# First mean return time in decoherent quantum walks

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1 First mean return time in decoherent quantum walks Péter Sinkovicz, János K. Asbóth, Tamás Kiss Wigner Research Centre for Physics Hungarian Academy of Sciences, April 0.

2 Problem statement Example: N=,,,... A ij = A ji transition amplitude First mean return time in decoherent quantum walks /

3 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i First mean return time in decoherent quantum walks /

4 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i First mean return time in decoherent quantum walks U 0i /

5 Problem Our numerical results Conclusion Arrival by absorption: 0i := ψ0 i U 0i or p := h0 U 0i First mean return time in decoherent quantum walks ψ i := [I 0ih0 ] U 0i q := hψ ψ i /

6 Conditional wave function: evolve these parts, which haven t come back where ψ t+ := [I 0 0 ] U ψ t p t := 0 U ψ t q t+ := ψ t+ ψ t+ p t probability: Prob(X t = 0 X n 0 if n < t) q t probability: Prob(X n 0 if n t) First mean return time in decoherent quantum walks /

7 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /

8 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = pt N= example p t : t pt t F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /

9 Mean return time T := tp t q t := Trρ(t) t= t=0 t=0 where q t+ + p t+ = q t and q 0 = pt N= example p t : t pt t T is an integer number equal with the graph size independent of A ij transition amplitude T = N F. A. Grünbaum,A. H. Werner, R. F. Werner, (0) First mean return time in decoherent quantum walks /

10 Classical case : symmetric, regular graph T Kemeny, G.; Snell, L. (90) First mean return time in decoherent quantum walks /

11 Our interest? decoherence x First mean return time in decoherent quantum walks 7 /

12 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks 8 /

13 Homogeneous decoherence For see a transition between Grünbaum et al. s and the Classical random walk we stick in a simplest decoherence. So one step of the process is the following: coherent time step decoherence C[ρ] = UρU measurement D[ρ] xy = dρ xy + ( d)ρ xx δ xy M[ρ] = [I 0 0 ] ρ [I 0 0 ] First mean return time in decoherent quantum walks 9 /

14 T numerical result U + homogeneous decoherence T = N N for d = unitary time evolution, and d = 0 is the classical case d 0 where d ρ t+ = M[T [ρ t ]] = M[D[C[ρ t ]]] d and D[ρ] = ρ dρ dρ... dρ ρ First mean return time in decoherent quantum walks 0 /

15 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks /

16 Kraus representation In order to map all quantum channel we can use Kraus representation for the decoherence channel d ρ := D[ρ] = K µ ρk µ µ= ρ m,µ = D m,n µ,ν ρ n,ν trace preserving (probability preserving): d K µk µ = I D stochastic µ= may we will get some redundancy, because the measure and the unitary time evolution can be defined by Kraus operators First mean return time in decoherent quantum walks /

17 numerical guess unital map T [ N I] = N I T = N T [ ] = D[C[ ]] measure less process is unital (leave the totally mixed state invariant) only if d K µ K µ = I D T stochastic µ= trace preserving and untial map quantum walk ρ T [ρ] classical walk λ t Wλ t First mean return time in decoherent quantum walks /

18 Numerical study in order to find out?? T = N I) homogeneous decoherence II) Kraus representation III) Master equation First mean return time in decoherent quantum walks /

19 Master equation For check we formulated our guess in another language: T [ ] can be substitute with the time evolution which has generated by the t ρ = i [H, ρ] + L(ρ) L(ρ) = i,j κ i,j [ i j ρ j i ( i i ρ + ρ j j ) ] Master equation. First mean return time in decoherent quantum walks /

20 Master equation For check we formulated our guess in another language: T [ ] can be substitute with the time evolution which has generated by the t ρ = i [H, ρ] + L(ρ) L(ρ) = i,j κ i,j [ i j ρ j i ( i i ρ + ρ j j ) ] Master equation. numerical guess unital map L(I) = 0 κ i,j = κ j,i T = N First mean return time in decoherent quantum walks /

21 Conclusion I) numerically studied systems Kraus representation unitary quantum walk homogeneous decoherence inhomogeneous, but symmetric decoherence uni-stochastic map Master equation L(I) = 0 (unital map) II) analytically we need proof for numerical guess T [I] = I unital T = N First mean return time in decoherent quantum walks /

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