Download A Simple Algorithm for Complete Factorization of an N

Quant. Phys. Lett. 6, No. 1, 73-77 (2017)
73
Quantum Physics Letters
An International Journal
http://dx.doi.org/10.18576/qpl/060110
A Simple Algorithm for Complete Factorization of an
N-Partite Pure Quantum State
Dhananjay P. Mehendale∗ and Pramod S. Joag
Department of Physics, Savitribai Phule University of Pune, Pune, India-411007.
Received: 7 Feb. 2017, Revised: 21 Mar. 2017, Accepted: 23 Mar. 2017
Published online: 1 Apr. 2017
Abstract: We present a simple algorithm to completely factorize an arbitrary N-partite pure quantum state. This complete factorization
of such a pure state also specifies its complete entanglement status : whether the given N-partite pure quantum state is completely
separable (N factors), or completely entangled (no factors), or partially entangled having entangled factors of different sizes which
cannot be factored further. The problem of deciding entanglement status of a bipartite pure quantum state is one of the initial problems
encountered in quantum information research and this problem is usually tackled using the well known Schmidt decomposition
procedure. One obtains Schmidt number of the state which decides the entanglement status of the state. In this paper we first develop a
simple criterion which when fulfilled enables us to factorize given N-partite pure quantum state as tensor product of an m-partite pure
quantum state and an n-partite pure quantum state where m + n = N. This criterion gives rise to an effective mechanical procedure in
terms of an easy algorithm to perform complete factorization of given N-partite pure quantum state and thus provides an easy method
to determine complete entanglement status of the state. In this paper we carry out our discussion for the case of N-qubit pure quantum
state instead of N-qudit case for the sake of simplicity of presentation. The extension to the case of N-qudit pure quantum state is
straightforward and follows by proceeding along similar lines. We just mention this extension to avoid repetition and only briefly
demonstrate it with the help of one of the examples discussed at the end of the paper.
Keywords: Multipartite pure quantum state, criterion for entanglement and separability, complete factorization
1 Introduction
One of the central issues in quantum information theory is
whether a given multipartite pure quantum state is
separable or entangled [1,2,3,4,5]. This important
question of deciding whether a given multipartite pure
quantum state is separable or entangled is completely
solved in this paper. In this paper, we present an
algorithm to completely factorize an arbitrary N-partite
pure quantum state, that is, it is factorized until no further
factorization is possible. An N-partite pure quantum state
may be completely separable, that is, it is a tensor product
of N states, each pertaining to one of the individual parts,
or it may be a product of M (M < N) states, each
belonging to one of the M subsystems, some of them
containing two or more parts. If the completely factorized
N-partite state has such a structure, then the states of the
subsystems containing more than one part appearing in
this factorization are necessarily entangled, otherwise,
they would have factorized further. In this paper we carry
out our discussion for the case of N-qubit pure quantum
∗ Corresponding
state instead of N-qudit case for the sake of simplicity of
presentation. The extension to the case of N-qudit pure
quantum state is straightforward and follows by
proceeding along similar lines. We just mention this
extension to avoid repetition and only briefly demonstrate
it with the help of one of the examples discussed at the
end of the paper.
2 The criterion for factorization
Notation: Let |ψ i be an N-qubit pure state :
|ψ i =
2N
∑ ars |rs i
(1)
s=1
expressed in terms of the computational basis. Here the
basis vectors |rs i are ordered lexicographically. That is,
the corresponding binary sequences are ordered
lexicographically: r1 = 00 · · · 00, r2 = 00 · · · 01, . . . ,
author e-mail: [email protected]
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74
D. P. Mehendale, P. S. Joag: A Simple Algorithm for Complete...
r2N = 11 · · · 11, so that |r1 i = |00 · · ·00i, |r2 i = |00 · · · 01i,
. . . , r2N = |11 · · · 11i. Let m, n be any integers such that
1 ≤ m, n < N and m + n = N. Let the corresponding two
sets of computational basis vectors ordered
lexicographically be |i1 i, . . . , |i2m i (each of length m) and
| j1 i, . . . , | j2n i (each of length n). Rewrite |ψ i thus :
lexicographically be |i1 i, . . . , |i2m i (each of length m) and
| j1 i, . . . , | j2n i (each of length n). Then we can write
"
#
m
n
|ψ i =
2m 2n
∑ ∑ aiu jv |iu i ⊗ | jv i.
(2)
u=1 v=1
Here in the symbol aiu jv , the suffix iu jv is the juxtaposition
of the binary sequences iu and jv in that order. Thus we
get a 2m × 2n matrix A = [aiu jv ] which will be called the
2m × 2n matrix associated to |ψ i.
Lemma 1: The state |ψ i given by (1) can be factored as
the product, |ψ1 i ⊗ |ψ2 i, of an m−qubit state |ψ1 i and an
n−qubit state |ψ2 i if and only if the 2m × 2n matrix A
associated to |ψ i can be expressed as BT C where B is a
1 × 2m matrix, C is a 1 × 2n matrix and BT is the transpose
of B.
Proof: With the above notation, let
|ψ 1 i =
and |ψ2 i =
2m
∑ biu |iui,
u=1
2
|iu i ⊗ ∑ aiu jv | jv i .
(4)
v=1
|ψ i =
=
2
2
∑
|iu i ⊗ ∑ ku ai p jv | jv i
u=1
v=1
2m
2n
u=1
v=1
∑ ku|iu i ⊗ ∑ ai p jv | jv i
= |ψ1 i ⊗ |ψ2 i,
2m
∑ ku|iu i
and |ψ2 i =
u=1
2n
∑ ai p jv | jv i.
v=1
Thus |ψ i factors as stated. Conversely, suppose |ψ i given
by (1) can be factored as the product of an m−qubit state
and an n−qubit state, in that order. Then by lemma 1, the
2m × 2n non-zero matrix A associated to |ψ i can be
expressed as BT C where B is a 1 × 2m matrix and C is a
1 × 2n matrix. Hence by lemma 2, rank(A) = 1.
This proves the theorem. u=1
2n
∑ c jv | jv i.
v=1
2m 2n
∑ ∑ biu c jv |iu i ⊗ | jvi.
(3)
u=1 v=1
Comparing (2) and (3) we see that |ψ i can be factored as
|ψ1 i ⊗ |ψ2i if and only if
aiu jv = biu c jv , for u = 1, . . . , 2m and v = 1 . . . , 2n
i.e. if and only if A = BT C where B = [biu ] and C = [c jv ].
We also need the following standard result:
Lemma 2: An a × b non-zero matrix A over complex
numbers can be expressed as BT C for some 1 × a matrix
B and 1 × b matrix C if and only if rank(A) = 1.
Now we can prove the
Theorem: The state |ψ i given by (1) can be factored as
the product, |ψ1 i ⊗ |ψ2 i, of an m−qubit state |ψ1 i and an
n−qubit state |ψ2 i if and only if the 2m × 2n matrix A
associated to |ψ i is of rank 1.
Proof: With the above notation, let |rw i be the first basic
vector such that arw 6= 0. Choose integers m, n such that
1 ≤ m, n < N and m + n = N. Let the corresponding two
sets of computational basis vectors ordered
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2
∑
Consider the associated 2m × 2n matrix A = [aiu jv ].
Suppose |rw i = |i p i| jq i so that the first non-zero element
of A is the qth element in the pth row, namely ai p jq . Thus
the pth row of A is non-zero. Now, suppose rank(A) = 1.
Then there exist numbers k1 , . . . , k2m such that k p = 1 and
rowu = ku row p (u = 1, . . . , 2m ) i.e. aiu jv = ku ai p jv ,
(u = 1, . . . , 2m , v = 1, . . . , 2n ). Hence (4) can be written as
"
#
m
n
where |ψ1 i =
Then the product |ψ1 i ⊗ |ψ2i is
| ψ 1 i ⊗ |ψ 2 i =
|ψ i =
3 Algorithm
We now proceed to present our algorithm, based on the
above theorem, for complete factorization of an arbitrary
N-qubit pure quantum state. Later we will indicate how
this algorithm could be modified to cater to an arbitrary Npartite pure quantum state. We use the above notation.The
steps of the algorithm are as follows.
(i) We express the given N-qubit pure state |ψ i in terms of
the computational basis as
|ψ i =
2N
∑ ars |rs i
s=1
where the basis vectors |rs i are ordered lexicographically
as before.
(ii) Now our aim is to check as first step (using the
Theorem just proved above) whether given |ψ i has a
linear (1-qubit) factor and an (N − 1)-qubit factor. In
order to find the corresponding 21 × 2N−1 matrix A
associated to |ψ i, we rewrite this state as
" N−1
#
|ψ i =
2
∑ |iu i ⊗
u=1
2
∑ aiuiv | jv i
v=1
.
Quant. Phys. Lett. 6, No. 1, 73-77 (2017) / www.naturalspublishing.com/Journals.asp
Here the basis vectors ordered lexicographically are |i1 i =
|0i, |i2 i = |1i (each of length 1) and | j1 i, . . . , | j2(N−1) i (each
of length N − 1). Hence the associated matrix is
a
a
a
· · · a01···11
.
A = 00···0 00···01 00···010
a10···0 a10···01 a10···010 · · · a11···11
(iii) Now there are two cases.
Case I
If rank(A) = 1, then by above theorem there exist
numbers k1 , k2 such that
|ψ i = |ψ1 i ⊗ |ψ2i,
where |ψ1 i =
2
2N−1
u=1
v=1
∑ ku |iu i and |ψ2 i =
∑ ai p jv | jv i,
with k p = 1 where the pth row of A is the first non-zero row
of A (p = 1 or 2). Thus in this case the state has a factor
|ψ1 i. In this case we go back to step (i) with |ψ i = |ψ2 i.
Case II
If rank(A) 6= 1, then by above theorem we do not get a
factor like |ψ1 i = k1 |i1 i + k2|i2 i with |ψ i = |ψ1 i ⊗ |ψ2 i. In
this case our aim is to check as next step whether given |ψ i
has a 2-qubit factor and an (N − 2)-qubit factor. For this we
proceed with the originally given state |ψ i as given in the
next step (iv).
(iv) In order to find the corresponding 22 × 2N−2 matrix A
associated to |ψ i, we rewrite this state as
" N−2
#
2
|ψ i =
2
2
∑ |iu i ⊗ ∑ aiu iv | jv i
u=1
.
v=1
Here the basis vectors ordered lexicographically are |i1 i =
|00i, |i2 i = 01, |i3 = |10i, |i4 i = |11i (each of length 2)
and | j1 i, . . . , | j2(N−2) i (each of length N − 2). Thus

a000···0
a010···0
A=
a100···0
a110···0
a000···01
a010···01
a100···01
a110···01
a000···010
a010···010
a100···010
a110···010
···
···
···
···

a001···11
a011···11 
.
a101···11 
a111···11
Again there are two cases.
Case I
If rank(A) = 1, then by above theorem there exist
numbers k1 , . . . , k4 such that
|ψ i = |ψ1 i ⊗ |ψ2i,
where |ψ1 i =
4
∑ ku |iu i
u=1
and |ψ2 i =
2N−2
∑ ai p jv | jv i,
75
|ψ i = |ψ1 i ⊗ |ψ2 i. In this case our aim is to check as next
step whether given |ψ i has a 3-qubit factor and an
(N − 3)-qubit factor. In order to find the corresponding
23 × 2N−3 matrix A associated to |ψ i, we rewrite this state
as
" N−3
#
3
|ψ i =
2
2
∑ |iu i ⊗ ∑ aiuiv | jv i
u=1
.
v=1
Here the basis vectors ordered lexicographically are
|i1 i = |000i, . . ., |i8 i = |111i (each of length 3) and
| j1 i, . . . , | j2(N−3) i (each of length N − 3). We again
construct associated matrix matrix A = [aiu jv ]. This,
again, leads to two separate cases, namely, whether the
rank of the associated matrix A is equal to unity or not
and so on. Thus, as above the algorithm continues until
|ψ i is completely factored.
4 Generalization to N-qudit case
For this remark we use the usual notation. A general Nqudit state with dimensions d1 , d2 , . . . , dN can be written
as
|ψ i = ∑ ai1 i2 ...iN |i1 i2 . . . iN i
i1 ,i2 ,...iN
where ik ∈ {0, 1, . . . , dk − 1}; k = 1, 2, . . . , N. To check
whether |ψ i has a linear factor (on left), we re-write this
state as
|ψ i = |0i ⊗
∑
i2 ,...,iN
a0i2 ...iN |i2 . . . iN i + |1i ⊗
+ · · · + |d1 − 1i ⊗
∑
i2 ,...,iN
∑
i2,...,iN
a1i2 ...iN |i2 . . . iN i
a(d1 −1)i2 ...iN |i2 . . . iN i.
This allows us to write down the d1 by (d2 × d3 × · · · × dN )
matrix A associated to |ψ i and |ψ i facotrs as
|ψ i =
d1 −1
!
∑ ki |ii
i=0
⊗
∑
i2 ,...,iN
!
a pi2 ...iN |i2 . . . iN i ,
if and only if rank(A) = 1. Here k p = 1 and row p is the first
non-zero row of A. If rank(A) 6= 1, we check whether a
two partite state factors out and so on. From this point, the
algorithm proceeds exactly as in the N-qubit case. Please
see Example (iv) below.
v=1
with k p = 1 where the pth row of A is the first non-zero
row of A. Thus in this case the state has a factor |ψ1 i. In
this case we go back to step (i) with |ψ i = |ψ2 i.
Case II
If rank(A) 6= 1, then by above theorem we do not get a
factor like |ψ1 i = k1 |i1 i + · · · + k4 |i4 i with
5 Examples
(i) Consider following example ([6], page 423)
1
1
|ψ i = √ |01i − √ |10i.
2
2
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76
D. P. Mehendale, P. S. Joag: A Simple Algorithm for Complete...
We proceed as per our algorithm and first check whether
|ψ i has a linear factor (on left). For this we rewrite |ψ i
thus:
two qubit factor to given |ψ i. For this we construct the
following associated matrix
 A 1: 1 
0 2 2 0
0 0 0 0


A =
0 0 0 0
0 21 12 0
1
1
|ψ i = 0|00i + √ |01i − √ |10i + 0|11i.
2
2
1
1
= |0i ⊗ 0|0i + √ |1i + |1i ⊗ − √ |0i + 0|1i
2
2
Therefore the associated matrix A is
"
#
0 √12
A =
− √12 0
Clearly, rank(A) = 2 > 1, so that |ψ i has no linear factor
and therefore the state |ψ i is entangled.
(ii) Consider the following two qubit state
1
1
1
1
|ψ i = √ |00i − √ |01i + √ |10i − √ |11i.
3
3
6
6
To check whether |ψ i has a linear factor (on left), we
rewrite |ψ i thus:
1
1
1
1
|ψ i = |0i ⊗ √ |0i − √ |1i + |1i ⊗ √ |0i − √ |1i
3
3
6
6
Therefore the associated matrix A is
" 1
#
√ − √1
3
A = 13
√ − √1
6
6
√
Clearly, row2 = 1/( 2) row1 so that rank(A) = 1 and
1
1
1
1
1
|ψ i = |0i ⊗ √ |0i − √ |1i + |1i ⊗ √ √ |0i − √ |1i .
3
3
2
3
3
√
Hence with k1 = 1, k2 = 1/( 2), |ψ i factors into two
linear factors as follows:
1
1
1
|ψ i = |0i + √ |1i ⊗ √ |0i − √ |1i .
3
3
2
Hence the state |ψ i is separable.
(iii) Consider the following four qubit state,
1
|ψ i = [|0001i + |0010i + |1101i + |1110i]
2
First, to check whether there exists
construct the associated matrix A :
1 1
0 2 2 000 0
A =
0 0 0 0 0 21 12
a linear factor we
0
0
Clearly, rank(A) is greater than 1, therefore, clearly |ψ i
has no linear factor and therefore it is entangled. We now
continue as per algorithm to check whether there exists a
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Clearly, rank(A) = 1 and |ψ i factors into two 2-qubit
factors:
1
1
|ψ i = √ [|00i + |11i] ⊗ √ [|01i + |10i] .
2
2
By applying our method we can readily see that these two
factor states of |ψ i are both entangled states.
(iv) Consider the following 3-partite state comprising a
qubit, a qutrit and, a qudit with d = 4.
r
r
r
3
35
7
5 35
i|001i +
|003i +
i|022i
|ψ i =
4 241
241
2 241
r
r
15
9
5
5
−
|101i + 3
i|103i − √
|122i.
4 241
241
2 241
As per the algorithm to check whether there exists a
linear factor, we rewrite |ψ i thus:
i
√
√
√
1 h
|ψ i = √
|0i ⊗ 5 35i|01i + 4 35|03i + 6 7i|22i
4 241
i
√
√
1 h
|1i ⊗ −15 5|01i + 12 5i|03i − 18|22i
+ √
4 241
Therefore the associated 2 by (3 × 4) matrix A is
q

q
q
35
35
7
3
i 0
0
·
·
·
0
0 45 241
2
241 i
q241
q
A=
5
5
−9
√
0 −15
4
241 0 3
241 i 0 · · · 0 2 241
where columns six to nine consist of zeros.
By applying the algorithm, |ψ i factors thus:
3i
|ψ i = |0i + √ |1i ⊗
7

0
,
0
!
√
√
√
6 7
5 35
4 35
√
|03i + √
i|22i .
i|01i + √
4 241
241
4 241
To check the second factor, say |ψ2 i, for a linear factor, we
rewrite it thus:
√
√
1 h
|0i ⊗ 0|0i + 5 35i|1i + 0|2i + 4 35|3i
|ψ2 i = √
4 241
+ |1i ⊗ 0|0i + 0|1i + 0|2i + 0|3i
i
√
+ |2i ⊗ 0|0i + 0|1i + 6 7i|2i + 0|3i
Therefore the associated 3 by 4 matrix A is
 √
√ 
0
5
35i
0
4
35
1 
,
0 0
0
0
A= √
4 241 0 0 6√7i 0
Quant. Phys. Lett. 6, No. 1, 73-77 (2017) / www.naturalspublishing.com/Journals.asp
Here rank(A) = 2 > 1, so that |ψ2 i is entangled. Thus the
state |ψ i is entangled and it has one linear factor and one
bipartite entangled factor.
Conclusion:
We believe that our algorithm is a very useful tool to
understand the structure of a multipartite pure quantum
state vis-a-vis entanglement.
Acknowledgement
One of us (DPM) thanks Dr. M. R. Modak, S. P. College,
Pune-411030, India, for some useful discussions.
References
[1] Werner R.F. Phys.Rev.A 40 4277 (1989).
[2] Alber G., Beth T, Horodecki M, Horodecki P, Horodecki
R, Rotteler M, Weinfurter H, Werner R and Zeilinger A
Quantum Information : An Introduction to Basic Theoretical
Concepts and Experiments (Berlin : Springer) (2001).
[3] Bouwmeester D, Ekert A, and Zeilinger A The Physics of
Quantum Information (Berlin : Springer) (2000).
[4] Nielsen A and Chuang I Quantum Computation and
Quantum Information(Cambridge : Cambridge university
Press) (2000).
[5] Peres A Quantum Theory : Concepts and Methods (Dordrecht
: Kluwer) (1993).
[6] Colin P. Williams, Explorations in Quantum Computing,
second edition, Springer-Verlag London Limited (2011).
77
Dhananjay
P.
Mehendale, Prof. Dhananjay
P.
Mehendale
served
as
associate
professor
in
S.
P.
College,
affiliated to Savitribai Phule
University of Pune, Pune,
India-411007. The subjects
of his interest and study are
Physics, Mathematics, and
Engineering and students of science and engineering have
successfully completed their project work under his
supervision. He taught various courses in Physics,
Electronic Science, and Computer Science departments,
and has done research in various areas of these subjects.
He worked on various science and engineering projects
and one of his projects won award in the national science
projects competition organized by Department of Science
and Technology. His research interest these days is topics
in Quantum Computation and Quantum Information and
this paper is an outcome of this new interest.
Pramod
S.
Joag
Prof. Pramod S. Joag
served as a professor
of Physics at Savitribai
Phule University of Pune,
Pune, India-411007. He is
presently with Indian Institute
of Science Education and
Research, Pune. His research
interests include quantum
correlations, nonlocality issues and foundations of
quantum mechanics. He has proposed geometric
measures of entanglement and quantum discord in
n-partite quantum states and states of n-partite fermionic
systems. He has given separability criteria for n-partite
quantum states including mixed states. He has given a
combinatorial approach to the quantum correlations in
multipartite quantum systems and used it to prove the
degree conjecture for the separability of multipartite
quantum systems. He has found an algorithm to
classically simulate spin s singlet state for an infinite
sequence of spins.
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Natural Sciences Publishing Cor.