Quantum Walk on Spin Networks by Marcelo Amaral, Raymond Aschheim and Klee Irwin
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Presentation from the 4th International Conference on the Nature and Ontology of Spacetime May 31, 2016
![Quantum walk on
spin network
(arXiv:1602.07653v1
M M Amaral,
Raymond
Aschheim and Klee
Irwin
Introduction
Particle interacting
LQG
Quantum walk
Entanglement
Entropy
A model of walker
position
topologically
encoded on spin
network
Application to
black hole
Discussion
Motivations
→ Feynman path integral quantization
Quantum (transition amplitudes)
W = D[]e
i
h
S
(1)
→ Feynman path integral discretization](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Quantum Walk on Spin Networks by Marcelo Amaral, Raymond Aschheim and Klee Irwin
- 1. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black 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
- 2. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Motivations → Feynman path integral quantization Quantum (transition amplitudes) W = D[]e i h S (1) → Feynman path integral discretization
- 3. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Motivations → Feynman checkerboard → quantum random walk → It from bit → From an ontological viewpoint we will see that dynamics and mass emerge from the spin network topology, and that quantum walk implements it.
- 4. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Spin network → Spin network graph Γ = (V (Γ), L(Γ)), with V (Γ) → v1, v2..., vertices and L(Γ) → l1, l2..., ({1 2, 1, 3 2...}) links → For the particle state space the relevant contribution comes 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 1 C. Rovelli; F. Vidotto, Phys. Rev. D 2010, 81, arXiv:0905.2983v2.
- 5. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Transition probabilities → If the link l starts at vertex m and ends at vertex n we can change the notation, relabelling the color of this link l between m and n as jmn and the wave function on the end points 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 transition probabilities 2 Pmn = jmn(jmn + 1) k jmk(jmk + 1) . (4) 2 J. M. Garcia-Islas, ARXIV gr-qc/1411.4383.
- 6. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Quantum walk → Consider the Nv -dimensional Hilbert space Hn, {|n , n = 1, 2, ..., Nv } and Hm, {|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 HNv n ⊗ HNv m spanned by these bases, by states at the previous |m and current |n steps, defined by 3 |ψn(t) = Nv m Pmn |n ⊗ |m , (5) 3 M. Szegedy, arXiv:quant-ph/0401053, arXiv:quant-ph/0401053v11.
- 7. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Quantum walk → For the evolution define a reflection, which we can interpret 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.
- 8. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Quantum walk → For the evolution define a reflection, which we can interpret 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) k jmk(jmk + 1) |n ⊗ |m . (9)
- 9. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Quantum walk → To do: - Continuum limit; - Two particle QW and entanglement; - Others Laplacians 4. 4 Johannes Thurigen tesis, arXiv:1510.08706v1
- 10. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entanglement Entropy → Consider the Schmidt decomposition. Take a Hilbert space H and decompose it into two subspaces H1 of dimension N1 and H2 of dimension N2 ≥ N1, so H = H1 ⊗ H2. (10) Let |ψ ∈ H1 ⊗ H2, and { ψ1 i } ⊂ H1, { ψ2 i } ⊂ H2, and positive real numbers {λi }, then the Schmidt decomposition read |ψ = N1 i λi ψ1 i ⊗ ψ2 i , (11) where λi are the Schmidt coefficients and the number of the terms in the sum is the Schmidt rank, which we label N. → With this we can calculate the Entanglement Entropy between the two subspaces SE = − i∈N λi logλi . (12)
- 11. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entanglement Entropy → We can now calculate the local Entanglement Entropy between the previous step and the current n step (similar for current and next). Identify the Schmidt coefficients λi with our Pmn. Note that the Schmidt rank N is the valence of the node. Then the local Entanglement Entropy on current step is SEn = − N m PmnlogPmn. (13) By maximizing Entanglement Entropy SEn = logNmax , (14) where Nmax is the largest valence.
- 12. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropic motion → For particular cases the particle will move to a place where the entropy is the largest. This can be though of as an entropic motion. → And we can compute the change of entropy with respect to position for this motion. In our case as we have a discrete system we have that the variation of entropy with respect to position is just a difference of the local entropies at neighbour vertices dS dx = |SEn − SEm | proportional to a small number identified with the particle mass M dS dx = |SEn − SEm | = αM, (15) where α is a constant of dimension [bit/mass].
- 13. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position encoded on spin network → From (4) and (13) we can compute the local entropy from a vertex as SEn = logσ − 1 σ N m jmn(jmn + 1)log (jmn(jmn + 1)) , (16) where σ = N m jmn(jmn + 1) of neighbour links. For example at a node {2, 3, 3}, j = {1, 3 2, 3 2} gives σ = 19 2 and SEn = 1.06187. At a node {2, 2, 2}, j = {1, 1, 1} gives σ = 6, SEn = 1.09861 which is the maximal possible local entropy.
- 14. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy map → The local entropy at each node is color coded. From equation (15) a massless particle move on same color and a massive particle moves along constant absolute color differences.
- 15. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position topologically encoded → The walker position, or the presence of a particle at one node is encoded by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor positions, piloted by the walk probability
- 16. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position topologically encoded → The walker position, or the presence of a particle at one node is encoded by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor positions, piloted by the walk probability
- 17. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position topologically encoded → The walker position, or the presence of a particle at one node is encoded by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor positions, piloted by the walk probability
- 18. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position topologically encoded → The walker position, or the presence of a particle at one node is encoded by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor positions, piloted by the walk probability
- 19. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Walker position topologically encoded → The walker position, or the presence of a particle at one node is encoded by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor positions, piloted by the walk probability
- 20. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Black hole isolated quantum horizons → We can propose that DQW is the black hole quantum horizon, where the particle mass is the black hole mass in a random quantum walk on a fixed spin network. → In the isolated quantum horizons formulation the entropy is usually calculated by considering the eigenvalues of the area operator A(j) and introducing an area interval δa = [A(j) − δ, A(j) + δ] of width δ of the order of the Planck length, with relation to the classical area a of the horizon. A(j) is given by A(j) = 8πγl2 ph l jl (jl + 1), (17)
- 21. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy → The entropy, in admensional form, is SBH = lnN(A), (18) with N(A) the number of microstates of quantum geometry on the horizon (area interval a) implemented by considering states with links sequences that implement the condition 8πγl2 ph Na l=1 jl (jl + 1) ≤ a, (19) related to the area, where Na is the number of admissible j that puncture the horizon area.
- 22. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy → Let’s investigate how the horizon area and related entropy can be emergent from maximal entanglement entropy of a DQW → Considering edge coloring, the maximal entanglement entropy occurs for states on nodes of large valence Nmax and sequence with jl = l 2 (l = 1, 2, ..., Nmax ). So we can rewrite condition (19) as Na i=1 ai = ac, (20) with ac = a 4πγl2 ph . (21) and each ai is calculated from ai = Nmax l=1 l(l + 2), (22)
- 23. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy → And considering the dominant contributions given by the over-estimate each ai and counting N(ac) that will give N(A), each ai is a integer l(l + 2) = (l + 1)2 − 1 ≈ l + 1, (23) which means that the combinatorial problem we need to solve is to find N(ac) such that (20) holds. With the assumption of DQW we need at least two elements.
- 24. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy So, we have to count partitions of ac in parts with 2 or more elements. Noting that all partitions of ac + 1 in parts with 2 or more elements can be obtained from the sum of partitions of 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 4l2 ph , (25) where φ = 1+ √ 5 2 is the golden ratio. 5 Because the cardinal N(ac ) of the set of ordered tuples of integers strictly greater than 1 summing to ac is the ath c Fibonacci number F(ac ).
- 25. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Entropy → Bekenstein-Hawking entropy is recovered by setting the Barbero-Immirzi parameter γ = log2(φ) π = 0.22 (26) with agree with results from LQG.
- 26. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Discussion Results from Loop Quantum Gravity suggest the interesting idea that we can apply the results and tools from quantum information and quantum computation to a quantum spacetime. This is a field of research very promising. In this work we start a project to apply this tools like DQW to spacetime. We considered a DQW of a quantum particle on a quantum gravitational field and studyied applications of related Entanglement Entropy.
- 27. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Discussion Relating this model with QuasiCrystal E8 model 6 are under investigation: the spin network can be choosen as the dual of a quasicrystal and the digital physics rules can be implemented by the quantum walk. 6 F. Fang; K. Irwin,“A Chiral Icosahedral QC and its Mapping to an E8 QC”, Aperiodic2015 poster, arXiv:1511.07786 [math.MG].
- 28. Quantum walk on spin network (arXiv:1602.07653v1 M M Amaral, Raymond Aschheim and Klee Irwin Introduction Particle interacting LQG Quantum walk Entanglement Entropy A model of walker position topologically encoded on spin network Application to black hole Discussion Discussion Thank you.