Open quantum random walks with decoherence on coins with $n$ degrees of freedom

0.2 MB·English

by  Sheng Xiong

Open quantum random walks with decoherence on
3
coins with n degrees of freedom
1
0
2

n
a
J
6 December 11, 2013
2

]
R
Sheng Xiong
P
. Department of Mathematics
h
t University of Pittsburgh, Pittsburgh, PA 15260
a
m
Wei-Shih Yang
[
Department of Mathematics

2
Temple University, Philadelphia, PA 19122
v
4
Email: [email protected], [email protected]
3
0
0 Abstract
.
1
0 In this paper, we define a new type of decoherent quantum random walks
3 with parameter 0 p 1, which becomes a unitary quantum random walk
1 ≤ ≤
(UQRW) when p = 0 and an open quantum random walk (OPRW) when
:
v
p = 1 respectively. We call this process a partially open quantum random walk
i
X (POQRW). We study the limiting distribution of a POQRW on Z1 subject to
r decoherence on coins with n degrees of freedom, which converges to a convex
a
combinationofnormaldistributionsifthesuperoperator satisfiestheeigen-
kk
L
value condition, that is, 1 is an eigenvalue of with multiplicity one and all
kk
L
other eigenvalues have absolute values less than 1. A Perron-Frobenius type of
theorem is provided in determining whether or not the superoperator satisfies
the eigenvalue condition. Moreover, we compute the limiting distributions of
characteristic equations of the position probability functions for n= 2 and 3.
1
1 Introduction
It has been given rise to a vast field of exploration for the behavior of quantum
open systems [2,3,5–7] since S. Attal et al. [1] recently introduced a new type of
an open quantum random walk (OQRW) on graphs, which is an exact quantum
analogue of classical Markov Chain. This new type of walks has exhibited many
interesting phenomenon since it is partially a quantum walk and partially a classical
walk; moreover, it has a speed-up property like a quantum walk behavior and yet it
hasaMarkovpropertylikeaclassical walk[3]. Inparticular, S.Attaletal.[1]pointed
out that there is a strong link between OQRWs and the well known unitary quantum
random walks (UQRW) and that it is difficult to produce the limit distribution for
OQRWs, due to lack of knowledge about the invariant measures of this Markov chain
and even their existence at time of their paper was written. But soon in [3] S.
Attal et al. obtained a Central Limit Theorem for the case of the nearest neighbors
homogeneous OQRW on Zd under an assumption that the superoperator admits a
unique invariant state. The variance is somewhat abstract, so Konno and Yoo [7]
furtherstudiedthelimitdistributionsofOQRWsonone-dimensionallatticespaceand
compute the distribution of the OQRWs concretely for many examples and thereby
obtain the limit distributions of them. Under an equivalent condition that 1 is an
eigenvalue of the superoperator with multiplicity one and all other eigenvalues have
absolute values less than 1(for convenience, throughout the paper, this condition is
called“eigenvalue condition”), the authors of this paper also [4] obtained the limit
distribution (a convex combination of Gaussian distribution) for UQRWs on Z. This
suggests us to further explore the strong link between these two types of quantum
walks and the limit distributions of OQRWs under the eigenvalue condition.
In this paper, we consider an OQRW resulting from a total decoherent on the
coin space of a UQRW while the dimension of the coin space can be arbitrarily large.
In such case, we investigate the transitions of how a unitary QRW is eventually
collapsed into an OQRW. Moreover, we study a type of QRWs where each step has
a probability p of decoherent on the coin space. If p > 0, this process will eventually
collapse into OQRW and exhibit the diffusive behavior. We call this process partially
open quantum walk with parameter p (POQRW). Here 0 p 1. If p = 0, it’s a
≤ ≤
UQRW; if p = 1, it is corresponding to an OQRW. Assuming the superoperator of
the POQRW satisfies the eigenvalue condition, we conclude that the POQRW also
converges to a convex combination of normal distributions.
The rest of our paper is organized as follow: in Section 2 we briefly summarize the
formalism of POQRWs. In Section 3 we prove a limiting theorem of POQRWs on Z1
with decoherence on coins with n degrees of freedom with the help of the Quantum
2
Fourier Transform and the generalized Gell-Mann matrices basis. In Section 4 we
prove a Perron-Frobenius type of theorem which is very useful in determining when
the superoperator satisfies the eigenvalue condition. In Section 5 we demonstrate
kk
L
some examples of POQRWs (n = 2,3) that satisfy the eigenvalue condition and give
explicit formulas for their limiting distributions. In particular, our results generalize
the Central Limit Theorem in [3].
2 The partially open quantum random walk on the
lattices Zd and decoherence
Let us consider a general open quantum random walk on Zd. Let e d be the
{ j}j=1
standard orthonormal basis of Zd and we put e = e ,j = 1,2,...,d. We denote
j+d j
−
the state space by a Hilbert space = , where denotes the position
P C P
H H ⊗ H H
space and denotes the coin space. The orthonormal basis of the position space
C
H
are x >, where x Zd, the basis of the coin space are ξ >,i = 1,2,...,n.
P C i
H | ∈ H |
We will assume that the walk starts at the origin. Let us describe the dynamics of
the quantum walker.
Let U = [u ,u ,...,u ]T be a n n unitary matrix and Π ,j = 1,2,...,m
1 2 n j
×
be orthogonal projection matrices which partition the matrix U into m matrices
B ,j = 1,2, ,m, where
j
···
B = [u ,u ,...,u ,0,0,...,0]T,
1 1 2 d1
B = [0,0,...,0,u ,u ,...,u ,0,0,...,0]T,
2 1+d1 2+d1 d1+d2
B = [0,0,...,0,u ,u ,...,u ,0,0,...,0]T,
j 1+Pji=−11di 2+Pji=−11di Pji=1di
for j = 1,2...,m, and m d = n. Note that
i=1 i
P
m m
B = Π U, B B = I, B = U.
j j j∗ j j
Xj=1 Xj=1
Let us define
Lj = x+e >< x B ,
x | j |⊗ j
Lj = Lj, for j = 1,2,...,m,
x
Xx
3
where m = 2d, and we also put L = m Lj, and L = L . Then L is a unitary
x j=1 x x x
operator on H. Let ψ0 ∈ H and ψnP= Lnψ0. Then {ψPn}∞n=0 is called a quantum
random walk on Zd.
In terms of density operator
M (ρ) = LρL
0 ∗
is the one-step dynamics of the density operator. So ρ(n) = Mn(ρ(0)) is the dynamics
0
of the quantum random walks. Note that
m
M (ρ) = LjρLj′∗.
0 x x′
jX,j′=1Xx,x′
Let
m
Mopen(ρ) = LjxρLjx∗.
Xj=1 Xx
Then ρ(n) = Mn (ρ(0)) is the dynamics of the open quantum random walks defined
open
by S. Attal, F. Petruccione and I. Sinayskiy [1] (2012).
Let us now define the partially open quantum random walk. Let
D = √pLj,j = 1,2,...,m,
j
and
D = √qL,
0
where 0 p,q 1,p+q = 1. We have m D D = I and therefore D m can
≤ ≤ j=0 j∗ j H { j}j=0
be viewed as measurements on = PP C. Let us define a positive map on :
H H ⊗H H
m
M(ρ) = M (ρ) = D ρD ,
p j j∗
Xj=0
where ρ is a density operator on , then Mn is called the POQRW on Zd
H { }∞n=1
associated with B . Note that if p = 0, it is a UQRW. If p = 1, it is corresponding
j
{ }
to an OQRW, as seen below.
ThesuperoperatorofopenquantumrandomwalksdefinedinS.Attal, N.Guillotin-
Plantard and C. Sabot [3] is
m
Mopen(ρ) = LjxρLjx∗,
Xj=1 Xx
4
and in our case,
m
M (ρ) = LjρLj ∗,
1 x x′
Xj=1 Xx,x′
so they are not exactly the same operators. But if ρ = x >< x ρ , where ρ
x| |⊗ x x
is a positive operator on HC such that xTr(ρx) = 1, Pthen ρ1 = Mopen(ρ) also has
the same form and P
M (ρ) = M (ρ), for all ρ = x >< x ρ .
open 1 x
| |⊗
Xx
Therefore, our case p = 1 includes the corresponding open quantum random walks.
Since we are dealing with partially open quantum random walks, the form of density
operator has a nontrivial off-diagonal (in x-space) entry after the initial step, we need
to consider general density operators for later iterations. Thus our formula M is a
1
generalization of the open quantum random walks.
In general, the density operator for quantum random walk in Fourier transforma-
tion basis is given by
dk dk
′
ρ = Z (2π)d Z (2π)d|k >< k′|⊗χkk′, (2.1)
where k = (k ,k ,...,k ), 0 k 2π, and dk = dk dk ...dk . Then after one step
1 2 d i 1 2 d
≤ ≤
the density operator becomes
m
ρ ρ = D ρD
→ ′ j j∗
Xj=0
m
= p(LjxρLjx′∗)+q (LxρLx′∗)
Xj=1 Xx′ Xx Xx′ Xx
m
dk dk
′
= Z (2π)d Z (2π)d|k >< k′|⊗ AjUkχkk′Uk∗′A∗j,
Xj=0
where A = √pΠ ,A = √qI,B = e ikejB ,U = m B , and ke denotes
j j 0 jk − j k j=1 jk j
the dot product of k and ej. Suppose the quantum wPalk starts at a pure state
0 > Φ >, then the initial state is
0
| ⊗|
dk dk
′
ρ = k >< k Φ >< Φ . (2.2)
0 Z (2π)d Z (2π)d| ′|⊗| 0 0|
5
Let the quantum random walk proceed for t steps. Then the state evolves to
dk dk
′
ρ = k >< k
t ′
Z (2π)d Z (2π)d| |⊗
A U A U Φ >< Φ U A U A .
jt k··· j1 k| 0 0| k∗′ ∗j1··· k∗′ ∗jt
j1X,...,jt
Let kk′ be the operator acting on the vector space of linear operators L( C) and
L H
χkk′ L( C), then it follows from the direct computation
∈ H
m
Lkk′χkk′ ≡ AjUkχkk′Uk∗′A∗j = p Bjkχkk′Bj∗k′ +qUkχkk′Uk′∗. (2.3)
Xj Xj=1
kk′ is a linear operator that maps from L( C) to L( C), that is kk′ L(L( C)).
L H H L ∈ H
kk′ is also called a superoperator, and
L
dk dk
′
k >< k A U χ U A
Z (2π)d Z (2π)d| ′|⊗ j k kk′ k∗′ ∗j
Xj
dk dk
′
= Z (2π)d Z (2π)d|k >< k′|⊗Lkk′χkk′.
In terms of the superoperator kk′,
L
dk dk
ρ = ′ k >< k t Φ >< Φ . (2.4)
t Z (2π)d Z (2π)d| ′|⊗Lkk′| 0 0|
The probability to reach a point x at time t is
p(x,t) = Tr [ x >< x I]ρ
t
{ | |⊗ }
1
= dk dk < k x >< x k > Tr t Φ >< Φ
(2π)2d Z Z ′ | | ′ {Lkk′| 0 0|}
1
= dk dk e ix(k k′)Tr t Φ >< Φ . (2.5)
(2π)d Z Z ′ − − {Lkk′| 0 0|}
3 The limiting distributions of quantum walks on
Z1 with decoherence on coins with n degrees of
freedom
Consider the Fourier transformation of p(x,t)
Pˆ(ν,t) < eiνx > = eiνxp(x,t). (3.1)
t
≡
Xx
6
To simplify < eiνx > , we use the properties of δ function
t
1
xme ix(k k′) = ( i)mδ(m)(k k ). (3.2)
− − ′
2π − −
Xx
Then
< eiνx > = eiνxp(x,t)
t
Xx
dk dk
= eiνx ′e ix(k k′)Tr t Φ >< Φ
Z 2π Z 2π − − {Lkk′| 0 0|}
Xx
dk dk
= ′ eix(ν+k′ k)Tr t Φ >< Φ
Z 2π Z 2π − {Lkk′| 0 0|}
Xx
dk dk
= ′2πδ(ν +k k)Tr t Φ >< Φ
Z 2π Z 2π ′ − {Lkk′| 0 0|}
1
= dkTr t Φ >< Φ . (3.3)
2π Z {Lk,k+ν| 0 0|}
Let Oˆ denote any operator on . Then the generating function of < eiνx > is given
C t
H
by
∞
G(z,ν) = zt < eiνx >
t
Xt=0
1 ∞
= dk ztTr t Oˆ
2π Z {Lk,k+ν }
Xt=0
1 1
= dkTr Oˆ .
2π Z {I z }
k,k+ν
− L
where z < 1. This definition makes sense since the spectrum of is less than
k,k+ν
| | L
or equal to 1 by Lemma (3.1) in [4].
Let the coin Hilbert space be n-dimensional spanned by orthogonal ba-
C
H
sis ξ ,ξ ,...,ξ , Π be the orthogonal projection onto the subspace spanned by
1 2 n 1
ξ ,ξ ,...,ξ , and Π be the orthogonal projection onto the subspace spanned by
1 2 n1 2
ξ ,ξ ,...,ξ . We set n = n n . Let U be a unitary operator on , suppose
n1+1 n1+2 n 2 − 1 HC
B = Π U and B = Π U. Then
1 1 2 2
B = ω B , B = ω B ,
1k k 1 2k k 2
7
where ω = eik and it follows U = B +B and
k k 1k 2k
Lkk′(ρ) = p(B1kρB1∗k′ +B2kρB2∗k′)+qUkρUk∗′ (3.4)
or
Lkk′(ρ) = p(ωkΠ1UρU∗Π∗1ωk′ +ωkΠ2UρU∗Π∗2ωk′)+qUkρUk∗′. (3.5)
In order to analyze the spectrum of the superoperator kk′, we will use the nor-
L
malized Gell-Mann basis. Let E be the matrix with 1 in the jk-th entry and 0
jk
elsewhere. Consider the space of d d complex matrices, Cd d, for a fixed d. Define
×
×
the following matrices
* For k < j, fd = E +E .
k,j kj jk
* For k > j, fd = i(E E ).
k,j − jk − kj
* For k = j, Let fd = hd = I , the identity matrix.
1,1 1 d
* For 1 < k = j < d, fd = hd = hd 1 0.
k,k k k− ⊕
* For k = j = d, fd = hd = 2 (hd 1 (1 d)).
d,d d d(d 1) 1− ⊕ −
q −
The collection of matrices fd , 1 k,j d are called the generalized Gell-
{ k,j ≤ ≤ }
Mann matrices in dimension d. Here means Matrix Direct Sum. The generalized
⊕
Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli
matrices. One can also check that they are orthogonal in the Hilbert-Schmidt inner
product on Cd d. By dimension count, one sees that they span the vector space of
×
d d complex matrices. When d = 2, they are Pauli matrices. When d = 3, they are
×
Gell-Mann matrices.
fn
Notes that fn are not normalized. We put γn = k,j , then γn , 1 k,j
{ k,j} k,j ||fkn,j|| { k,j ≤ ≤
n are orthonormal basis of n n complex matrices in the Hilbert-Schmidt inner
pr}oduct. For short notation, we×set γ n = γn = hnl . When we fixed the dimension
l l,l hn
n, we omit the superscript n. i.e, γ = γn ,γ|| =l|| γn. We order the basis γ by
k,j k,j l l ij
γ ,γ ,...,γ , γ ,γ ,...,γ ,...,γ ,γ ,...,γ . Now we are ready to prove
11 12 1n 21 22 2n n1 n2 nn
Lemma 3.1. Suppose U U(n),Π ,Π are defined as above and B is unital.
1 2 j
∈ { }
Then has the following form in terms of normalized Gell-Mann basis γ : the
k,k+ν ij
L
first column of satisfies
k,k+ν
L
< γ , (γ ) >= 0, k = j
kj k,k+ν 1
L ∀ 6
 2n2cosν + n1 n2ω , l = 1
 n −n ν
< γl,Lk,k+ν(γ1) >= 0, 2 ≤ l ≤ n1
2n1isinν , n +1 l n,
 √n(l 1)l 1 ≤ ≤
  −
8
where ω = eiν. The first row of satisfies
ν k,k+ν
L
1
< γ , (γ) >= ω δ 2isinν Tr(B γB ),
1 Lk,k+ν ν γγ1 − √n 2 2∗
where γ is a normalized basis. In particular, if ν = 0, then has the following
k,k
L
representation
1 0 0 0
0 
× × × .
0
 × × ×
0 
 × × ×
Proof: For n i n 1,
1
≤ ≤ −
1 0 0 0 0 0 0
··· ··· ···
0 1 0 0 0 0 0
··· ··· ···
... ... ... ... ... ... ... ... ... ...
 
 
0 0 1 0 0 0 0
 
 ··· ··· ··· 
1 0 0 0 1 ... 0 0 0
γi = ||hi|| ... ... ·.·.·. ... ... ... ... ... ·.·.·. ...
0 0 0 0 1 0 0
 
··· ··· ···
0 0 0 0 0 (i 1) 0
 
··· ··· − − ···
... ... ... ... ... ... ... ... ... ...
 
0 0 0 0 0 0 0
 ··· ··· ··· 
where h = i(i 1) and there are n rows above the line and the first i n 1
i 1 1
|| || − − −
p
9
rows below the line with one nonzero entry. It follows
1
Lkk′(γ1) = √n [p(ωk′−kΠ1UIU∗Π∗1 +ωk′−kΠ2UIU∗Π∗2)+q(ωk′−kΠ1UIU∗Π∗2 +ωk′−kΠ2UIU∗Π∗1)]
1
= (ωk′ kΠ1 +ωk′ kΠ2)
√n − −
ω 0 0 0 0 0
ν
··· ···
0 ω 0 0 0 0
 ν 
. . ··.· . . . ···. .
. . . . . . . .
. . . . . . . .
 
1  0 0 ω 0 0 0 
=  ··· ν ··· 
√n  0 0 0 ω 0 0 
 ν 
 0 0 ··· 0 0 ω ··· 0 
 ν 
 . . ··.· . . . ···. . 
 .. .. .. .. .. .. .. .. 
 
 0 0 0 0 0 ω 
 ··· ··· ν 
where ν = k k, and the block at (1,1) and (2,2) positions are n n and n n
′ 1 1 2 2
− × ×
respectively. Hence
1 1
< γ , (γ ) > = [n ω +(i n 1)ω (i 1)ω ]
i k,k+ν 1 1 ν 1 ν ν
L √n i(i 1) − − − −
−
1 p 1
= 2n isinν.
√n i(i 1) 1
−
p
For
1 0 0 0 0 0
··· ···
0 1 0 0 0 0
 
. . . ··.· . . ··.· .
. . . . . . . .
. . . . . . . .
 
1 0 0 1 0 0 
γ =  ··· ··· 
n h 0 0 0 1 0 0 
n  
|| ||  ··· ··· 
0 0 0 0 1 0
 
. . ··.· . . . ··.· 
.. .. .. .. .. .. .. 
 
0 0 0 0 0 (n 1)
 ··· ··· − − 
we have
1 1
< γ , (γ ) > = [n ω +(n 1)ω (n 1)ω ]
n k,k+ν 1 1 ν 2 ν ν
L √n n(n 1) − − −
−
1 p 1
= 2n isinν.
1
√n n(n 1)
−
p
10