equipped with the usual symplectic form. The distance between lattice sites i, j ∈ L is due to the periodic boundary conditions given by For future purposes, we also define distances between subsets of sites. For A, B ⊂ L we define their distance as Starting from an initial quantum state (neither restricted to be a pure state nor a Gaussian state)̺ 0 on L, we are concerned with its time evolution, under the HamiltonianĤ b i denotes a bosonic operator at site i. We take the Hamiltonian to be defined by We will work in units in which J = 1, = 1. This is the Hamiltonian of a harmonic chain, in particular approximating the Hamiltonian of a Bose-Hubbard model in the limit of a hopping that is dominant compared to the interaction. Note that local relaxation may be shown for general quadratic Hamiltonians with sufficiently local couplings (exponentially decaying in the coupling distance) on (not necessarily cubic) lattices of arbitrary spatial dimension [15]. The assumption thatĤ is quadratic, is essential, however, for the validity of the argument. Yet, the physical intuition that local relaxation is due to a mixing of excitations that essentially travel ballistically through the lattice should be expected to be valid more generally. In this sense, the present proof constitutes the mentioned "theoretical laboratory" in which this mechanism can be studied in great clarity. In particular, we will consider local properties of̺(t), and will see that in a sense this reduced state will eventually relax in a sense yet to be specified. To this end let S ⊂ L be a subset of sites of the lattice and define the reduced state associated with the degrees of freedom of this subset aŝ We write its characteristic function χ̺ S (t) : |S| → as the expectation value of the Weyl operator, i.e., as as a complex-valued phase space function uniquely defining the state. States the characteristic function of which are a Gaussian in phase space are referred to as being quasi-free or Gaussian states. One finds, after solving Heisenberg's equation of motion and using the cyclic invariance of the trace, for the time evolution of the characteristic function of the reduced state̺ S (t) where α ∈ |L| , the latter to be understood as a matrix exponent of the matrix A collecting the coupling coefficients. Note that this is exactly the setting of a sudden quench: One considers a ground or thermal state of some Hamiltonian (or any initial state) and the abrupt switch to the HamiltonianĤ, and follows the free time evolution under this new Hamiltonian. Note that the weak assumptions on the decay of correlations also allows for initial ground states of critical models.

Moments
For operatorsÔ defined on the Hilbert space associated with the lattice model, we write expectation values as For vectors of phase space coordinates α, α ′ ∈ |L| we define the anti-hermitian operatorb and instances of second and fourth moments ofb(α) as For A ⊂ L we define the vector α A ∈ L by as the indicator function and write for certain second moments and functions of indicator functions.

Assumptions on initial state and locality of dynamics
In this section, we state the assumptions on the considered initial states and highlight the role of Lieb-Robinson bounds in the argument.

Assumptions on the initial state
The assumptions on the initial state essentially express some degree of clustering or decay of correlations. This clustering can be assumed to be quite weak, and even a slow algebraic decay of correlations is allowed for. We will also include some natural conditions on the initial state that significantly simplify the argument. In particular, this includes an assumption of the initial state to commute with the total particle number operator. In any bosonic system involving massive particles, this natural requirement will always be satisfied. More specifically, we assume that the initial state is such that 1 Then, for all α ∈ L and all A, B ⊂ L we have b (α) = 0, and The first assumption concerns two-point correlations.

Assumption 1 (Two-point correlations)
Let the initial state̺ 0 be such that there exist absolute constants This is a natural assumption on the initial state in a way that does not depend on the system size. This assumption may be relaxed to decay stronger than the spatial dimension of the lattice (i.e., in the setting at hand the two may be replaced by a one), resulting, however, in a much more involved proof [15]. We note a first immediate consequence of Assumption 1 that will be used in the main theorem. We also encounter the recurrence time: the system has to be sufficiently large for the bounds on the second moments to hold.

Lemma 1 (Variances) Under Assumption 1 and for
for all A, B ⊂ L such that 4e|t| ≤ d A,S . Here, and c 1 , ǫ 1 , and µ 1 are as in Assumption 1.
The second assumption concerns four-point correlations.

Assumption 2 (Four-point correlations)
Let the initial stateρ 0 be such that there exist absolute constants for all i, j, k, l. Here, P (i, j, k, l) contains all permutations of the indices i, j, k, l.
We finally put a clustering condition on the initial state.

Assumption 3 (Clustering)
Let the initial state̺ 0 be such that there exist absolute constants c 5 , µ 2 > 0 such that for all A, B ⊂ L and all L.
Note that this condition may again be relaxed [15].

Lieb-Robinson bounds and locality of dynamics
Lieb-Robinson bounds are upper bounds to group velocities of excitations travelling through a quantum lattice system. They define a causal cone of a local excitation, outside of which any influence of this excitation is exponentially suppressed. As such, they provide an upper bound to the speed of any non-negligible information propagation by time evolution of a quantum lattice model [20,21,22,23]. Originally formulated for spin systems, in the present context, we need to invoke an instance that is suitable for the harmonic, infinite-dimensional individual constituents at hand [24,25]. Subsequently we will formulate a useful related bound defining a causal cone for matrix entries of the propagator itself, which will be used in the main theorem.

Lemma 3 (Lieb-Robinson bound) For all times, all i, j ∈ L and all
For all t, L such that L ≥ |t| 7/6 ≥ 1 one has for all i, j ∈ L that This bound shows the exponential smallness of the matrix entries of C = e −itA in the distance d i,j away from the main diagonal at a given time. Also, Eqs. (29) signify bounds needed to prove convergence due to mixing within the causal cone for large times. The proof will be given in the appendix.

Quantum central limit theorems
Lindeberg central limit theorems are central limit theorems that do not rely on an identically distributed assumption concerning the considered random variables. Here we formulate a quantum version thereof, which will in the main theorem be extended to a quantum version of a central limit theorem that in addition does not rely on independent random variables. We start by noting a number of useful facts on quantum characteristic function in order to proceed to formulate the quantum central limit theorem itself.

Some facts about characteristic functions
In this section, we will relate quantum characteristic functions to classical characteristic functions, by invoking Bochner's theorem. For a similar correspondence of quantum to classical characteristic functions, see, e.g., Ref. [26]. For a fixed vector of phase space One finds φ (n) (0) = 0 for odd n ∈ AE and where φ (n) denotes the n-th derivative of φ. Using the Cauchy-Schwarz inequality, one finds for even m ∈ AE the following chain of inequalities. We suppress the argument α ofb defined as in Eq. (13), use the fact thatb † = −b and write̺ 0 = n ̺ n |ψ n ψ n | for the spectral decomposition of the state̺ 0 . We hence find for the m-th derivative Furthermore, φ(0) = 1, and φ is positive semi-definite and continuous at the origin [27]. It follows from Bochner's theorem that φ is a classical characteristic function of a classical random variable. Hence, if φ (2) (0) exists, φ is twice continuously differentiable on the entire real line, i.e., as φ (1) (0) = 0, one has from Taylor's theorem Hence, for all α ∈ L as α was arbitrary.
If φ (4) (0) exists, φ is four times continuously differentiable on the entire real line, i.e., one has from Taylor's theorem

A Lindeberg-type quantum central limit theorem
We now turn to the actual quantum instance of a Lindeberg-type central limit theorem. Let A 1 , . . . , A n ⊂ L be mutually disjoint sets. These take the role of independent but not necessarily identically distributed random variables. Consider If |z i − 1| ≤ 1/2 for all i, a complex logarithm is defined by the Mercator series. We have where Hence, Now, let x, y, z ∈ Ê. We set out to show that |e x+iy − e z | ≤ |x − z + iy|e max{x,z} .
This holds for x = z, i.e., we may let x = z w.l.o.g. Then i.e., using the mean value theorem, there is a c between x, z such that Now, we recall that |z i − 1| ≤ 1/2 and |z i | ≤ 1, i.e., Hence, using the fact that σ Ai = −|σ Ai |, we find which yields the following theorem.

Theorem 1 (Lindeberg quantum central limit theorem) Let
A dynamical quantum central limit for initially uncorrelated states̺ 0 = ⊗ L i=1̺ i follows directly from this theorem. The subsequent technicalities are due to the fact that we allow for correlations in the initial state.

Main theorem
We now turn to stating the main theorem. We arrive at a statement showing local convergence to a Gaussian state in trace-norm, under the weak assumptions on the initial state specified above. We first introduce a variant of a Bernstein-Spohn blocking argument, and then make use of the above quantum Lindeberg central limit theorem, to show convergence to a Gaussian characteristic function in phase space. We then show that this implies closeness of̺ S (t) to a Gaussian state-so a maximum entropy state for given second moments.

The Bernstein-Spohn blocking argument
We divide-for a given time-the lattice into two parts: Into sites inside the causal cone, where mixing occurs, and sites outside the causal cone, the influence of which is exponentially suppressed. Inside the cone, we in turn split the region into several subsets, the role of which will soon become obvious. This choice of a blocking is derived from the "room and corridor" argument on classical lattice systems presented in Ref. [2,3]. To this end let Then Now define the set as the sites that are significantly outside the causal cone, compare Lemma 3. Define also the sets Then L = A ∪ B ∪ T -the entire lattice is covered by these sets-and S ⊂ A 1 . What is more, The sets A and B cover hence the complement of T in an alternating fashion, the first set A 1 containing the subset S of interest for which we aim at showing relaxation. The goal is now to choose b sufficiently small (such that the influence of sites in B is small due to the large separation of the B i ), but large enough such that the A i are sufficiently far apart (and therefore approximately independent) and the Lindeberg central limit theorem yields closeness of χ̺ S (t) to a Gaussian function in phase space. We will see that for sufficiently large |t|, this is achieved by the choice for which one can prove the following Lemma.

Lemma 4 (Correlations) Let Assumptions 1 and 2 hold and let t, L be such that
where µ = µ 1 /(6(µ 1 + 1)). Then, and for A, B, T as above, one has The proof is again presented in the appendix. Here, the constant µ 1 is defined as specified in the assumptions on the initial state, c 2 , c 4 as in the Lemmata following the assumptions, and is the boundary of S.

The main theorem
Let A, B, T as above and writeD A =D(α A ) and similarly for other the sets. We have (we employ the triangle inequality three times) The first term is bounded by (we write again̺ 0 = n ̺ n |ψ n ψ n | for the spectral decomposition of̺ 0 and employ the Cauchy-Schwarz inequality) To bound the second term in Eq. (54) we let I = {m ∈ AE : m even, 1 ≤ m ≤ n}, J = {1, . . . , n}\I. Then, using the fact that d Ai,Aj ≥ b for i = j, we find under and, using the fact that d Ai,Aj ≥ a for i, j ∈ I, i = j, This means that which bounds the second term in Eq. (54). Now, for x, y ∈ Ê there is a x 0 between x and y such that e x − e y = (x − y)e x0 , i.e., |e x − e y | = |x − y|e x0 ≤ |x − y|e max{x,y} .
Hence, for the third term in Eq. (54) we find (we recall that σ L , σ Ai ≤ 0) If |σ Ai | ≤ 1, the last term in Eq. (54) may be bounded using the Lindeberg-type central limit theorem and we finally have where bounds on the moments may be read off from Lemma 4. This yields the main theorem.
This theorem indeed shows convergence of characteristic functions of reduced states associated with some sublattice S to Gaussian characteristic functions in phase space. Note that the size of this sublattice may even slowly grow in time; one would still end up with a Gaussian characteristic function. Surely, in any finite system there will be recurrences, but with increasing L, these recurrence times t rec can be made arbitrarily large.
The characteristic function corresponds to the Gaussian state̺ G (t) with second moments and one has 3 Note that, if the initial correlations are translationally invariant, γ(0) and A commute, i.e., the second moments are conserved, γ(t) = γ(0). By virtue of Lemma 5, which we prove in the next section, trace-norm convergence of states is also inherited from this main theorem. We state this corollary for a sublattice S that is constant in time.

Corollary 1 (Convergence of reduced states to Gaussian states)
Let |S| and γ(0) be bounded by an absolute constant. Under the assumptions of Theorem 2 and for any ε > 0 there exists a recurrence time t rec > 0, which increases polynomially with L, and a relaxation time t relax such that for all |t| ∈ [t relax , t rec ], where̺ G (t) is the Gaussian state with characteristic function e σL/2 .
This means that in the sense of trace-norm closeness, local reduced states become Gaussian -and hence maximum entropy-states.

Convergence of quantum states
In this section, we will show that pointwise convergence of characteristic functions is inherited by trace-norm convergence on the level of quantum states. This argument generalizes the one of Ref. [27]-showing weak convergence for states from pointwise convergence of characteristic functions-and of Ref. [28] -relating trace-norm and weak convergence. The central idea is to consider two sets: On the one hand, this is the subset of phase space that supports, in a sense yet to be made precise, "most" of the characteristic function. On the other hand, this is the subspace of Hilbert space with the property that the projection of the state onto it is a positive operator that eventually has almost unit trace. One can then bound the error made when neglecting the complement of these sets and in turn relate the relevant sets in phase and state space. Inside these relevant sets, one can relate weak convergence to pointwise convergence of characteristic functions and use the equivalence of norms. Here we present an argument going beyond the results in Refs. [27,28] and use a gentle measurement Lemma of Ref. [29].
where α ∈ and L k n are associated Laguerre polynomials, i.e., p n,m (|α| 2 ) is a polynomial of degree min{m, n} + |n − m|/2 = (n + m)/2 in |α| 2 . Here {|n } n=0,1,... is the standard number basis of the Hilbert space associated with a single mode. We can make use of this in the situation of having N modes: Define for some T > 0 as the region far away in 2-norm from the origin of phase space and M := N \M as its complement. For a continuous function f : N → with |f (α)| ≤ 2 consider the bound (we write p n,m (α) = i p ni,mi (|α i | 2 ) and |n = |n 1 · · · n N ) These findings may be summarized as follows. For all N, n i , m i ∈ AE and every ǫ > 0 there is a c 0 > 0 and a compact set C ⊂ N such that for all continuous f : N → with |f (α)| ≤ 2.
This will take the role of a projection onto the relevant part of state space. For stateŝ ̺ 1 ,̺ 2 one finds (see the Appendix for a proof) What is more, using the Heisenberg-Weyl-correspondence, one has I.e., combining Eqs. (72,74), one has that for every N ∈ AE, ǫ > 0 and M ⊂ AE N there is a constant c 0 > 0 and a compact C ⊂ N such that for all N −mode states̺ 1 ,̺ 2 . Now, from the gentle measurement lemma of Ref. [29] it follows that and forn i =b † ib i one finds (we writeM = AE N \M ) where the last inequality holds for the particular choice of Hence, to summarize, for every m, N ∈ AE and ǫ > 0 there is a constant c 0 > 0 and a compact C ⊂ N such that for all N −mode states̺ 1 ,̺ 2 .

Trace-norm bounds
We summarize the above findings in the following Lemma. It shows that closeness in trace-norm is inherited by pointwise closeness of characteristic functions. The last line corresponds essentially a bound to the local energy per mode, which can in all relevant settings be bounded from above. Note that all constants on the right hand side of the inequality can be made explicit.
Lemma 5 (Trace-norm convergence from pointwise phase space convergence) For every ǫ, c 1 > 0 and N ∈ AE there is a constant c 0 > 0 and a compact C ⊂ N such that for all states̺ 1 ,̺ 2 on a collection of N sites with, for all i, Using Theorem 2, the pointwise distance between characteristic functions and ǫ may hence be subsumed in ε to yield the trace norm estimate in Corollary 1.

Summary, discussion and relationship to notions of typicality
The above theorem shows that weakly clustering states locally relax to maximum entropy states. This holds true in the strong sense of trace-norm convergence, to states that have maximum entropy for given second moments. Physically speaking, this means that locally, the system appears as if it had entirely relaxed, without the need of an external environment. Every part of the system forms the environment of the other, and merely mediated by local interactions, the system relaxes. This is a proven instance of a situation in which under non-equilibrium quenched dynamics, closed systems do locally relax. As has been mentioned before, it can hence be viewed as a proof of the "local relaxation conjecture", generalizing the findings of Ref. [7], and constituting to the knowledge of the authors a first proof of local relaxation in a situation not requiring product or Gaussian, meaning quasi-free, initial states. That is to say, the statement shows that for a large class of initial states, locally, the system quickly looks relaxed and stays like that for long times. We finally mention a potential link of the above results to the concept of "typicality": The term typicality refers-roughly speaking-to the idea that when quantum states are randomly chosen from a suitable ensemble of large-dimensional pure states, almost all realizations will share properties like reduced states of subsystems having large entropy, or being close to a Gibbs state. This effect is a manifestation of the phenomenon of concentration of measure. The simplest such situation is realized when drawing random pure state according to the Haar measure on H = d ⊗ D : Then the von-Neumann entropy of the d-dimensional subsystem will, for large D, be almost certainly be almost maximum, with any deviations being exponentially suppressed in probability. Similarly, if states are drawn from an ensemble of pure states that are eigenvectors of a Hamiltonian with eigenvalues close to a limiting point in the spectrum, defining an energy, then one can specify technical conditions under which one can prove that almost certainly, subsystems will locally look like the reduction of a Gibbs state to that subsystem [17,18]. That is to say, almost all states appear to have high local entropy. This is strictly proven only in case of finite-dimensional constituents, although it is to be expected that the same intuition carries over also to the situation of infinite-dimensional constituents. In the light of this observation, it appears plausible that starting from an "untypical initial situation", systems will eventually be driven to a "typical one" by virtue of unitary time evolution.
Needless to say, time evolution defines merely a one-dimensional manifold, parameterized by time, and from the above argument alone, there is no guarantee that the system will indeed arrive at a typical situation (although steps can be taken that show that for most times, albeit unknown ones, the system appears locally typical [19] in a non-equilibrium situation). Here-for a class of models-a significantly stronger statement is proven, in that after a short time, the system will certainly come arbitrarily close to a maximum entropy state under the constraints of second moments and will stay there. It would be interesting to take steps to combine such kinematical and dynamical approaches towards understanding equilibration in closed quantum systems.

Acknowledgements
We would like to thank T.J. Osborne, M.M. Wolf, and U. Schollwöck for discussions. This work has been supported by the EU (QAP, QESSENCE, COMPAS, MINOS), and the EURYI grant scheme.

Lieb-Robinson bound (Lemma 3)
Under periodic boundary conditions, the entries of C take the form C i,j = C i−j = C di,j , As A i,j = 0 for d i,j > 1, we find from Taylor's theorem the estimate i.e., for A ⊂ L we find from Eq. (11) that i∈A |α i | ≤ i∈A j∈S where the last inequality holds for 4e|t| ≤ d A,S . Furthermore, e 2it cos(2πk/L) e 2πikl/L =: where g(φ) = e 2it cos(φ) can be written as where J n is a Bessel function of the first kind. Hence, for |t| ≥ 1 and where, for |t| ≥ 1 and 0 ≤ l ≤ L/2, which we can assume as i.e., for L ≥ |t| 7/6 ≥ 1 (we use a bound obtained in Ref. [30]),

Moments
We will need the following bound. Let ǫ, µ > 0 and A ⊂ L. Then

Distances
For all k ∈ A i , l ∈ A j we have (we pick s ∈ S such that d l,s = d l,S ) (i − 1)(a + b) ≤ d k,S ≤ d k,s ≤ d k,l + d l,s = d k,l + d l,S < d k,l + j(a + b) − b (103) and (we pick s ∈ S such that d k,s = d k,S ) i.e., d Ai,Aj = min k∈Ai l∈Aj Furthermore d S,T = min l∈T d S,l ≥ n(a + b) ≥ τ 2 . (106)

Cardinalities
For i ∈ L\S pick s i ∈ ∂S such that d i,S = d i,si = d i,∂S . Then i.e., Furthermore,

Closeness of quantum states
We have for any states̺ 1 ,̺ 2 , treating the relevant part of state space. Within that relevant part, we can make use of