Quantum Walk on Spin Networks by Marcelo Amaral, Raymond Aschheim and Klee Irwin

Quantum walk onspin network

(arXiv:1602.07653v1)

M M Amaral,Raymond

Aschheim and KleeIrwin

Introduction

Particle interactingLQG

Quantum walk

EntanglementEntropy

A model of walkerpositiontopologicallyencoded on spinnetwork

Application toblack hole

Discussion

Quantum walk on spin network(arXiv:1602.07653v1)

M M Amaral, Raymond Aschheim and Klee Irwin

Fourth International Conference on the Nature and Ontology of Spacetime

May 31, 2016


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Motivations

→ Feynman path integral quantizationQuantum (transition amplitudes)

W =

∫D[]e

ihS (1)

→ Feynman path integral discretization


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Motivations

→ Feynman checkerboard → quantum random walk

→ It from bit→ From an ontological viewpoint we will see that dynamicsand mass emerge from the spin network topology, and thatquantum walk implements it.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Spin network

→ Spin network graph Γ = (V (Γ), L(Γ)),with V (Γ)→ v1, v2..., verticesand L(Γ)→ l1, l2..., ({12 , 1,

32 ...}) links

→ For the particle state space the relevant contributioncomes from the position on the vertices of graph Γ.

→ Hamiltonian operator on a fixed spin network 1

〈ψ|H|ψ〉 = κ∑l

jl(jl + 1) (ψ(lf )− ψ(li ))2 , (2)

lf are the final points of the link l and li the initial points

1C. Rovelli; F. Vidotto, Phys. Rev. D 2010, 81, arXiv:0905.2983v2.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Transition probabilities

→ If the link l starts at vertex m and ends at vertex n wecan change the notation, relabelling the color of this link lbetween m and n as jmn and the wave function on the endpoints as ψ(vm), ψ(vn). Now, for H, we have

〈ψ|H|ψ〉 = κ∑l

jmn(jmn + 1) (ψ(vn)− ψ(vm))2 , (3)

→ Random walk associated with (3) have transitionprobabilities 2

Pmn =jmn(jmn + 1)∑kjmk(jmk + 1)

. (4)

2J. M. Garcia-Islas, ARXIV gr-qc/1411.4383.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Quantum walk

→ Consider the Nv -dimensional Hilbert spaceHn, {|n〉 , n = 1, 2, ...,Nv} andHm, {|m〉 ,m = 1, 2, ...,Nv},where Nv is the number of the vertex V (Γ) .

→ The state of the walk is given in the product HNvn ⊗HNv

m

spanned by these bases, by states at the previous |m〉 andcurrent |n〉 steps, defined by 3

|ψn(t)〉 =Nv∑m

√Pmn |n〉 ⊗ |m〉 , (5)

3M. Szegedy, arXiv:quant-ph/0401053, arXiv:quant-ph/0401053v11.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Quantum walk→ For the evolution define a reflection, which we caninterpret with the “coin” operator

C = 2∑n

|ψn〉〈ψn| − I , (6)

and a swap operation

S =∑n,m

|m, n〉〈n,m| , (7)

and we have the unitary evolution

U = CS , (8)

that defines the DQW.

→ It is given by equations (5 - 8) with P given by (4).Namely, for equation (5)

|ψn(t)〉 =Nv∑m

√√√√ jmn(jmn + 1)∑kjmk(jmk + 1)

|n〉 ⊗ |m〉 . (9)


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Quantum walk→ For the evolution define a reflection, which we caninterpret with the “coin” operator

C = 2∑n

|ψn〉〈ψn| − I , (6)

and a swap operation

S =∑n,m

|m, n〉〈n,m| , (7)

and we have the unitary evolution

U = CS , (8)

that defines the DQW.→ It is given by equations (5 - 8) with P given by (4).Namely, for equation (5)

|ψn(t)〉 =Nv∑m

√√√√ jmn(jmn + 1)∑kjmk(jmk + 1)

|n〉 ⊗ |m〉 . (9)


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Quantum walk

→ To do:

- Continuum limit;

- Two particle QW and entanglement;

- Others Laplacians 4.

4Johannes Thurigen tesis, arXiv:1510.08706v1


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entanglement Entropy→ Consider the Schmidt decomposition. Take a Hilbertspace H and decompose it into two subspaces H1 ofdimension N1 and H2 of dimension N2 ≥ N1, so

H = H1 ⊗H2. (10)

Let |ψ〉 ∈ H1 ⊗H2, and {∣∣ψ1

i

⟩} ⊂ H1, {

∣∣ψ2i

⟩} ⊂ H2, and

positive real numbers {λi}, then the Schmidt decompositionread

|ψ〉 =

N1∑i

√λi∣∣ψ1

i

⟩⊗∣∣ψ2

i

⟩, (11)

where λi are the Schmidt coefficients and the number of theterms in the sum is the Schmidt rank, which we label N.→ With this we can calculate the Entanglement Entropybetween the two subspaces

SE = −∑i∈N

λi logλi . (12)


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entanglement Entropy

→ We can now calculate the local Entanglement Entropybetween the previous step and the current n step (similar forcurrent and next). Identify the Schmidt coefficients λi withour Pmn. Note that the Schmidt rank N is the valence of thenode. Then the local Entanglement Entropy on current stepis

SEn = −N∑m

PmnlogPmn. (13)

By maximizing Entanglement Entropy

SEn = logNmax , (14)

where Nmax is the largest valence.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropic motion

→ For particular cases the particle will move to a placewhere the entropy is the largest. This can be though of asan entropic motion.→ And we can compute the change of entropy with respectto position for this motion. In our case as we have a discretesystem we have that the variation of entropy with respect toposition is just a difference of the local entropies atneighbour verticesdSdx = |SEn − SEm |proportional to a small number identified with the particlemass M

dS

dx= |SEn − SEm | = αM, (15)

where α is a constant of dimension [bit/mass].


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Walker position encoded on spin network→ From (4) and (13) we can compute the local entropyfrom a vertex as

SEn = logσ − 1

σ

N∑m

jmn(jmn + 1)log (jmn(jmn + 1)) , (16)

where σ =N∑mjmn(jmn + 1) of neighbour links. For example

at a node {2, 3, 3}, j = {1, 32 ,32} gives σ = 19

2 andSEn = 1.06187. At a node {2, 2, 2}, j = {1, 1, 1} givesσ = 6, SEn = 1.09861 which is the maximal possible localentropy.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy map

→ The local entropy at each node is color coded. Fromequation (15) a massless particle move on same color and amassive particle moves along constant absolute colordifferences.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Walker position topologically encoded

→ The walker position, or the presence of a particle at onenode is encoded by a triangle. Its move is a couple of 3-1and 1-3 Pachner moves on neighbor positions, piloted by thewalk probability


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Black hole isolated quantum horizons

→ We can propose that DQW is the black hole quantumhorizon, where the particle mass is the black hole mass in arandom quantum walk on a fixed spin network.

→ In the isolated quantum horizons formulation the entropyis usually calculated by considering the eigenvalues of thearea operator A(j) and introducing an area intervalδa = [A(j)− δ,A(j) + δ] of width δ of the order of thePlanck length, with relation to the classical area a of thehorizon. A(j) is given by

A(j) = 8πγl2ph∑l

√jl(jl + 1), (17)


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy

→ The entropy, in admensional form, is

SBH = lnN(A), (18)

with N(A) the number of microstates of quantum geometryon the horizon (area interval a) implemented by consideringstates with links sequences that implement the condition

8πγl2ph

Na∑l=1

√jl(jl + 1) ≤ a, (19)

related to the area, where Na is the number of admissible jthat puncture the horizon area.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy→ Let’s investigate how the horizon area and related entropycan be emergent from maximal entanglement entropy of aDQW→ Considering edge coloring, the maximal entanglemententropy occurs for states on nodes of large valence Nmax andsequence with jl = l

2 (l = 1, 2, ...,Nmax). So we can rewritecondition (19) as

Na∑i=1

ai = ac , (20)

withac =

a

4πγl2ph. (21)

and each ai is calculated from

ai =Nmax∑l=1

√l(l + 2), (22)


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy

→ And considering the dominant contributions given by theover-estimate each ai and counting N(ac) that will giveN(A), each ai is a integer√

l(l + 2) =√

(l + 1)2 − 1 ≈ l + 1, (23)

which means that the combinatorial problem we need tosolve is to find N(ac) such that (20) holds. With theassumption of DQW we need at least two elements.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy

So, we have to count partitions of ac in parts with 2 or moreelements. Noting that all partitions of ac + 1 in parts with 2or more elements can be obtained from the sum of partitionsof ac and ac − 1,

N(ac + 1) = N(ac) + N(ac − 1), (24)

it is straightforward5 to see that

logN(A) =log(φ)

πγ

a

4l2ph, (25)

where φ = 1+√5

2 is the golden ratio.

5Because the cardinal N(ac) of the set of ordered tuples of integersstrictly greater than 1 summing to ac is the athc Fibonacci number F (ac).


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Entropy

→ Bekenstein-Hawking entropy is recovered by setting theBarbero-Immirzi parameter

γ =log2(φ)

π= 0.22 (26)

with agree with results from LQG.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Discussion

Results from Loop Quantum Gravity suggest the interestingidea that we can apply the results and tools from quantuminformation and quantum computation to a quantumspacetime. This is a field of research very promising. In thiswork we start a project to apply this tools like DQW tospacetime. We considered a DQW of a quantum particle ona quantum gravitational field and studyied applications ofrelated Entanglement Entropy.


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Discussion

Relating this model with QuasiCrystal E8 model 6 are underinvestigation: the spin network can be choosen as the dualof a quasicrystal and the digital physics rules can beimplemented by the quantum walk.

6F. Fang; K. Irwin, “A Chiral Icosahedral QC and its Mapping to anE8 QC”, Aperiodic2015 poster, arXiv:1511.07786 [math.MG].


(arXiv:1602.07653v1)

M M Amaral,Raymond


Introduction


Quantum walk

EntanglementEntropy



Discussion

Discussion

Thank you.

Quantum Walk on Spin Networks by Marcelo Amaral, Raymond Aschheim and Klee Irwin

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Transcript of Quantum Walk on Spin Networks by Marcelo Amaral, Raymond Aschheim and Klee Irwin