PHYSICAL REVIEW A, VOLUME 65, 022107
Preserving entanglement under perturbation and sandwiching all separable states
Robert B. Lockhart1,* and Michael J. Steiner2,†
1
Mathematics Department, United States Naval Academy, Annapolis, Maryland 21402
2
Naval Research Laboratory, Washington, D.C. 20375
共Received 22 September 2000; published 14 January 2002兲
Every entangled state can be perturbed and stay entangled. For a large class of pure entangled states, which
includes all bipartite and all maximally entangled ones, we show how large the perturbation can be. Maximally
entangled states can be perturbed the most. For each entangled state in our class, we construct hyperplanes
which sandwich the set of all separable states. As the number of particles, or the dimensions of the Hilbert
spaces for two of the particles increases, the distance between two of these hyperplanes goes to zero.
DOI: 10.1103/PhysRevA.65.022107
PACS number共s兲: 03.65.Ud, 03.67.⫺a
I. INTRODUCTION
Quantum systems display many properties, which are not
observed in the macroscopic world. One of the most fascinating is entanglement. Starting with the fundamental paper
of Einstein et al. 关1兴 and continuing through the works of
Bell 关2兴 and others to the present day, it has played a key role
in debates over the foundations, completeness, and interpretation of quantum mechanics 关3–5兴.
Today entanglement is playing a key role in the burgeoning field of quantum information 关6,7兴. It is fundamental to
teleportation 关8 –10兴, secure key distribution 关11,12兴, dense
coding 关10兴, quantum error correction, 关13,14兴 and other applications 关15兴. Thus increasingly, researchers around the
world are working with, or at least trying to work with, entangled states.
Obviously then, it is important to know how much one
can perturb an entangled state and still have entanglement.
Such considerations arise when one takes experimental error
or decoherence into account in applications. For instance,
while one wants to work with maximally entangled states in
quantum communication, not all is lost if the maximally entangled states are perturbed, provided they remain entangled
to some extent. Indeed one can purify ensembles of entangled states to get subensembles of highly entangled states
关16兴. A striking example of being able to use states which
remained entangled though not maximally so while the experiment ran, is the recent one at Aarhus in which two clouds
of atoms were entangled for half a millisecond 关17兴.
With regard to decoherence, there are well-known decoherence channels 关18兴 in which a maximally entangled state
quickly becomes the totally mixed state. The question naturally arises as to how long it maintains any entanglement at
all. To answer such a question one needs to know the path
the decohering state takes and a neighborhood of the original
state in which all states are entangled.
In this paper, for a large class of entangled pure states, we
construct open neighborhoods of pure and mixed entangled
states. The class of entangled states we do this for includes
all bipartite entangled pure states, all multiparticle maxi-
*Email address: rbl@usna.edu
†
Email address: mjs@mike.nrl.navy.mil
1050-2947/2002/65共2兲/022107共4兲/$20.00
mally entangled states, and many others. In particular, suppose we have p particles with the ␣th one modeled on
Cn ␣ , ␣ ⫽1, . . . ,p. Then our system of p particles is modeled
j
on the Hilbert space CN ⫽Cn 1 丢 ¯ 丢 Cn p . Let 兵 兩 ␺ ␣␣ 典 其 be an
j
j
orthonormal basis for Cn ␣ . Then 兵 兩 ␺ 11 , . . . , ␺ pp 典 其 is an orthoN
normal basis for C . Without loss of generality, assume n 1
⭐¯⭐n p . The entangled states we consider in this paper are
the ones of the form
n1
␺ ⫽ 兺 v j 兩 ␺ 1j , . . . , ␺ pj 典 ,
j⫽1
共1兲
n
1
where ⌺ j⫽1
兩 v j 兩 2 ⫽1 and no v j equals 1. 共If a v j ⫽1, the ␺
would not be entangled.兲
Note that by using a Schmidt decomposition, every bipartite state can be expressed in this form. So too is every maximally entangled state on Cn 丢 ¯ 丢 Cn of this form. Indeed
they are the ones with each v j ⫽1/冑n. Finally, notice that by
using local operations, i.e., acting on CN by U(n 1 ) 丢 ¯
n
n
j
j
丢 U(n p ), any state ␸ ⫽⌺ j 1⫽1 ¯⌺ j p⫽1 w j ¯ j 兩 ␺ 11 , . . . , ␺ p p 典
1
p
1
p
can be expressed in the form of Eq. 共1兲, provided for each j i
there is at most one w j 1 ... j p , which is not 0.
The neighborhood, G ␺ , of entangled states we construct
for ␺ lies in the set of all mixed and pure states, since this is
the physically reasonable thing to do. Thus we need to express ␺ in terms of its density matrix E ␺ . For each E ␺ , we
find the distance to the nearest pure product states. What we
find is the following:
Theorem 1. Let E ␺ be the entangled, pure state
n1
⌺ j,k⫽1
v j¯
v k 兩 ␺ 1j , . . . , ␺ pj 典具 ␺ 1k , . . . , ␺ pk 兩 . The closest, pure,
product states to E ␺ are a distance 冑2(1⫺ 兩 v j 0 兩 2 ) away from
E ␺ , where 兩 v j 0 兩 ⫽max兵兩v1 兩, . . . , 兩 v p 兩 其 . An example of such a
closest, pure product state is the projection, S ␺
j
j
j
j
⫽ 兩 ␺ 1 0 , . . . , ␺ p 0 典具 ␺ 1 0 , . . . , ␺ p 0 兩 .
We shall give the proof of this theorem in the next section. For now, let us describe how it is used to construct G ␺ ,
which is quite simple. Take C ␺ to be the N⫺1 dimensional
hyperplane which contains S ␺ and is perpendicular to the
line, L ␺ , connecting E ␺ with (1/N)I, the totally mixed state.
Similarly, let F ␺ be the parallel hyperplane, which contains
any projection that is furthest away from E ␺ . 共These are
65 022107-1
©2002 The American Physical Society
ROBERT B. LOCKHART AND MICHAEL J. STEINER
PHYSICAL REVIEW A 65 022107
precisely the projections that commute with E ␺ , a separable
example of which is R⫽ 兩 ␺ 11 ,..., ␺ 1p⫺1 ␺ 2p 典具 ␺ 11 ,..., ␺ 1p⫺1 ␺ 2p 兩 .兲
Then we have the following theorem.
Theorem 2. All separable states either lie on one
of the hyperplanes C ␺ or F ␺ or lie between them. Thus
every state outside the sandwich formed by C ␺ and F ␺
is entangled. This region, G ␺ , outside the C ␺ , F ␺
sandwich is an open, connected neighborhood of
n1
E ␺ ⫽⌺ j,k⫽1
v j v k 兩 ␺ 1j , . . . , ␺ pj 典具 ␺ 1k , . . . , ␺ pk 兩 . A state Q is in
G ␺ , if and only if, 具 Q,E ␺ 典 ⫽Tr(QE ␺ )⬎ 兩 v j 0 兩 2 where, as before, 兩 v j 0 兩 ⫽max兵兩v1兩, . . . , 兩 v p 兩 其 .
Again we shall postpone the proof until the next section
and instead shall now make a few remarks and give one last
theorem.
Remark 3. G ␺ will be largest when 兩 v j 0 兩 2 is smallest.
n
1
Since ⌺ j⫽1
兩 v j 兩 2 ⫽1, this occurs when 兩 v 1 兩 2 ⫽¯⫽ 兩 v p 兩 2
⫽1/n 1 . Thus it is these states, which are the maximally entangled ones when n 1 ⫽¯⫽n p , that can withstand the
greatest amount of perturbation and still be in G ␺ and so
remain entangled.
Remark 4. Since the plane C ␺ contains the separable state
S ␺ , there is no larger neighborhood of E ␺ consisting solely
of entangled states, given by an inequality, 具 Q,E ␺ 典 ⬎K, than
G ␺ . In this sense G ␺ is the largest neighborhood of E ␺ consisting solely of entangled states. It may, however, not contain the largest ball of entangled states centered at E ␺ .
Remark 5. It is well known 关19–22兴 that if E ␺ is a maximally entangled state on Cn 丢 ¯ 丢 Cn , then the separable
state on the line L ␺ , which connects E ␺ with (1/N)I, that is
closest to E ␺ is W(s)⫽(1⫺s)(1/N)I⫹sE ␺ , where s⫽(1
⫹n p⫺1 ) ⫺1 . When p⫽2, it is easy to compute that this state
lies in the hyperplane C ␺ . This has two important consequences 共a兲 of all separable states, not just those on L ␺ , the
state W(s) is the closest to E ␺ , and 共b兲 the neighborhood,
G ␺ , contains the largest open ball of entangled states centered at E ␺ . Thus in this case G ␺ is the largest physically
usable neighborhood of E ␺ consisting solely of entangled
states. When p⬎2, the state W(s) lies inside the sandwich
formed by C ␺ and F ␺ . This means G ␺ might not, in this
case, contain the largest ball of entangled states centered at
E ␺ . It also means that, in this case, W(s) is not the closest
separable state to E ␺ . Indeed, simple geometry shows the
line that contains E ␺ and intersects the line connecting W(s)
with S ␺ perpendicularly, intersects that line at a separable
state which is closer to E ␺ than W(s).
Remark 6. From the last example given in the remark just
made, it should be clear that we do not claim all states between C ␺ and F ␺ are separable. Many are entangled. In fact,
numerical simulation for low dimensional bipartite cases indicates that a large percentage of the states inside the sandwich are entangled. However, there are no separable states
outside the sandwich.
To finish this introduction, we shall state our last
theorem. Basically it says that for a system modeled on
Cn 1 丢 ¯ 丢 Cn p with n 1 ⭐¯⭐n p , the thickness of the thinnest
sandwich that contains all separable states goes to 0 as n p⫺1
or p increases to infinity. This means that for systems with a
large number of particles, or with at least two particles modeled on large dimensional Hilbert spaces, all separable states
cluster near a hyperplane that contains the totally mixed
state. Before stating the theorem we have to make the following definition.
Definition 7. For the set of integers 兵 n 1 ,...,n p 其 ,
and any partition ␲ of the set into two
let
f (␲)
subsets,
兵兵 n ␲ 1 ,...,n ␲ k 其 , 兵 n ␲ k⫹1 ,...,n ␲ p 其其 ,
⫽min(n␲1¯n␲k,n␲k⫹1¯n␲p). Then ␬ (n 1 ,....,n p ) is the maximum over all partitions of 兵 n 1 ,...,n p 其 into two subsets of
1/f ( ␲ ).
For example, if the system consists of p qubits,
then ␬ ⫽2 ⫺m , if p⫽2 2m and ␬ ⫽2 ⫺(m⫺1) if p⫽2 2m⫺1 .
On the other hand if, for instance, the system is modeled
on C2 丢 C3 丢 C4 丢 C30, then ␬ ⫽1/24. In all cases,
␬ ⭐(n 1 n 3 ¯n p⫺1 ) ⫺1
if
p
is
odd
and
␬
⭐max关(np⫺1n1n3¯np⫺2)⫺1,(npn2¯np⫺3)⫺1兴 if p is even.
Theorem 8. Consider a quantum system modeled on
Cn 1 丢 ¯ 丢 Cn p . There exist parallel hyperplanes that are a distance ␬ 冑N/(N⫺1) apart and that have the property that all
separable states either lie on one of the planes or lie between
them. In particular, for every separable state T the largest ball
of separable states centered at T must have a radius no bigger
than ␬ 冑N/(N⫺1).
II. PROOFS OF THEOREMS
In this section we prove our theorems, starting with the
n1
first. For ␺ ⫽⌺ j⫽1
v j 兩 ␺ 1j ,..., ␺ pj 典 , the associated projection is
n1
E ␺ ⫽⌺
v j¯
v k 兩 ␺ 1j ,..., ␺ pj 典具 ␺ 1j ,..., ␺ pj 兩 . For ␮ ⫽1, . . . ,p,
j,k⫽1
n
take A ␮ to be the projection ⌺ j ␮ k
␮ , ␮ ⫽1
j
k
a ␮ j¯
a 兩 ␺ ␮␮ 典具 ␺ ␮␮ 兩 on
␮ ␮k␮
Cn ␮ and A to be the pure product projection A 1 丢 ¯ 丢 A p .
Then
n
n
A⫽⌺ j 1 k
1, 1 ⫽1
j
¯⌺ j p ,k
p
j
p ⫽1
a 1 j 1 ¯a p j¯
a ¯¯
a pk p
p 1k 1
j
j
⫻ 兩 ␺ 1 1 , . . . , ␺ p p 典具 ␺ 1 1 , . . . , ␺ p p 兩 .
We want to find the smallest distance from such an A to
E ␺ . To do so, first note the square of the distance from A to
E ␺ is
储 E ␺ ⫺A 储 2 ⫽ 具 E ␺ ⫺A,E ␺ ⫺A 典
⫽ 具 E ␺ ,E ␺ 典 ⫺2 Re具 E ␺ ,A 典 ⫹ 具 A,A 典 .
共2兲
Since E ␺ and A are positive semidefinite Hermitian operators, their inner product is real and equals Tr(E ␺ A). Hence
储 E ␺ ⫺A 储 2 ⫽2 关 1⫺Tr(E ␺ A) 兴 . This will be minimum when
n1
n1 ¯
¯
Tr(E ␺ A)⫽⌺ k⫽1
a 1k , . . . ,a
is maxiv k¯
pk ⌺ j⫽1 v j a 1 j ¯a p j
mum.
n1 ¯
Setting ⌽⫽⌺ j⫽1
v j a 1 j , . . . ,a p j , we see that Tr(E ␺ A)
n
⫽⌽⌽̄⫽ 兩 ⌽ 兩 2 .
In
turn
兩⌽兩2⫽兩⌺ 1 ¯
v j a 1 j ¯a p j 兩 2
n
n
j⫽1
1
1
⭐(⌺ j⫽1
兩 v j 兩兩 a 1 j 兩 ¯ 兩 a p j 兩 ) 2 ⫽(⌺ j⫽1
兩 v j 兩 r 1 j ¯r p j ) 2 , where r ␮ j
⫽ 兩 a ␮ j 兩 . This last expression is equivalent to 兩 具 ␳ ,V ␤ 典 兩 2 ,
where V is the n 1 ⫻n 1 diagonal matrix with 兩 v j 兩 as the diagonal entries and ␳ and ␤ are the n 1 dimensional vectors with
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PRESERVING ENTANGLEMENT UNDER PERTURBATION . . .
PHYSICAL REVIEW A 65 022107
the components ␳ j ⫽r 2 j ¯r p j and ␤ j ⫽r 1 j . Using the
Cauchy-Schwarz inequality and the definition of the operator
norm of a matrix, we obtain the inequality 兩 具 ␳ ,V ␤ 典 兩 2
2
2
储 ␤ 储 2 . Since V is a diagonal matrix, 储 V 储 op
⭐ 储 ␳ 储 2 储 V 储 op
n1
2
2
2
⫽max 兩v j兩 ⫽兩v j0兩 . Furthermore, by assumption ⌺ j⫽1 r ␮ j ⫽1
and so 储 ␤ 储 2 ⫽1 and 储 ␳ 储 2 ⭐1. Thus Tr(E ␺ A)⫽ 兩 具 ␳ ,V ␤ 典 兩 2
j
j
j
j
⭐ 兩 v j 0 兩 2 . Noting that if S ␺ ⫽ 兩 ␺ 10 , . . . , ␺ p0 典具 ␺ 11 , . . . , ␺ pp 兩 ,
2
then Tr(E ␺ S ␺ )⫽ 兩 v j 0 兩 , we obtain the proof of the first theorem.
The proof of the second, basically, uses simple vector
operations and facts from trigonometry. For two states K
and Q, take V(K,Q) to be the vector with tail at K and
head at Q. As above, take S ␺ to be any of the closest pure
product states to E ␺ and consider the triangle whose sides
are V((1/N)I,E ␺ ), V((1/N)I,S ␺ ), and V(S ␺ ,E ␺ ). Since E ␺
and S ␺ are rank 1 projections, the length of the first two sides
is 冑(N⫺1)/N. Moreover, we have just proven the length
of the third side is 冑2(1⫺ 兩 v j 0 兩 2 ). Since this is true
regardless of S ␺ , the projection of V((1/N)I,S ␺ ) onto
V((1/N)I,E ␺ ) will be the same for all S ␺ . This means
that all S ␺ lie in the hyperplane, C ␺ , which is perpendicular
to V((1/N)I,E ␺ ). This hyperplane divides the set of states
into two regions: 共i兲 one which contains the plane and all
states on the (1/N)I side of the plane, and 共ii兲 G ␺ , which is
the open, connected set that includes E ␺ and all states on
that side of C ␺ . A state, Q, is in G ␺ , if and only if the
projection of V((1/N)I,Q) onto V((1/N)I,E ␺ ), is longer
than the projection of V((1/N)I,S ␺ ) onto V((1/N)I,E ␺ ),
i.e., 具 Q⫺(1/N)I,E ␺ ⫺(1/N)I 典 ⬎ 具 S ␺ ⫺(1/N)I,E ␺ ⫺(1/N)I 典
⫽ 兩 v j 0 兩 2 ⫺(1/N). Expanding the left-hand side of this
inequality and using the fact that if P is rank 1, then
具 P,(1/N)I 典 ⫽Tr((1/N)I P)⫽1/N, we get 具 Q,E ␺ 典 ⫺1/N
⬎ 兩 v j 0 兩 2 ⫺1/N.
This proves the inequality in the theorem and it also
shows why there are no separable states in G ␺ . Indeed due
to the convexity of the set of separable states, if there were a
separable state in G ␺ , then there would have to be a pure,
separable state in G ␺ . But this last inequality shows that
such a state would be closer to E ␺ than is possible by theorem 共1兲.
The same reasoning can be applied to F ␺ . By Eq. 共2兲 we
see that the projections 共separable or entangled兲 which are
furthest from E ␺ are those whose inner product with E ␺ is 0.
These are precisely the ones that commute with E ␺ . Since
they all must lie on F ␺ , it follows that there can be no states
共separable or entangled兲 on the side of F ␺ that does not
contain E ␺ . Hence all separable states either lie on C ␺ or F ␺
or lie between them.
As for the last theorem, first note that the projection
of V((1/N)I,S ␺ ) onto V((1/N)I,E ␺ ) has length ( 兩 v j 0 兩 2
⫺(1/N)) 冑N/(N⫺1) and the projection of a furthest away
pure state onto V((1/N)I,E ␺ ) has length 1/N 冑N/(N⫺1).
Thus the distance between C ␺ and F ␺ is 兩 v j 0 兩 2 . Since
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n
1
⌺ j⫽1
兩 v j 兩 2 ⫽1, the minimum 兩 v j 0 兩 is 1/n 1 . To obtain the last
theorem we express CN as the tensor product of CN 1
and CN 2 , where N 1 ⫽n ␲ 1 ,...,n ␲ k and N 2 ⫽n ␲ k⫹1 ,...,n ␲ p ,
with ␲ being the partition that makes 1/f ( ␲ ) maximum.
Hence CN 1 ⫽Cn ␲ 1 丢 ¯ 丢 Cn ␲ k and CN 2 ⫽Cn ␲ k⫹1 丢 ¯ 丢 Cn ␲ p .
Any state that is separable in Cn 1 丢 ¯ 丢 Cn p is also
separable in CN 1 丢 CN 2 . However, these latter states are all
sandwiched between hyperplanes C ␺ and F ␺ , which are associated with an entangled state for which 兩 v j 兩 2 ⫽1/N 1 . It
follows from what we said a moment ago that for such a state
the distance between C ␺ and F ␺ is 1/N 1 冑N/(N⫺1)
⫽ ␬ N/(N⫺1).
ACKNOWLEDGMENT
This work was supported by a grant from the Office of
Naval Research.
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ROBERT B. LOCKHART AND MICHAEL J. STEINER
PHYSICAL REVIEW A 65 022107
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