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Distances between Probability Measures

Let G be a finite quantum group and M_p(G) be the set of states on the \mathrm{C}^\ast-algebra F(G).

The algebra F(G) has an invariant state \int_G\in\mathbb{C}G=F(G)^\ast, the dual space of F(G).

Define a (bijective) map \mathcal{F}:F(G)\rightarrow \mathbb{C}G, by

\displaystyle \mathcal{F}(a)b=\int_G ba,

for a,b\in F(G).

Then, where \|\cdot\|_1^{F(G)}=\int_G|\cdot| and \|\cdot\|_\infty^{F(G)}=\|\cdot\|_{\text{op}}, define the total variation distance between states \nu,\mu\in M_p(G) by

\displaystyle \|\nu-\mu\|=\frac12 \|\mathcal{F}^{-1}(\nu-\mu)\|_1^{F(G)}.

(Quantum Total Variation Distance (QTVD))

Standard non-commutative \mathcal{L}^p machinary shows that:

\displaystyle \|\nu-\mu\|=\sup_{\phi\in F(G):\|\phi\|_\infty^{F(G)}\leq 1}\frac12|\nu(\phi)-\mu(\phi)|.

(supremum presentation)

In the classical case, using the test function \phi=2\mathbf{1}_S-\mathbf{1}_G, where S=\{\nu\geq \mu\}, we have the probabilists’ preferred definition of total variation distance:

\displaystyle \|\nu-\mu\|_{\text{TV}}=\sup_{S\subset G}|\nu(\mathbf{1}_S)-\mu(\mathbf{1}_S)|=\sup_{S\subset G}|\nu(S)-\mu(S)|.

In the classical case the set of indicator functions on the subsets of the group exhaust the set of projections in F(G), and therefore the classical total variation distance is equal to:

\displaystyle \|\nu-\mu\|_P=\sup_{p\text{ a projection}}|\nu(p)-\mu(p)|.

(Projection Distance)

In all cases the quantum total variation distance and the supremum presentation are equal. In the classical case they are equal also to the projection distance. Therefore, in the classical case, we are free to define the total variation distance by the projection distance.

Quantum Projection Distance \neq Quantum Variation Distance?

Perhaps, however, on truly quantum finite groups the projection distance could differ from the QTVD. In particular, a pair of states on a M_n(\mathbb{C}) factor of F(G) might be different in QTVD vs in projection distance (this cannot occur in the classical case as all the factors are one dimensional).

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Just back from a great workshop at Seoul National University, I am just going to use this piece to outline in a relaxed manner my key goals for my work on random walks on quantum groups for the near future.

In the very short term I want to try and get a much sharper lower bound for my random walk on the Sekine family of quantum groups. I believe the projection onto the ‘middle’ of the M_n(\mathbb{C}) might provide something of use. On mature reflection, recognising that the application of the upper bound lemma is dominated by one set of terms in particular, it should be possible to use cruder but more elegant estimates to get the same upper bound except with lighter calculations (and also a smaller \alpha — see Section 5.7).

I also want to understand how sharp (or otherwise) the order n^n convergence for the random walk on the dual of S_n is — n^n sounds awfully high. Furthermore it should be possible to get a better lower bound that what I have.

It should also be possible to redefine the quantum total variation distance as a supremum over projections \sim subsets via G \supset S\leftrightarrow \mathbf{1}_S. If I can show that for a positive linear functional \rho that |\rho(a)|\leq \rho(|a|) then using these ideas I can. More on this soon hopefully. No, this approach won’t work. (I have since completed this objective with some help: see here).

The next thing I might like to do is look at a random walk on the Sekine quantum groups with an n-dependent driving probability and see if I can detect the cut-off phenomenon (Chapter 4). This will need good lower bounds for k\ll t_n, some cut-off time.

Going back to the start, the classical problem began around 1904 with the question of Markov:

Which card shuffles mix up a deck of cards and cause it to ‘go random’?

For example, the perfect riffle shuffle does not mix up the cards at all while a riffle shuffle done by an amateur will.

In the context of random walks on classical groups this question is answered by the Ergodic Theorem 1.3.2: when the driving probability is not concentrated on a subgroup (irreducibility) nor the coset of a normal subgroup (aperiodicity).

Necessary and sufficient conditions on the driving probability \nu\in M_p(\mathbb{G}) for the random walk on a quantum group to converge to random are required. It is expected that the conditions may be more difficult than the classical case. However, it may be possible to use Diaconis-Van Daele theory to get some results in this direction. It should be possible to completely analyse some examples (such as the Kac-Paljutkin quantum group of order 8).

This will involve a study of subgroups of quantum groups as well as normal quantum subgroups.

It should be straightforward to extend the Upper Bound Lemma (Lemma 5.3.8) to the case of compact Kac algebras. Once that is done I will want to look at quantum generalisations of ‘natural’ random walks and shuffles.

I intend also to put the PhD thesis on the Arxiv. After this I have a number of options as regard to publishing what I have or maybe waiting a little while until I solve the above problems — this will all depend on how my further study progresses.

 

Slides of a talk given at the Topological Quantum Groups and Harmonic Analysis workshop at Seoul National University, May 2017.

Abstract A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis & Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. These ideas are generalised to the case of finite quantum groups.

After a long time I have finally completed my PhD studies when I handed in my hardbound thesis (a copy of which you can see here).

It was a very long road but thankfully now the pressure is lifted and I can enjoy my study of quantum groups and random walks thereon for many years to come.

I have finally finished the first draft of my PhD thesis. My advisor Dr Stephen Wills is presently reading through it and will get back to me with his comments in the next few weeks. The project was successful in that I managed to prove the Diaconis-Shahshahani Upper Bound Lemma for finite quantum groups… how successful my application of the Lemma to concrete examples is probably open to debate. First draft of abstract and introduction — without references — below the fold.

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Let \mathbb{G} be a finite quantum group described by A=\mathcal{C}(\mathbb{G}) with an involutive antipode (I know this is true is the commutative or cocommutative case. I am not sure at this point how restrictive it is in general. The compact matrix quantum groups have this property so it isn’t a terrible restriction.) S^2=I_A. Under the assumption of finiteness, there is a unique Haar state, h:A\rightarrow \mathbb{C} on A.

Representation Theory

A representation of \mathbb{G} is a linear map \kappa:V\rightarrow V\otimes A that satisfies

\left(\kappa\otimes I_A\right)\circ\kappa =\left(I_V\otimes \Delta\right)\circ \kappa\text{\qquad and \qquad}\left(I_V\otimes\varepsilon\right)\circ \kappa=I_V.

The dimension of \kappa is given by \dim\,V. If V has basis \{e_i\} then we can define the matrix elements of \kappa by

\displaystyle\kappa\left(e_j\right)=\sum_i e_i\otimes\rho_{ij}.

One property of these that we will use it that \varepsilon\left(\rho_{ij}\right)=\delta_{i,j}.

Two representations \kappa_1:V_1\rightarrow V_1\otimes A and \kappa_2:V_2\rightarrow V_2\otimes A are said to be equivalent, \kappa_1\equiv \kappa_2, if there is an invertible intertwiner between them. An intertwiner between \kappa_1 and \kappa_2 is a map T\in L\left(V_1,V_2\right) such that

\displaystyle\kappa_2\circ T=\left(T\otimes I_A\right)\circ \kappa_1.

We can show that every representation is equivalent to a unitary representation.

Timmermann shows that if \{\kappa_\alpha\}_{\alpha} is a maximal family of pairwise inequivalent irreducible representation that \{\rho_{ij}^\alpha\}_{\alpha,i,j} is a basis of A. When we refer to “the matrix elements” we always refer to such a family. We define the span of \{\rho_{ij}\} as \mathcal{C}\left(\kappa\right), the space of matrix elements of \kappa.

Given a representation \kappa, we define its conjugate, \overline{\kappa}:\overline{V}\rightarrow\overline{V}\otimes A, where \overline{V} is the conjugate vector space of V, by

\displaystyle\overline{\kappa}\left(\bar{e_j}\right)=\sum_i \bar{e_i}\otimes\rho_{ij}^*,

so that the matrix elements of \overline{\kappa} are \{\rho_{ij}^*\}.

Timmermann shows that the matrix elements have the following orthogonality relations:

  • If \alpha and \beta are inequivalent then h\left(a^*b\right)=0, for all a\in \mathcal{C}\left(\kappa_\alpha\right) and b\in\mathcal{C}\left(\kappa_\beta\right).
  • If \kappa is such that the conjugate, \overline{\kappa}, is equivalent to a unitary matrix (this is the case in the finite dimensional case), then we have

\displaystyle h\left(\rho_{ij}^*\rho_{kl}\right)=\frac{\delta_{i,k}\delta_{j,l}}{d_\alpha}.

This second relation is more complicated without the S^2=I_A assumption and refers to the entries and trace of an intertwiner F from \kappa to the coreprepresention with matrix elements \{S^2\left(\rho_{ij}\right)\}. If S^2=I_A, then this intertwiner is simply the identity on V and so the the entries \left[F\right]_{ij}=\delta_{i,j} and the trace is d=\dim V.

Denote by \text{Irr}(\mathbb{G}) the set of unitary equivalence classes of irreducible unitary representations of \mathbb{G}. For each \alpha\in\text{Irr}(\mathbb{G}), let \kappa_\alpha:V_{\alpha}\rightarrow V_{\alpha}\otimes A be a representative of the class \alpha where V_\alpha is the finite dimensional vector space on which \kappa_\alpha acts.

Diaconis-Van Daele Fourier Theory

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In Group Representations in Probability and Statistics, Diaconis presents his celebrated Upper Bound Lemma for a random walk on a finite  group G driven by \nu\in M_p(G). It states that

\displaystyle \|\nu^{\star k}-\pi\|^2\leq \frac{1}{4}\sum d_\rho \text{Tr }(\widehat{\nu(\rho)}^k(\widehat{\nu(\rho)}^\ast)^k),

where the sum is over all non-trivial irreducible representations of G.

In this post, we begin this study by looking a the (co)-representations of a quantum group A. The first thing to do is to write down a satisfactory definition of a representation of a group which we can quantise later. In the chapter on Diaconi-Fourier Theory here we defined a representation of a group as a group homomorphism

\rho:G\rightarrow GL(V)

While this was perfectly adequate for when we are working with finite groups, it might not be as transparently quantisable. Instead we define a representation of a group as an action

\Phi:V\times G\rightarrow V.

such that the map \rho(g):V\rightarrow V\rho(g)x=\Phi(x,g) is linear.

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Let G be a group and let A:=C^*(G)  be the C*-algebra of the group G. This is a C*-algebra whose elements are complex-valued functions on the group G. We define operations on A in the ordinary way save for multiplication

\displaystyle (fg)(s)=\sum_{t\in G}f(t)g(t^{-1}s),

and the adjoint f^*(s)=\overline{f(s^{-1})}. Note that the above multiplication is the same as defining \delta_s\delta_t=\delta_{st} and extending via linearity. Thence A is abelian if and only if G is.

To give the structure of a quantum group we define the following linear maps:

\Delta:A\rightarrow A\otimes A\Delta(\delta_s)=\delta_s\otimes\delta_s.

\displaystyle \varepsilon:A\rightarrow \mathbb{C}\varepsilon(\delta_s)=1.

S:A\rightarrow AS(\delta_s)=\delta_{s^{-1}}.

The functional h:A\rightarrow \mathbb{C} defined by h=\mathbf{1}_{\{\delta_e\}} is the Haar state on A. It is very easy to write down the j_n:

\displaystyle j_n(\delta_s)=\bigotimes_{i=0}^n\delta_s.

To do probability theory consider states \varepsilon,\,\phi on A and form the product state:

\displaystyle \Psi=\varepsilon\otimes\bigotimes_{i=1}^\infty \phi.

Whenever \phi is a state of A such that \phi(\delta_s)=1 implies that s=e, then the distribution of the random variables j_n converges to h.

At the moment we will use the one-norm to measure the distance to stationary:

d(\Psi\circ j_n,h)=\|\Psi\circ j_n-h\|_1.

A quick calculation shows that:

d(\Psi\circ j_n,h)=\sum_{s\neq e}|\phi(\delta_s)|^n.

When, for example, \phi(\delta_s)=2/m^2 when s are transpositions in S_m, then we have

d(\Psi\circ j_n,h)={m\choose 2}\left(\frac{2}{m^2}\right)^n.

Taken from Random Walks on Finite Quantum Groups by Franz & Gohm.

Theorem

Let \phi be a state on a finite quantum group A. Then the Cesaro mean

\displaystyle \phi_n=\frac{1}{n}\sum_{k=1}^n\phi^{\star n}n\in\mathbb{N}

converges to an idempotent state on A, i.e. to a state \pi such that \pi\star\pi=\pi.

Proof : Let \phi' be an accumulation point of \{\phi_n\}_{n\geq0}, this exists since the states on A form a compact set. We have

\|\phi_n-\phi\star \phi_n\|=\frac{1}{n}\|\phi-\phi^{n+1}\|\leq \frac{2}{n}.

I have no idea where the equality comes from.

Choose sequence \{n_k\}_{k\geq 0} such that \phi_{n_k}\rightarrow \phi', we get \phi\star\phi'=\phi' and similarly \phi'\star \phi=\phi'. By linearity this implies \phi_n\star\phi'=\phi'=\phi'\star \phi_n. If \phi'' is another accumulation point of \{\phi_n\}_{n\geq 0} and \{m_{\ell}\}_{\ell\geq 0} a sequence such that \phi_{m_\ell}\rightarrow\phi'', then we get \phi''\star\phi'=\phi'=\phi'\star\phi'' and thus \phi'=\phi'' by symmetry (??). Therefore the sequence \{\phi_n\}_{n\geq0} has a unique accumulation point, i.e. it converges \bullet

Remark

If \phi is faithful, then the Cesaro limit \pi is the Haar state on A (prove this).

Remark

Due to cyclicity the sequence \{\phi^{\star n}\}_{n\geq 0} does not converge in general. Take, for example, the state \phi=\eta_2 (p.28) on the Kac-Paljutkin quantum group A, then we have

\eta_2^{\star n}=\left\{\begin{array}{ccc}\eta_2&\text{if}& n\text{ is odd}\\ \varepsilon&\text{if}&n\text{ is even}\end{array}\right.,

but

\displaystyle \lim_{n\rightarrow\infty}\frac1{n}\sum_{k=1}^n\eta_2^{\star k}=\frac{\varepsilon+\eta_2}2.

Taken from Random Walks on Finite Quantum Groups by Franz & Gohm

In this section we will show how one can recover a classical Markov chain from a quantum Markov chain. We will apply a folklore theorem that says one gets a classical Markov process, if a quantum Markov process can be restricted to a commutative algebra.

We might like to do more. This result recovers a Markov chain — if the quantum process is in fact a random walk on a finite quantum group can we recover the group, the transition probabilities (yes), the driving probability measure?? 

Conjecture

If we restrict a random walk on a finite quantum group to a commutative subalgebra we can recover a random walk on a finite group.

For random walks on quantum groups we have the following result.

Theorem 6.1

Let A be a finite quantum group \{j_n\}_{n\geq 0} a random walk on a finite dimensional A-comodule algebra B, and B_0 a unital abelian sub-*-algebra of B. The algebra B_0 is isomorphic to the algebra of functions on a finite set, say B_0\cong F(X) where X={1,\dots,d}.

If the transition operator T_\phi of \{j_n\}_{n\geq 0} leaves B_0 invariant, then there exists a classical Markov chain \{\xi_n\}_{n\geq 0} with values in X, whose probabilities can be computed as time-ordered moments of \{j_n\}_{n\geq 0}, i.e.

P(\xi_0=i_0,\dots,\xi_\ell=i_\ell)=\Psi\left(j_0\left(\mathbf{1}_{\{i_0\}}\right)\cdots j_\ell\left(\mathbf{1}_{\{i_\ell\}}\right)\right)

for all \ell\geq 0 and i_0,\dots,i_\ell\in X.

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