Quantum recurrence and matrix Schur functions

Quantum recurrenceand

matrix Schur functions

Luis Velazquez

Departamento de Matematica Aplicada & IUMA, U Zaragoza

Joint works with:

Reinhard Werner, Albert Werner

Institut fur Theoretische Physik, Leibniz Universitat Hannover

Jean Bourgain

Institute for Advanced Study, Princeton

Alberto Grunbaum, Jon Wilkening

Department of Mathematics, UC Berkeley

Recurrence and expected return time

A key concept in the study of random processes is the idea of recurrence.

A random process is recurrent if it returns to the initial state with probability one.

Otherwise it is called transient.

In principle the recurrence of a random process could depend on the initial state,

so we should talk about recurrent or transient states for a given random system.

Recurrent states can have different return speeds, which can be measured by the

expected return time.

Recurrent states can have a finite or infinite expected return time. The last ones

are the recurrent states which are closest to the transience.

Random Walks: Polya recurrence

RW = discrete classical random process discrete-time Markov chain

on a countable state space X

• Transition matrix P = (Px,y)x,y∈X Px,y ≥ 0∑

y∈X Px,y = 1

Px,y = probability of transition x→ y in one step

• Return probabilityin n steps

pn = p(x)n =∑

x1,...,xn−1

Px,x1Px1,x2 · · ·Pxn−1,x = (P n)x,x

•First timereturn probabilityin n steps

qn = q(x)n =∑

x1,...,xn−1 6=x

Px,x1Px1,x2 · · ·Pxn−1,x

q =∞∑n=1

qn Total return probability (Polya number)

x is recurrent ⇔ q = 1

τ =∞∑n=1

nqn Expected return time

Random Walks: Polya recurrence

The study of recurrence is greatly aided by the use of generating functions

p(z) =∞∑n=0

pnzn q(z) =

∞∑n=1

qnzn

A key result relates the first return prob. qn to the more accesible return prob. pn

q(z) = 1−1

p(z)Renewalequation

This allows us to rewrite the recurrence notions in terms of pn instead of qn

q =∞∑n=1

qn = q(1) = 1−1

p(1)Polya number

x is recurrent ⇔∞∑n=0

pn =∞

τ =∞∑n=1

nqn = q′(1) = limz→1

p′(z)

p(z)2Expected return time

Example: Translation invariant birth-death process

X = Z · · · •x−1

d←− •x

b−→ •x+1· · · b, d > 0

b+ d = 1

In this case explicit results are available, giving for any state x

p(z) =1√

1− 4bdz2q = 1−

√1− 4bd

The Pólya numberqas a function of

the probability transitionb

0.2 0.4 0.6 0.8 1.0b

0.2

0.4

0.6

0.8

1.0

q

All the states are recurrent if b = d =1

2and transient otherwise

Random Walks: Spectral characterization of recurrence

For reversible Markov chains the transition matrix P is symmetrizablespectral theory−−−−−−−−−−→

For any state x ∈ X there exists a spectral measure mx on [−1,1] such that

n-th moment

∫ 1

−1

tn dmx(t) = (P n)x,x return prob. of x in n steps

This allows us the use of spectral techniques for RW, but also provides a link to

the machinery of orthogonal polynomials on R.

How is the recurrence of a state x codified in its spectral measure m = mx?

x is recurrent ⇔∫ 1

−1

dm(t)

1− t=∞

τ =1

m({1})Expected return time

Finite expected return time ⇔ 1 is a mass point

Quantum Walks: Why?

• What is a QW? Simplified model of a quantum system in evolution

• Why are QW important? Simplest setting to study the quantum behaviour:

◦ Superposition: quantum systems can be simultaneously in different states

◦ Parallelism: quantum systems may spread much faster than classical ones

◦ Entanglement: quantum systems arbitrarily far from each other can be linked

Quantum Walks: Why?






The exponentially increasing interest in QW rests on the discovery of astonishing

applications to our daily life of this “magic” quantum behaviour of nature:super-phenomena (photosynthesis extreme efficiency), quantum computation,quantum cryptography, quantum teleportation, . . .

Quantum Walks: Why?






The exponentially increasing interest in QW rests on the discovery of astonishing

applications to our daily life of this “magic” quantum behaviour of nature:super-phenomena (photosynthesis extreme efficiency), quantum computation,quantum cryptography, quantum teleportation, . . .

RW QW

Randomness Ignorance Intrinsic

States @ superposition ∃ superposition

Evolution Markov process Unitary process

Measurement Does not disturb evolution Projection which alter evolution

Path counting Path counting

Methods Fourier analysis Fourier analysis

Spectral self-adjoint - OP line Spectral unitary - OP circle

Quantum Walks

QW = discrete quantum evolution discrete-time unitary process on a complex

Hilbert space H with inner product 〈·|·〉

Quantum Walks



• States

If the elements x ∈ X are all the possible outcomes of a measurement

the measurable states form a countable orthonormal basis {φx}x∈X of H

Quantum Walks



• States



A system can be in a complex superposition ψ =∑

x∈X cxφx and then

any measurement of X yields a random result with Prob(x) = |cx|2

Quantum Walks



• States






Consequences

I ‖ψ‖2 =∑

x∈X |cx|2 = Prob(X) = 1 ⇒ ψ must be unitary!

Quantum Walks



• States






Consequences

I ‖ψ‖2 =∑


I ψ ≡ eiζψ same state ⇒ State space = complex projective CP(H)

Quantum Walks



• States






Consequences

I ‖ψ‖2 =∑


I ψ ≡ eiζψ same state ⇒ State space = complex projective CP(H)

The non trivial geometry of the state space has measurable consequences!

Quantum Walks

• Measurement = Orthogonal projector

If a system is in a state ψ, the measurement of a concrete value x can result in:

I The value x is found: Then ψcollapse−−−−−→ φx

Quantum Walks




I The value x is NOT found: Then ψcollapse−−−−−→ projection of ψ onto φ⊥x

Quantum Walks





cx = 〈φx|ψ〉 is called the amplitude for the transition ψ → φx

Prob(ψ → φx) = |〈φx|ψ〉|2

Quantum Walks







• Evolution = Unitary operator

The one-step evolution is given by ψ → Uψ where U is a unitary operator on HUnitarity ensures conservation of total probability ‖ψ‖2 = Prob(X) = 1

Quantum Walks









〈φx|Unψ〉 is the amplitude for the transition ψ → φx in n steps

Prob(ψn steps−−−−−→ φx) = |〈φx|Unψ〉|2

Quantum Walks









〈φx|Unψ〉 is the amplitude for the transition ψ → φx in n steps

Prob(ψn steps−−−−−→ φx) = |〈φx|Unψ〉|2

Prob = |amplitude|2

Quantum Walks: Recurrence

Given a unitary step U and an initial state ψ we search for the return probability to ψ

• Return amplitudein n steps

µψn = 〈ψ|Unψ〉 Prob(ψn steps−−−−−→ ψ) = |µψn |2





However, to avoid overcounting, the total return probability needs the first time ones:

Q = projector onto ψ⊥ ≡ collapse for NO measurement of ψ







ψstep 1−−−→ Uψ








measurement−−−−−−−−→NO ψ

QUψ









QUψstep 2−−−→ U(QU)ψ











(QU)2ψ











(QU)2ψstep 3−−−→ · · ·











(QU)2ψstep 3−−−→ · · ·

•First timereturn amplitudein n steps

aψn = 〈ψ|U(QU)n−1ψ〉 Prob(ψn steps−−−−−→1st time

ψ) = |aψn |2











(QU)2ψstep 3−−−→ · · ·



ψ) = |aψn |2

Rψ =∞∑n=1

|aψn |2 Total return probability

ψ is recurrent ⇔ Rψ = 1

τψ =∞∑n=1

n|aψn |2 Expected return time

Example: Translation invariant coined walk

Consider a particle with spin in an infinite 1D lattice with the one-step amplitudes

X = Z× {↑, ↓} · · · •↓

c21←−c22

↑•↓

c11−→c12

↑• · · ·

x−1 x x+1

C =

(c11 c12

c21 c22

)unitary coin

Example: Translation invariant coined walk

Consider a particle with spin in an infinite 1D lattice with the one-step amplitudes

X = Z× {↑, ↓} · · · •↓

c21←−c22

↑•↓

c11−→c12

↑• · · ·

x−1 x x+1

C =

(c11 c12

c21 c22

)unitary coin

Explicit calculations are possible, giving for any basis state ψ = |x, ↑〉, |x, ↓〉

Rψ =(1 + 2ρ2)ρa+ (1− 4ρ2) arcsin a

π2a4

[a = |c12|

ρ =√

1− a2

]

The return probabilityRΨ

as a function ofthe probability transition Èc12È2

0.2 0.4 0.6 0.8 1.0a2

0.2

0.4

0.6

0.8

1.0

RΨ

In contrast to the classical case, no recurrent states appear

Quantum Walks: Spectral characterization of recurrence

In contrast to the classical case the transition matrix U is always unitaryspectral theory−−−−−−−−−−→

For any state ψ ∈ H there exists a spectral measure µψ on T ≡ |z| = 1 such that

n-th moment

∫Ttn dµψ(t) = 〈ψ|Unψ〉 = µψn return amplitude to ψ in n steps




n-th moment


This leads to a fruitful link between QW and OP on T, but also opens the question:

How is the recurrence of a state ψ codified in its spectral measure µψ?




n-th moment




The answer comes from the study of the generating functions

Sψ(z) =∞∑n=0

µψnzn =

∫T

dµψ(t)

1− tzStieltjes function of µψ

gψ(z) =∞∑n=1

aψnzn ???




n-th moment




The answer comes from the study of the generating functions

Sψ(z) =∞∑n=0

µψnzn =

∫T

dµψ(t)

1− tzStieltjes function of µψ

gψ(z) =∞∑n=1

aψnzn = zfψ(z) fψ(z) = Schur function of µψ!

Quantum Walks and Schur functions

A Schur function is a function f : D→ D analytic on the unit disk D ≡ |z| < 1



They are in one-to-one correspondence with the measures µ on the unit circle T via

f(z) =1

z

F(z)− 1

F(z) + 1F(z) =

∫T

t+ z

t− zdµ(t)




f(z) =1

z

F(z)− 1

F(z) + 1F(z) =

∫T

t+ z

t− zdµ(t)

This class of functions is a classical ingredient in complex analysis and OP on T

The NEW result is the discovery of the quantum role of Schur functions:

Schur Taylor coefficients = First time return amplitudes of QW




f(z) =1

z

F(z)− 1

F(z) + 1F(z) =

∫T

t+ z

t− zdµ(t)

This class of functions is a classical ingredient in complex analysis and OP on T

The NEW result is the discovery of the quantum role of Schur functions:

Schur Taylor coefficients = First time return amplitudes of QW

The identification of gψ(z) = zfψ(z) as a Schur function is a consequence of

gψ(z) = 1−1

Sψ(z)

Quantumrenewal equation

Like the classical one!But for amplitudes

instead of probabilities

and is KEY for the spectral analysis of quantum recurrence


Denoting 〈f, g〉 =∫ 2π

0

f(eiθ) g(eiθ)dθ

2πand ‖ · ‖ the corresponding norm:

Unitary step U , state ψ −→ Measure µψ −→ Schur function fψ

gψ(z) = zfψ(z) =∞∑n=1

aψnzn Generating function of

first time return amplitudes aψn

Rψ =∞∑n=1

|aψn |2 = ‖gψ‖2 = ‖fψ‖2 Total return probability

ψ is recurrent ⇔ ‖fψ‖ = 1 ⇔ |fψ| = 1 a.e. in T ⇔ µψ is singular

(fψ is inner)

τψ =∞∑n=1

n|aψn |2 =1

ilimr→1〈gψ(reiθ), ∂θgψ(reiθ)〉 = ] supp µψ

Expected

return time


CONSEQUENCES

• The recurrent states constitute the singular subspace of the unitary step U .Thus, an a.c. spectrum for U is equivalent to the absence of recurrent states.


CONSEQUENCES


τψ = ] supp µψ


CONSEQUENCES


τψ = ] supp µψ

• τψ <∞ iff µψ is a finite collection of mass points. Hence, the states with a finite

expected return time are those spanned by finitely many eigenvectors of U .In particular, a continuous spectrum for U means that no state can return in afinite expected time.


CONSEQUENCES


τψ = ] supp µψ



• In contrast to the classical case, the expected return time is quantized.Despite the epithet “quantum” of these walks, this is totally unexpected a prioribecause time does not come from an eigenvalue problem in Quantum Mechanics.


CONSEQUENCES


τψ = ] supp µψ




• The above quantization has a topological reason: τψ <∞ iff gψ is inner, and then

τψ becomes the winding number of the boundary values gψ(eiθ): T→ T.


CONSEQUENCES


τψ = ] supp µψ




• The above quantization has a topological reason: τψ <∞ iff gψ is inner, and then

τψ becomes the winding number of the boundary values gψ(eiθ): T→ T.

• Quantum recurrence paradox 1

First time return probabilities can be higher than return probabilities !!!Nothing prevents |aψn |2 > |µψn |2. Indeed, there exist QW with a state ψ such thataψn 6= 0 for all n, but µψn = 0 for n ≥ 3.

Quantum Walks: Subspace recurrence

Example Given a state ψ = α|x, ↑〉+ β|x, ↓〉 of a coinedwalk we can be interested in the return probability tothe whole site x (instead of to the single state ψ), i.e.the return probability to the subspace span{|x, ↑〉, |x, ↓〉}

· · ·↑•↓←→

↑•↓←→

↑•↓· · ·

x−1 x x+1

These kind of questions, so natural as the previous ones, lead to the notion ofsubspace recurrence as a generalization of the state recurrence already analyzed.



· · ·↑•↓←→

↑•↓←→

↑•↓· · ·

x−1 x x+1


Now we start with a unitary step U and a finite-dimensional subspace V of H.Given any initial state ψ ∈ V , we search for its return probability to V .

State recurrence corresponds to the particular case dim V = 1.



· · ·↑•↓←→

↑•↓←→

↑•↓· · ·

x−1 x x+1


Now we start with a unitary step U and a finite-dimensional subspace V of H.Given any initial state ψ ∈ V , we search for its return probability to V .

State recurrence corresponds to the particular case dim V = 1.

A precise expression of the total return probability to V can be obtained followingan approach similar to the case dim V = 1. This simply requires the substitutions

Projector onto ψ −→ P = Projector onto V

Projector onto ψ⊥ −→ Q = Projector onto V ⊥

which account for the possible collapses when checking the return to V (instead of ψ).


Remind that for state recurrence





ψ) = |aψn |2


In the case of subspace recurrence


µVn = PUnP Prob(ψn steps−−−−−→ V ) = ‖µVnψ‖2


aVn = PU(QU)n−1P Prob(ψn steps−−−−−→1st time

V ) = ‖aVnψ‖2







V ) = ‖aVnψ‖2

Now µVn , aVn are operators on V that we identify with matrices choosing a basis of V







V ) = ‖aVnψ‖2


RV (ψ) =∞∑n=1

‖aVnψ‖2 Total V -return probability

ψ is V -recurrent ⇔ RV (ψ) = 1

τV (ψ) =∞∑n=1

n‖aVnψ‖2 Expected V -return time







V ) = ‖aVnψ‖2


RV (ψ) =∞∑n=1

‖aVnψ‖2 Total V -return probability

ψ is V -recurrent ⇔ RV (ψ) = 1

τV (ψ) =∞∑n=1

n‖aVnψ‖2 Expected V -return time

We can rewrite RV (ψ) = 〈ψ|RV ψ〉 as a quadratic form in V giving by

RV =∞∑n=1

(aVn )†aVn

Example: Site recurrence in a translation invariant coined walk

Consider the site subspace Vx = span{|x, ↑〉, |x, ↓〉} in the previous 1D model

X = Z× {↑, ↓} · · ·↑•↓←→

↑•↓←→

↑•↓· · ·

x−1 x x+1

C =

(c11 c12

c21 c22

)unitary coin

Example: Site recurrence in a translation invariant coined walk

Consider the site subspace Vx = span{|x, ↑〉, |x, ↓〉} in the previous 1D model

X = Z× {↑, ↓} · · ·↑•↓←→

↑•↓←→

↑•↓· · ·

x−1 x x+1

C =

(c11 c12

c21 c22

)unitary coin

RVx =

(R 00 R

)is a scalar matrix with R =

ρa+ (1− 2ρ2) arcsin aπ2a2

[a = |c12|

ρ =√

1− a2

]

R RΨ

Comparison betweensite return probabilityR

and state return probabilityRΨ

for the basis states

0.2 0.4 0.6 0.8 1.0a2

0.2

0.4

0.6

0.8

1.0

Comparison between state and site recurrence for ψ = |x, ↑〉, |x, ↓〉Note that Rψ ≤ R = RVx(ψ) as one should naively expect

Quantum Walks and matrix Schur functions

The spectral characterization of subspace recurrence requires the association of amatrix measure and a matrix Schur function with each subspace.



As in the case of a single state, the unitarity of U ensures for any subspace V ⊂ Hthe existence of a spectral matrix measure µV on T such that

n-th moment

∫Ttn dµV (t) = PUnP = µVn return (matrix) amplitude to V in n steps




n-th moment


This allows to associate with V a matrix Schur function fV

fV (z) = z−1(F V (z)− I)(F V (z) + I)−1 F V (z) =

∫T

t+ z

t− zdµV (t)

which is analytic on the unit disk D and satisfies ‖fV ‖ ≤ 1 in D.




n-th moment


This allows to associate with V a matrix Schur function fV

fV (z) = z−1(F V (z)− I)(F V (z) + I)−1 F V (z) =

∫T

t+ z

t− zdµV (t)

which is analytic on the unit disk D and satisfies ‖fV ‖ ≤ 1 in D.

The theory of matrix Schur functions, linked to the matrix version of OP on T, hasexperienced a strong development influenced by the needs of electrical engineering,signal transmission and processing, prediction theory for stochastic processes, . . .

A list to which we should add from now the issue of recurrence in QW:Matrix Schur functions are the math objects which best codify subspace recurrence.

Quantum Walks: Spectral characterization of subspace recurrence

Denoting 〈〈f , g〉〉 =∫ 2π

0

f(eiθ)†g(eiθ)dθ

2πand |‖ · |‖ the corresponding matrix “norm”:

Unitary step U , subspace V −→ Matrix measure µV −→ Matrix Schur function fV

gV (z) = zf †V (z) =∞∑n=1

aVn zn Generating function of first time

return matrix amplitudes aVn

RV = |‖gV |‖2 = |‖f †V |‖2 RV (ψ) = 〈ψ|RV ψ〉

Total V -return

probability

Any ψ ∈ V is

V -recurrent⇔ |‖fV |‖ = 1 ⇔ fV is unitary a.e. in T ⇔ µV is singular

(fψ is inner)

τ rV =

1

i〈〈gV (reiθ), ∂θgV (reiθ)〉〉 τV (ψ) = lim

r→1〈ψ|τ r

V ψ〉Expected

V -return time

τV (ψ) <∞ for all ψ ∈ V ⇔ µV is a finite combination of mass points

Quantum Walks: State recurrence ←→ Subspace recurrence

Despite the previous analogies, there are many differences and unexpected relations

• Recurrent states need a singular subspace for U

• NO, unless V -recurrent states cover all of V





• States returning in a finite expected time need eigenvectors of U

• NO, unless the states returning to V in a finite expected time cover all of V







When the the expected return time is finite:

• τψ = winding number of gψ(eiθ) is an INTEGER TOPOLOGICAL INVARIANT

• NO. However,









• NO. However,

I τV (ψ) is a GEOMETRICAL INVARIANT of the curve gV (eiθ)ψ which reflects

the non-trivial geometry of the projective space of states (Berry’s phase)









• NO. However,

I τV (ψ) is a GEOMETRICAL INVARIANT of the curve gV (eiθ)ψ which reflects

the non-trivial geometry of the projective space of states (Berry’s phase)

I The V -AVERAGE of τV (ψ) is linked to a true TOPOLOGICAL INVARIANT∫V

τV (ψ) dψ =winding number of det gV (e

iθ)

dim V(RATIONAL NUMBER)


• Quantum recurrence paradox 2

State return probabilities can be higher than subspace return probabilities !!!

state return probabilitysite return probability

0 0.25 0.5 0.75

0.6

0.7

0.8

0.9

1

The above figure shows an example of this kind of phenomenon in the case ofa 1D coined walk.

It represents as a function of t the state and site return probability of the statesψ(t) = cos t|x, ↑〉+ sin t|x, ↓〉 lying in the same site x.

We can see that the state return probability (in blue) is occasionally bigger thanthe site return probability (in red).

statedim 2dim 3site

0 0.25 0.5 0.75

0.2

0.24

0.28

0.32

0.36

0.4

0.44

statedim 2dim 3site

0 0.25 0.5 0.75

0.2

0.24

0.28

0.32

0.36

0.4

0.44

Figure 1: Similar figures for a 2D QW in a square lattice. They compare, for a certaincurve of states ψ(t), the return probability to some nested subspaces of dimension 1 (thestate), 2, 3, and 4 (the site). The return probability does not necessarily increases whenenlarging the subspace.

statedim 2site

0 0.25 0.5 0.75

0.1

0.2

0.3

0.4

statedim 2site

0 0.25 0.5 0.75

0.1

0.2

0.3

0.4

Figure 2: The QW lives now in a 2D hexagonal lattice. The figures compare the returnprobability of a curve of states to some nested subspaces of dimension 1 (the state),2 and 3 (the site). The relation between the cases of dimension 1 and 2 is dramaticbecause it is most of the times the opposite of what one should naively expect.

Quantum Walks: Comments on recurrence

τψ INTEGER or INFINITE

Quantum recurrence paradoxes

Experimental validation?





∫V

τV (ψ) dψ RATIONAL or INFINITE is equivalent to

∞∑n=1

nTr[PU(QU)n−1P ] INTEGER or INFINITE

for any unitary U and orthogonal projectors P , Q = I − P

New result in Operator Theory?





∫V

τV (ψ) dψ RATIONAL or INFINITE is equivalent to

∞∑n=1

nTr[PU(QU)n−1P ] INTEGER or INFINITE

for any unitary U and orthogonal projectors P , Q = I − P

New result in Operator Theory?

Dirac equation

Continuous time version?

Quantum Walks: Returns to Schur and OP theory

Schur / OP −−−−−−−−→ QW


Schur / OP −−−−−−−−→ QW

Schur / OP?←−−−−−−−− QW


Schur / OP −−−−−−−−→ QW

Schur / OP?←−−−−−−−− QW

KHRUSHCHEV ←−−−−−−− FEYNMANformulas diagrams


Schur / OP −−−−−−−−→ QW

Schur / OP?←−−−−−−−− QW

KHRUSHCHEV ←−−−−−−− FEYNMANformulas diagrams

ILAS Meeting, Providence, RI, USA

June 3-7, 2013

Quantum recurrence and matrix Schur functions

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