Download Entanglement or Separability

Bachelor Thesis
Entanglement or Separability
an introduction
Lukas Schneiderbauer
December 22, 2012
Quantum entanglement is a huge and active research field these days. Not only the
philosophical aspects of these ’spooky’ features in quantum mechanics are quite interesting,
but also the possibilities to make use of it in our everyday life is thrilling. In the last few
years many possible applications, mostly within the ’Quantum Information’ field, have been
developed.
Of course to make use of this feature one demands tools to control entanglement in
a certain sense. How can one define entanglement? How can one identify an entangled
quantum system? Can entanglement be measured? These are questions one desires an
answer for and indeed many answers have been found.
However today entanglement is not yet fully in control by mathematics; many problems
are still not solved. This paper aims to provide a theoretical introduction to get a feeling
for the mathematical problems concerning entanglement and presents approaches to handle
entanglement identification or entanglement measures for simple cases.
The reader should be aware of the fact that this paper constitutes in no way the claim
to be a summary of all available methods, there exist many more than demonstrated in the
following pages.
Student ID number
Degree course
assisted by
0907633
Physics
ao. Univ.-Prof. i.R. Dr. Reinhold Bertlmann
Contents
1 Introduction
3
2 Preliminary definitions
2.1 Composite quantum systems . . . . . . . . . . . . . . . . . . . . .
2.2 Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Reduced density operator for a bipartite quantum system
2.3 Entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 A pure correlated composite state . . . . . . . . . . . . . .
2.4 Hilbert-Schmidt space . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Qudit systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Qubit systems . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Pure bipartite qudit states
3.1 Schmidt decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Von-Neumann entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
7
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4 Mixed bipartite qudit states
4.1 A criterion for non-entanglement . . . . . . . . . .
4.2 Generalization of the Von-Neumann entropy . . . .
4.2.1 Requirements for entanglement measures . .
4.2.2 Entanglement measures . . . . . . . . . . .
4.3 Generalization of the Schmidt rank . . . . . . . . .
4.4 Entanglement witnesses . . . . . . . . . . . . . . .
4.4.1 Entanglement Witness Theorem (EWT) . .
4.4.2 Positive Map Theorem (PMT) . . . . . . .
4.4.3 Positive Partial Transpose (PPT) Criterion
4.4.4 Bertlmann-Narnhofer-Thirring Theorem . .
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6 The choice of factorization
6.1 The choice of factorization for pure states . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The choice of factorization for mixed states . . . . . . . . . . . . . . . . . . . . . . . .
22
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7 Conclusion
24
References
24
5 Multipartite states
5.1 Some entanglement measures . . . . . . . .
5.1.1 Geometric measure of entanglement
5.1.2 Measure of entanglement by Barnum
5.2 An entanglement witness . . . . . . . . . . .
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1 INTRODUCTION
1 Introduction
Quantum entanglement, named by Erwin Schrödinger1 , is a quantum mechanics phenomenon which
was first outlined by Albert Einstein, Boris Podolski and Nathan Rosen in 1935 and led to the famous EPR paradoxon.[6] Their result was that quantum mechanics can’t be a correct theory since
entanglement would lead to a violation of the classical principle of local realism:
»We are thus forced to conclude that the quantum-mechanical description of physical
reality given by wave functions is not complete.« - Einstein, Podolski, Rosen [6]
On the other hand quantum mechanics was succussfully confirmed by experiments. Therefore the
idea of ’hidden variables’ arose, meaning quantum theory is not fundamental but a statistical theory
which covers the fundamental theory. This makes it possible to deny a violation of local realism as
matter of principle and at the same time to ’believe’ in the success of the theory.
1964 John Stewart Bell provided a way to determine experimentally whether hidden variables exist
or not (see his Bell inequalities[4]). Surprisingly the experiments turned out to deny the existence of
such hidden variables.
Although the interpretation of quantum theory may vary, today quantum mechanics is widely accepted as fundamental theory by physicists.
This work is an introduction developing criteria to identify entangled quantum systems for specific
cases. To start at an uniform level it first provides some fundamental mathematical definitions in
section 2. Mathematical entities with physical meaning like the Hilbert space or a density operator
are defined. Also a (mathematical) answer to the question »What is entanglement?« is given. The
following derivations are based on these definitions.
Next it faces one of the simplest non trivial problems in section 3: a pure bipartite qudit state. Two
important concepts, the Schmidt-decomposition and the Von-Neumann entropy, are introduced which
will prove to be useful in further studies. The Bell states will function as examples.
In chapter 4 mixed bipartite qudit systems are discussed. First a simple but important criterion for
non-entanglement is derived. Generalization approaches of the Von-Neumann entropy and the Schmidt
rank are presented as entanglement measures which will include the entanglement of formation, the
relative entropy of entanglement, the entanglement of distillation, entanglement cost and a HilbertSchmidt measure. Also the concept of entanglement witnesses with a few applications like the Positive
Partial Transpose Criterion is displayed.
Furthermore a small outlook for multipartite systems is given in section 5. Two more measures of
entanglement are presented and a specific entanglement witness for an N -partite system is given.
And last but not least the issue of the possibility to choose different algebra factorizations with
respect to the consequences to entanglement is discussed in chapter 6.
1
The original German name for this ’spooky’ feature was »Verschränkung«.
3
2 PRELIMINARY DEFINITIONS
2 Preliminary definitions
2.1 Composite quantum systems
A quantum system is represented by a Hilbert space H. Let’s consider a number of such systems,
denoted by HA , HB and so forth.
Definition 2.1. It is postulated that the composite system of these subsystems HA , HB , ... is
represented by their Product-Hilbert space HAB...
HAB... = HA ⊗ HB ⊗ ...
(2.1)
An operator O in a composite system S AB... is denoted by OAB... . For a product state |ΨA i⊗|ΨB i⊗...
one also writes |ΨA , ΨB , ...i.
2.2 Density operator
Definition 2.2. Given an ensemble {|ϕi i, pi } of N possible pure states |ϕi i with probability pi one
defines the density operator ρ
N
N
X
X
ρ :=
pi |ϕi ihϕi |,
pi = 1
(2.2)
i=1
i=1
Note that the |ϕi i are not the eigenstates of ρ, therefore not orthogonal in general. This density
operator ρ represents a mixed state 2 (see e.g. [2]) of a quantum system and has the following important
properties:
hϕ|ρ|ϕi ≥ 0 ∀|ϕi ∈ H ⇐⇒ ρ† = ρ
(2.3)
tr(ρ) = 1
(2.4)
Proof. Eq.
P (2.3) is obvious and eq. (2.4) can be shown by just using the definition of the trace:
tr(ρ) ≡ i hΨi |ρ|Ψi i with an arbitrary ON basis {Ψi }.
2.2.1 Reduced density operator for a bipartite quantum system
Definition 2.3. Let ρAB be a density operator on HAB . Then the reduced density operator ρA is
defined as
ρA := trB (ρAB ),
(2.5)
P
AB |ΨB i with
whereas trB (Z AB ) is called a partial trace, which is defined by trB (Z AB ) := n hΨB
n |Z
n
A
arbitrary basis {|Ψi i}. The result is an operator on H . More to partial traces can be found in [2].
The reduced density operator ρA can be envisioned as the state in the subsystem S A . All probability
predictions for local measurements (the related observable is in the form of A ⊗ I) on system S A can
be allocated to the reduced density operator.
2
A mixed state is a generalization of a pure state. To see this, set N = 1 in eq. (2.2) and one gets the density operator
of a pure state, namely ρpure = |ϕihϕ|.
4
2.3 Entangled states
2 PRELIMINARY DEFINITIONS
2.3 Entangled states
Definition 2.4. A composite state is called correlated if and only if its density operator ρAB... can
not be written as a product operator, i.e.
ρAB... 6= ρA ⊗ ρB ⊗ ...
(2.6)
Since ρ represents a mixed state in general, correlation alone may not necessarily imply a deviation
from classical views. To classify a non classical effect we go on with further definitions.
Definition 2.5. A state is called separable if and only if the density operator ρAB... can be written as
ρAB... =
n
X
B
pr ρA
r ⊗ ρr ⊗ ...
(2.7)
r=1
Note, that for n = 1, ρAB... is not correlated. The state for n 6= 1, that is a separable correlated
state, is called a classical correlated state. From the definition it is clear that the family of separable
states is a convex set 3 .
Definition 2.6. A state is called entangled if and only if the state is correlated and not separable, i.e.
AB...
ρ
6=
n
X
B
pr ρA
r ⊗ ρr ⊗ ...
(2.8)
r=1
It may be useful to take a look at a special case: a pure and correlated state.
2.3.1 A pure correlated composite state
Claim. A pure correlated composite state is an entangled one.
Proof. Let ρ ≡ ρAB... be the density operator of a pure state |Ψi ≡ |ΨAB... i, i.e. ρ = |ΨihΨ|. Choose
a vector |Φi, so that hΨ|Φi = 0.
First, we make the attempt to decompose ρ to a convex sum 4 of other density operators:
ρ =
=⇒ hΦ|ρ|Φi = 0 =
n
X
r=1
n
X
λr ρr
λr hΦ|ρr |Φi
r=1
Since all hΦ|ρr |Φi are positive (see eq. (2.3)) and λr are positive, the above equation holds, if and
only if hΦ|ρr |Φi = 0.
Now complete |Ψi to an orthonormal basis {|φk i}, |φ1 i ≡ |Ψi and look at the matrix elements
of ρ and ρr in this basis: hφi |ρr |φj i. We conclude, that all elements vanish, except for hΨ|ρ|Ψi =
hΨ|ρr |Ψi = 1, because of eq. (2.4). Therefore ρ = ρr ∀r ∈ [1; n]. Thus a decomposition is not possible.
Furthermore let’s assume, ρ is correlated, i.e. ρ 6= ρ1 ⊗ ρ2 ⊗ ..., then ρ cannot be written in the form
of eq. (2.7). Hence it must be entangled.
That implies:
|ΨAB... i =
6 |ΦA i ⊗ |ΦB i ⊗ ... ⇐⇒ |ΨAB... i entangled
3
4
A set C is said to be convexP
if (1 − t) x +
Ptny ∈ C, ∀(x, y) ∈ C, ∀t ∈ [0,1].
The convex sum of ρ is ρ = n
r=1 λr ρr ,
r=1 λr = 1, λr > 0
5
(2.9)
2.4 Hilbert-Schmidt space
2 PRELIMINARY DEFINITIONS
2.4 Hilbert-Schmidt space
e the set
Definition 2.7. Let H be a Hilbert space describing a quantum mechanical system and H
e
of operators acting on H. Then the set H forms a Hilbert space for itself. This space is called a
e is defined by
Hilbert-Schmidt space. The scalar product on A denoted by hB, Ci (B, C ∈ H)
hB, Ci := tr(B † C)
p
and induces the Hilbert Schmidt norm ||B|| := hB, Bi .
(2.10)
2.5 Qudit systems
Definition 2.8. A quantum system with d linear independent states is called Qudit system and it is
described in a d-dimensional Hilbert space Hd .
2.5.1 Qubit systems
A qubit system is an important specific case of a qudit system, namely a system with d = 2. The
system is represented by a 2-dimensional Hilbert space H2 . To name a few physical examples:
• Spin- 12 -particle
• Polarization of single photons
• Quantum dots
Bell states
It is not the purpose to discuss Bell states in detail here, however, it makes a good example for an
entangled bipartite qubit state as we will see later on.
Definition 2.9. Let |0i and |1i be the eigenstates of the Pauli operator 5 σ3 acting on H2 , then we
define the so called Bell basis also known as Bell states: |Φ± i, |ψ± i ∈ HAB = H2A ⊗ H2B .
1
|Φ± i := √ (|0, 0i ± |1, 1i),
2
1
|ψ± i := √ (|0, 1i ± |1, 0i)
2
(2.11)
One could calculate its reduced density operators (see eq. (2.5)) which are in the form:
1
ρA = ρB = I
2
5
(2.12)
Since the Pauli operators are not the subjects here, a detailed description is omitted, but can be referred in e.g. [2].
6
3 PURE BIPARTITE QUDIT STATES
3 Pure bipartite qudit states
After the basic but important definitions we will start discussing pure bipartite systems. A very
important concept is the so called Schmidt decomposition which can be applied to all pure bipartite
states, sadly only for those.
In preparation for more complicated matters, e.g. mixed states, the Von-Neumann entropy and its
properties will be discussed.
3.1 Schmidt decomposition
For a pure bipartite qudit state |ΨAB i ∈ Hd it is useful to introduce a decomposition named the
Schmidt decomposition, bi-orthogonal expansion or polar expansion. (see [7])
Theorem 3.1. Let |ΨAB i be a normed state in the composite system S AB in the Product-Hilbert-Space
HAB = HA ⊗ HB with dimension dim HA = a and dim HB = b . The reduced density operators of
the subsystems S A and S B are given by ρA = trB (ρAB ) and ρB = trA (ρAB ) with the density operator
ρAB = |ΨAB ihΨAB |.
B
Then |ΨAB i can be written in the form (3.1) with normed eigenstates |uA
n i, |wn i and eigenvalues
A
B
A
B
pn = pn of ρ and ρ , i.e.
A
ρA |uA
n i = pn |un i,
pA
n ∈R
ρB |wnB i = pn |wnB i,
AB
|Ψ
i=
k
X
√
pB
n ∈R
B
pn |uA
n , wn i,
pn > 0
(3.1)
n=1
Proof. First, expand |ΨAB i into the basis of eigenstates of ρA and ρB :
AB
|Ψ
i=
a,b
X
i,j
B
AB
B
huA
i |uA
i , wj |Ψ
i , wj i
|
{z
}
(3.2)
=:cij
The reduced density operator ρA can be written as a decomposition of his eigenstates:
(
a
a
X
X
>0 0≤n≤k
A
A
A
ρ =
pn |un ihun |, pn
,
pn = 1
=0 k+1≤n≤a
n=1
n=1
(3.3)
On the other side ρA must fulfill its role as a reduced density operator:
3.2
ρA = trB (ρAB ) = trB (|ΨAB ihΨAB |) =
a,b X
a,b
X
i,j
=
a,b X
a,b X
b
X
i,j
=
k,l
a,b X
a,b
X
i,j
k,l
A
B
B
|uA
i ihuk |⊗|wj ihwl |
z
}|
{
B
A
B
,
w
ihu
,
w
cij ckl trB (|uA
i
j
k
l |)
k,l
B
B
A
cij ckl hwm
|wjB ihwlB |wm
i|uA
i ihuk |
m
A
cij ckl hwlB |wjB i |uA
i ihuk |
|
{z
δij
=
a,b X
a
X
i,j
}
A
cij ckj |uA
i ihuk |
(3.4)
k
By comparison of eq. (3.3) and (3.4) one finds that cmn = 0 for m 6= n and c2nn = pA
n . After the
A = pB ≡ p . With these results it can
very similar calculation for ρB , one gets c2nn = pB
.
Thus
p
n
n
n
n
immediately be seen that eq. (3.2) reduces to eq. (3.1).
7
3.2 Von-Neumann entropy
3 PURE BIPARTITE QUDIT STATES
Definition 3.1. The number k in eq. (3.1) is called the Schmidt rank of |ΨAB i, the eigenvalues {pn }
are called Schmidt coefficients.
Claim. |ΨAB i is not entangled if and only if the Schmidt rank k = 1.
Proof. We know from eq. (2.9), that a pure state is entangled, if and only if the state is correlated.
B
AB i can only be written in the form (2.9) if k = 1.
Since {|uA
n i} and {|wn i form a basis, |Ψ
Claim. k = 1 ⇐⇒ tr((ρA )2 ) = tr((ρB )2 ) = 1.
Proof. k = 1 means, all eigenvalues {pn } of ρA = ρB ≡ ρ∗ vanish except of one. The trace of ρ∗ is 1
(see eq. (2.4)). Now suppose
∀pn : pn < 1 =⇒ p2n < pn . Therefore
Pk 2 k > 1. We know, pn > 0. Then
2
2
the trace tr((ρ∗) ) = n pn < 1. Hence the trace tr((ρ∗) ) = 1 ⇐⇒ k = 1.
Let us summarize this important and useful statement: For a pure bipartite qudit state one can
recognize entanglement by evaluating the trace of the squared reduced density operators.
What we have found is a nice criterion to detect entanglement.
Another interesting side product of the Schmidt decomposition is the gained knowledge, that a
subsystem of a pure entangled state cannot be pure (remember that a pure state means rk(ρA ) = 1
and this is equal to k = 1).
Example. Schmidt decomposition and Bell states
Using the above statement and the results from eq. (2.12) we can directly calculate the trace of the
reduced Bell states:
1
1
tr((ρA )2 ) = tr( I) =
4
2
This finding is indeed interesting. It tells us, that the Bell states must be entangled.
3.2 Von-Neumann entropy
Another important tool to get hints about existent entanglement is the Von-Neumann entropy.[1]
Definition 3.2. The Von-Neumann entropy S(ρ) is defined as
S(ρ) := −tr(ρ log ρ)
(3.5)
One can think of it as the quantum version of the entropy Ŝ known in thermodynamics. To make
this clear, one can for example show that the Von-Neumann entropy of the density matrices of the
micro canonical, the canonical and the grand canonical ensemble coincides with the entropy Ŝ. As the
thermodynamic entropy, the Von-Neumann entropy is a measure of information.
For practical reasons it is often useful to express S in the eigenvalues of ρ:
Claim. The Von-Neumann entropy S(ρ) can be written in the form
S(ρ) = −
d
X
(λi log λi )
i=1
d
X
i=1
with the eigenvalues λi of ρ.
8
λi = 1
(3.6)
3.2 Von-Neumann entropy
3 PURE BIPARTITE QUDIT STATES
Proof. Evaluate the trace with respect to eigenbasis {ψi } of ρ:
S(ρ) = −tr(ρ log ρ) = −tr(log ρ ρ)
= −
= −
d
X
i=1
d
X
i=1
hψi | log ρ ρ|ψi i = −
d
X
λi hψi | log ρ|ψi i
i=1
d
X
λi log λi hψi |ψi i = −
| {z }
=1
λi log λi
i=1
which is the claimed formula.
The Von-Neumann entropy S(ρ) has among others the following properties:
1. The entropy of a pure state is its minimum value.
ρ = |ΨihΨ| ⇐⇒ S(ρ) = 0
(3.7)
2. The entropy of a density operator with rank d fulfills
0 ≤ S(ρ) ≤ log d
(3.8)
Proof. To see the first property, one recognizes that ρ has only one eigenvalue λ1 = 1 if and only if ρ
is a pure state. With this finding one can evaluate eq. (3.6), which tells S(ρ|pure ) = 0.
PropertyP
2 can be proven by calculating extremal values of function (3.6). We need to include the
constraint i λi = 1 in the discussion, therefore we rewrite the expression as
d
X
λi = 1 =
i=1
d−1
X
λi + λd =⇒ λd (λi6=d ) = 1 −
i=1
d−1
X
λi
i=1
and thus the entropy reads
S(λl6=d ) = −
d−1
X
λi log λi − pd (λl6=d ) log pd (λl6=d )
i=1
With
∂λd
∂λl
= −1 the derivate of S(λl6=d ) emerges as
∂S
∂λl
= −
d−1
X
∂λi
∂λi
∂λd
∂λd
(
log λi +
)−
log λd −
∂λl
∂λl
∂λl
∂λl
i=1
= − log λl − 1 + log λd + 1 = − log λl + log λd
This is zero if λmax
= λd =⇒ λmax
= d1 , hence the local maximum S max is determined.
l
l
d
S
max
X1
1
1
= S( ) = −
log = log d
d
d
d
i=1
Since S max is the only local maximum (on the whole domain including the boundaries) it is a global
one.
The global minimum value S min = 0 is obvious (all 0 ≤ λi ≤ 1).
9
3.2 Von-Neumann entropy
3 PURE BIPARTITE QUDIT STATES
We already know from section 2.3.1, that a pure state can only possess non classical correlations (or
none at all). Let’s take a look at the valuable example of a pure qubit state.
Example. Von-Neumann entropy and Bell states
Let’s calculate the entropy for our example, the Bell states ρAB and its reduced operator ρA = ρB .
To find S(ρAB ), we can use property (3.6) of the entropy S; since ρAB is pure per definition:
S(ρAB ) = 0
This result shows no new findings, it just confirms the obvious: The pure Bell state is a state with
maximum information (as every pure state is of course).
The reduced density operator ρA apparently possesses the eigenvalues { 12 , 12 }. We use eq. (3.6) and
get the entropy:
S(ρA ) = log 2 = log d
With property (3.8) one notices that this is the maximum value of the entropy. As a consequence, the
sub state ρA is a maximal undetermined state.
With these considerations on an information theory level, one is tempted to define a measure of
entanglement:
Definition 3.3. Consider a pure state |ΨAB i with the corresponding density matrix ρAB and reduced
density matrices ρA and ρB . Then define the entropy of entanglement E(ρAB ) = E(Ψ)
0 ≤ E(Ψ) := S(ρA ) = S(ρB ) ≤ 1
(3.9)
and call it a measure of entanglement of the state |ΨAB i. A state with E(Ψ) = log d, d = dim H is
called maximal entangled (see the Bell state).
This is the first step to the more general approach of entanglement measures as we will see in chapter
4. Such measures also play a very important role for mixed states.
10
4 MIXED BIPARTITE QUDIT STATES
4 Mixed bipartite qudit states
As mixed states describe a more complicate matter than pure states, there are less concrete criteria
available.
This chapter begins with a derivation of a specific criterion for non-entanglement. Then entanglement
measures are introduced as a generalization of the Von-Neumann entropy. After a renewed discussion
about the Schmidt rank we will proceed with establishing a more geometrical approach by the so called
entanglement witnesses.
4.1 A criterion for non-entanglement
In section 3.1 we have seen that a sub-state6 cannot be pure if its pure super-state is entangled. We
will generalize this now.[2]
Theorem 4.1. Let S AB be a bipartite qudit system described in HAB = HdA ⊗ HeB and let ρAB be a
mixed density operator acting on HAB in this system. The corresponding reduced density operator of
the subsystem S A is ρA .
Furthermore if ρA is pure then the state ρAB is separable ( ⇐⇒ not entangled).
Proof. The mixed density operator ρAB can be written in its spectral basis:
X
X
AB
ρAB =
sn |ΨAB
sn = 1,
sn > 0
n ihΨn |,
n
(4.1)
n
A
B
For simplicity set |ΨAB
n i ≡ |Ψn i. Now expand |Ψn i to the spectral basis of ρ and ρ : {ui } and
{wj }: (cp. eq. (3.2))
X
|Ψn i =
cn,ij |ui , wj i
(4.2)
i,j
Using the expansion from eq. (4.2), eq. (4.1) reads
XXX
ρAB =
sn cn,ij cn,kl |ui , wj ihuk , wl |
|
{z
}
n
i,j
k,l
(4.3)
|ui ihuk |⊗|wj ihwl |
To find some constraints for the coefficients cn,ij , we evaluate the reduced density operator ρA in an
arbitrary ON basis {|ϕr i}:
ρA = trB (ρAB )
=hwl |wj i=δij
=
XXX
=
XXX
n
n
i,j
i,j
zX
}|
{
sn cn,ij cn,kl
hϕr |wj ihwl |ϕr i |ui ihuk |
r
k,l
sn cn,ij cn,kj |ui ihuk |
Because {|ui i} is the eigenbasis of ρA , it can also be written as:
X
ρA =
rn |un ihun |
n
6
(4.4)
k
It may be a slight abuse of the used vocabulary, nevertheless it’s a good aberration of “state of a subsystem”.
11
(4.5)
4.2 Generalization of the Von-Neumann entropy
4 MIXED BIPARTITE QUDIT STATES
By comparison of eq. (4.4) and eq. (4.5), cn,ij = 0 for i 6= j. Suppose now, that ρA is a pure state,
i.e. |ρA i = |u1 ihu1 |. Hence all cn,ij are zero except cn,11 . This implies that eq. (4.3) reduces to
X
ρAB =
sn c2n,11 (|u1 ihu1 | ⊗ |w1 ihw1 |)
(4.6)
n
which is the form of a separable state (see eq. (2.7)).
To sum up, this means: If a system is in a pure state it cannot be entangled with another
system.
4.2 Generalization of the Von-Neumann entropy
We have seen in section 3.2 that for pure bipartite states the Von-Neumann entropy is a measure of
entanglement. For mixed states this measure fails to distinguish classical and quantum correlations.
There are however some attempts to generalize the Von-Neumann entropy. The so called entropy
measures are designed to coincide on the special case of pure bipartite states and to be equal to the
entropy of the reduced density operators.
What succeeds now are the requirements for a general entanglement measure[5] followed by definitions of some specific cases.[5],[16]
Note that not all of these provide a rule to do a practical evaluation. Generally the calculation might
not even be possible.
4.2.1 Requirements for entanglement measures
The following properties for a “good” entanglement measure E are of course not mandatory and the
discussion for such requirements is still open. Actually some of the measures introduced below do not
fulfill all of the following properties.
1. ρ is separable ⇐⇒ E(ρ) = 0
2. Normalization
0 ≤ E(ρ) ≤ log d
(4.7)
for an entangled state of a d-dimensional system.
3. LOCC (local operations and classical communication) cannot increase E.7
4. Continuity
E(ρ) − E(σ) → 0
for ||ρ − σ|| → 0
(4.8)
5. Additivity: n identical copies of the state ρ should contain n times the entanglement of one copy.8
E(ρ⊗n ) = n E(ρ)
(4.9)
E(ρ ⊗ σ) ≤ E(ρ) + E(σ)
(4.10)
6. Subadditivity:
7. Convexity: The entanglement measure should be a convex function, i.e.
E(λρ + (1 − λ) σ) ≤ λ E(ρ) + (1 − λ) E(σ)
with 0 < λ < 1.
7
8
A measure which decreases under LOCC is called entanglement monotone.
A⊗m := A ⊗ A ⊗ ... ⊗ A
|
{z
}
m times
12
(4.11)
4.2 Generalization of the Von-Neumann entropy
4 MIXED BIPARTITE QUDIT STATES
4.2.2 Entanglement measures
Entanglement of formation P
Definition 4.1. Let ρAB = i pi ρi , ρi ≡ |ϕi ihϕi | be a mixed bipartite state, then the i-th entropy
of entanglement Ei ≡ E(ρi ) for the pure states ρi is given by eq. (3.9), i.e. E(ρi ) = S(trB (ρi )). The
entanglement of formation also known as entanglement of creation is then defined as
X
EF (ρAB ) := min
pi Ei
(4.12)
{dec}
i
where min means the minimum over all possible decompositions of ρAB .
{dec}
To express this formula in words: Form the average entanglement entropy for pure states over a
decomposition. Do this with all possible decompositions and pick the minimum.
Relative entropy of entanglement
Definition 4.2. Let ρAB be a mixed bipartite state (see eq. (2.7)) and σ AB a separable one in the
system S AB . Then one defines the relative entropy of entanglement ER (ρAB ).
ER (ρAB ) :=
min
σ AB ∈S AB
tr[ρAB (log ρAB − log σ AB )]
(4.13)
To understand the structure of this formula let’s compare it with the original entropy S(ρ) =
−tr(ρ log ρ). ER looks like “tr(ρ log σρ )” 9 . So one could speak of this as the entropy of a state relative
to an arbitrary separable one. As above, form this entropy for all available separable states and take
the minimum.
Entanglement of distillation
Proposition. Suppose ρA is the reduced density operator of ρAB . For such a ρA one can find a pure
super state |φAB i ∈ HA ⊗ HB , where ρA is again the corresponding reduced density operator.
This process is called purification. It makes clear that a reduced density operator is not unique. The
statement follows from the Schmidt decomposition theorem in section 3.1.
Definition 4.3. The entanglement of distillation ED (ρAB ) tells us about the number mout of maximal
entangled states |φAB i that can be purified from nin copies of a given state ρAB (written as (ρAB )⊗nin ).
More precisely it is defined as the ratio mout over nin in the limit of infinitely many states (ρAB )⊗nin .
mout
nin →∞ nin
ED (ρAB ) := lim
(4.14)
A mixed state, that can be distilled, i.e. ED > 0 is called free entangled. Its counterpart is bound
entanglement, for ED = 0.
Entanglement cost
In contrast to the entanglement of distillation the entanglement cost quantifies the amount of pure-state
entanglement needed to create ρAB using local operations.
Definition 4.4. The entanglement cost EC (ρAB ) is therefore defined as
EC (ρAB ) :=
min
nout →∞ nout
lim
where min is the number of maximal entangle states |φAB i required to prepare (ρAB )⊗nout .
9
Mathematically this expression is of course only valid in the sense of definition (4.13).
13
(4.15)
4.3 Generalization of the Schmidt rank
4 MIXED BIPARTITE QUDIT STATES
The entanglement cost has not been computed for any mixed state yet, however the calculation can
be done for our examples, the Bell states.[17]
Example. Let ρp a mixture of the two Bell states |Φ+ i and |Φ− i (see eq. (2.11)), i.e.
ρp ≡ (1 − p) |Φ+ ihΦ+ | + p |Φ− ihΦ− |
1
p ∈ [0; ]
2
Then the entanglement cost EC is given by
1 p
EC (ρp ) = H2 ( + p (1 − p))
2
where H2 (x) = S ∗ (x, 1 − x) with Shannon entropy S ∗ (p1 , p2 ) = −
P
i pi
log pi .10
Remarkable is that ED and EC are extremal measures, any ’sensible’ entanglement measure should
lie in between.
Hilbert-Schmidt measure
Similar to the relative entropy of entanglement we can pursue an even more geometrical approach by
using the Hilbert-Schmidt distance ||ρ1 − ρ2 ||.
Definition 4.5. For an arbitrary state ρAB the Hilbert-Schmidt measure is defined as
EH (ρAB ) := min ||ρAB − σ AB ||
σ AB ∈S
(4.16)
where S is the set of all separable states in HA ⊗ HB .
It is intuitively clear that EH serves as a measure of entanglement since it evaluates the distances
to all separable states and takes their minimum.
As will be shown in chapter 4.4.4 the Hilbert-Schmidt measure plays an important role connecting
entanglement measures with entanglement witnesses.
4.3 Generalization of the Schmidt rank
Similarly to the generalization approach for the entanglement of formation (4.2.2), one can define the
Schmidt number as generalization of the Schmidt rank (discussed in section 3.1).[5]
P
Definition 4.6. Let ρAB = i pi ρi , ρi ≡ |ϕki i ihϕki i | be a mixed bipartite state. Every pure state ρi
has its Schmidt rank ki . Set kmax ≡ max ki . The Schmidt number r is defined as the minimum of the
highest Schmidt rank over all possible decompositions.
r := min kmax
{dec}
(4.17)
The Schmidt number r cannot be higher than the smaller of the dimensions of the two subsystems.
It is called a measure of entanglement and useful in connection with Schmidt witnesses which are
discussed below.
4.4 Entanglement witnesses
A quite geometrical approach is the use of so called entanglement witnesses. Entanglement witnesses
cover a huge area by themselves. Again this chapter is not meant to be a summary of the whole topic
but specific topics are chosen.
A central role plays of course the Entanglement Witness Theorem (EWT).[11]
10
Proofing this would need a couple of preparations and is omitted.
14
4.4 Entanglement witnesses
4 MIXED BIPARTITE QUDIT STATES
4.4.1 Entanglement Witness Theorem (EWT)
The Entanglement Witness Theorem states the following:
Theorem 4.2. A state ρent ∈ H is entangled if and only if there exists a Hermitian operator A ∈ H
which satisfies
hρent , Ai < 0
hσ, Ai ≥ 0
∀σ ∈ S
(4.18)
where S is the set of all separable states. A is then called an entanglement witness.
This can be derived via the Hahn-Banach Theorem[12] of functional analysis. A geometrical representation of this theorem states the following:[10]
Theorem 4.3. Let C be a convex closed set in a topological vector space X, and let b ∈ X but b ∈
/ C.
Then there exists a hyperplane that separates b from the set C.
Proof. of Theorem 4.2. If we identify the above vector spaces X ≡ H and sets C ≡ S we know according
Theorem 4.3 that there exists a hyperplane which separates entangled and separable states. Since such
a hyperplane must exist there exists an A ∈ H which satisfies hρ, Ai = 0 where ρ is an arbitrary vector
in H. Theorem 4.2 follows straightaway with figure 4.1a (exchange A by −A if necessary).
Definition 4.7. An entanglement witness A is called optimal, that is Aopt , if there exists a separable
state σ̃ ∈ S such that
hσ̃, Aopt i = 0
(4.19)
It is called optimal since it detects more entangled states than normal witnesses would do as figure
4.1b illustrates.
(a) A plane in Euclidean space
(b) An optimal entanglement witness.
Figure 4.1: Geometrical illustration of entanglement witnesses, Source: [10]
4.4.2 Positive Map Theorem (PMT)
From the EWT another useful theorem can be derived, namely the Positive Map Theorem (PMT):
Theorem 4.4. Let Λ be a positive map acting on the Hilbert space. Then a bipartite state ρ is separable
if and only if
(I ⊗ Λ)ρ ≥ 0
∀ positive maps Λ.
(4.20)
15
4.4 Entanglement witnesses
4 MIXED BIPARTITE QUDIT STATES
Proof. The ⇐=- direction is shown easily.[10] One applies (I ⊗ Λ) to a separable state 2.7 and gets
X
B
(I ⊗ Λ)ρ =
pi ρA
i ⊗ Λρi .
i
Since Λ is positive, ΛρB
i is positive as well and therefore (I ⊗ Λ)ρ is positive.
The proof for the =⇒ - direction is more complex and provided by Michael, Pawel and Ryszard
Horodecki [8].
In other words that means if we can find a positive map Λ where (I ⊗ Λ)ρ < 0, i.e Λ is not completely
positive11 , we know that ρ is entangled.
Example. An often stated example is the partial transposition
Bell state |Φ+ i (see eq. (2.11)). The matrix representation of
ρB ≡ |Φ+ ihΦ+ | and its partial transposition reads
1
1
1
2 0 0 2
2
0 0 0 0 0
 
(I ⊗ T )ρB = (I ⊗ T ) 
0 0 0 0 = 0
1
1
0
2 0 0 2
PT : (I ⊗ T ). Let’s test it on the
its corresponding density operator
0
0
0
1
2
0
0
0
1
2

0
0
.
0
1
2
Calculating the eigenvalues leads to {− 12 , 21 , 12 , 21 } which means (I ⊗ T )ρB is not positive, therefore ρB
is entangled (as we already know).
4.4.3 Positive Partial Transpose (PPT) Criterion
Theorem 4.4 can be used to create a more specific criterion for the special case of a bipartite qubit
system. It is a necessary and sufficient criterion developed by Peres and Horodecki [8].
Theorem 4.5. A state ρ acting on H2 ⊗ H2 , H2 ⊗ H3 or H3 ⊗ H2 is separable if and only if its partial
transposition ρTB is a positive operator, i.e.
ρTB ≡ (I ⊗ T )ρ ≥ 0.
(4.21)
Definition 4.8. A state satisfying eq. (4.21) is called a PPT state.
The fact that this is indeed a necessary condition can immediately be seen recalling the PMT
(Theorem 4.4) and the stated example of the partial transposition. As shown in [10], to prove that the
criterion is also sufficient, one needs a theorem by Stormer and Woronowitz [14],[19] which is repeated
here for the sake of completeness.
Theorem 4.6. Any positive map Λ that maps operators on Hilbert spaces H2 ⊗H2 , H2 ⊗H3 or H3 ⊗H2
can be decomposed in the following way
CP
Λ = ΛCP
1 + Λ2 T
(4.22)
where ΛCP
and ΛCP
are completely positive maps.
1
2
Proof. Let ρ be a state which satisfies (I ⊗ T )ρ ≥ 0. Since ΛCP
and ΛCP
are completely positive it
1
2
must be true that
CP
(I ⊗ ΛCP
1 )ρ + (I ⊗ Λ2 )(I ⊗ T )ρ ≥ 0
11
A positive map Λ is called completely positive if the map (Id ⊗ Λ) is still a positive map ∀d.
16
4.4 Entanglement witnesses
4 MIXED BIPARTITE QUDIT STATES
which can obviously be written as
CP
(I ⊗ (ΛCP
1 + Λ2 T ))ρ ≥ 0
(4.23)
. Now the need of Theorem 4.6 gets apparent. It tells us that eq. (4.23) reduces to
(I ⊗ Λ)ρ ≥ 0
where Λ is a positive map. Now since any Λ can be decomposed in terms of (4.22) the PMT (theorem
4.4) can be used which tells us that ρ is separable if and only if (I ⊗ T )ρ ≥ 0.
4.4.4 Bertlmann-Narnhofer-Thirring Theorem
The Bertlmann-Narnhofer-Thirring Theorem connects the Hilbert-Schmidt measure defined in section
4.2.2 with the concept of entanglement witnesses. More specific it tells that the Hilbert-Schmidt
measure of an entangled state equals the maximal violation of an inequality which basically defines
the entanglement witnesses.
Before going into details we will proof a Lemma which will considerably ease the proof of the theorem
itself. After that the mentioned inequality will be defined and the theorem will be stated.
Lemma. Let ρ1 and ρ2 be two arbitrary states and an operator C be defined as12
C(ρ1 , ρ2 ) :=
ρ1 − ρ2 − hρ1 , ρ1 − ρ2 i I
||ρ1 − ρ2 ||
(4.24)
Also let ρ0 be the ’nearest’ separable state of ρent , i.e.
||ρ0 − ρent || = min ||σ − ρent ||
σ∈S
e := C(ρ0 , ρent ) is an optimal entanglement
where S denotes again the set of all separable states, then C
witness of ρent as defined by eq. (4.18) and (4.19).
e has to fulfill hρent , Ci
e < 0 and
Proof. In order to be an entanglement witness of ρent the operator C
e ≥ 0 ∀σ ∈ S. That this is satisfied can be shown in the following manner:
hσ, Ci
Looking at figure 4.2 it is obvious that
hρp − ρ1 ,
ρ1 − ρ2
i=0
||ρ1 − ρ2 ||
defines a hyperplane through ρ1 orthogonal to ρ1 − ρ2 .
By evaluating hρ, C(ρ1 , ρ2 )i, which is
hρ, C(ρ1 , ρ2 )i = hρ,
ρ1 − ρ2
ρ1 − ρ2
ρ1 − ρ2
i − hρ1 ,
i hρ, Ii = hρ − ρ1 ,
i
||ρ1 − ρ2 ||
||ρ1 − ρ2 || | {z }
||ρ1 − ρ2 ||
=1
one recognizes that the above hyperplane definition reduces to13
hρp , Ci = 0
, similar one defines the left-hand and right-hand states ρl and ρr (see figure 4.2) as:
hρl , Ci < 0
hρr , Ci > 0
12
13
h., .i is defined by eq. (2.10) .
In this context omitting the arguments of C(., .) means evaluating C(ρ1 , ρ2 ).
17
(4.25)
4.4 Entanglement witnesses
4 MIXED BIPARTITE QUDIT STATES
Figure 4.2: A hyperplane orthogonal to ρ1 − ρ2 , Source: [10]
e ≡ C(ρ0 , ρent ). Since ρ0 is the nearest separable state of ρent it lies on the boundary
Now consider C
e = 0 is tangent to S. Also the plane is orthogonal to
of S. Therefore the plane defined by hρp , Ci
e
e
e = 0. Thus C
e is indeed an optimal
ρent − ρ0 . So we have hρent , Ci < 0, hσ, Ci ≥ 0 ∀σ ∈ S and hρp , Ci
entanglement witness.
It is not hard to see that the definition (4.18) of an entanglement witness can be rewritten as
hσ, Ai − hρent , Ai ≥ 0
∀σ ∈ S
(4.26)
This inequality is also called a generalized Bell inequality (GBI). One can define the maximal violation
of the GBI as follows.
Definition 4.9. The maximal violation of the GBI is given by
B(ρent ) :=
max
min (hσ, Ai − hρent , Ai)
σ∈S
A, ||A−aI||≤1
(4.27)
where a is the coefficient corresponding to the identity matrix when expanding A into the Pauli-Matrix
basis {σi ⊗ σj }. However, the specific choice of the used norm will not be of any importance to us.
Theorem 4.7. The Hilbert-Schmidt measure EH (ρent ) of an entangled state ρent equals the maximal
violation of the GBI B(ρent ):
EH (ρent ) = B(ρent )
(4.28)
Proof. The distance ||ρ1 − ρ2 || of two states can be written as
||ρ1 − ρ2 || = hρ1 − ρ2 ,
ρ1 − ρ2
ρ1 − ρ2
i = hρ1 − ρ2 ,
+ c Ii
||ρ1 − ρ2 ||
||ρ1 − ρ2 ||
where c is a complex constant because (ρ1 − ρ2 , c I) = c (tr(ρ1 ) − tr(ρ2 )) = 0 since tr(ρ) = 1 ∀ρ.
2 −ρ2 i
Choosing c = − hρ||ρ1 ,ρ1 −ρ
yields per definition to
2 ||
ρ1 − ρ2 − hρ1 , ρ1 − ρ2 i I
i
||ρ1 − ρ2 ||
= hρ1 , C(ρ1 , ρ2 )i − hρ2 , C(ρ1 , ρ2 )i
||ρ1 − ρ2 || = hρ1 − ρ2 ,
Using eq. (4.29) the Hilbert-Schmidt measure EH (ρent ) can be written as:
e − hρent , Ci
e =♣
EH (ρent ) = min ||ρent − σ|| = ||ρent − ρ0 || = hρ0 , Ci
σ∈S
18
(4.29)
4.4 Entanglement witnesses
4 MIXED BIPARTITE QUDIT STATES
From the definition of an optimal entanglement witness it is clear that maxA hρent , Ai = hρent , Aopt i
e is an optimal
(Aopt has to be parallel to ρent ) and hρ0 , Aopt i = 0 since ρ0 is on the plane. Since C
entanglement witness we can rewrite EH (ρent ) further as
♣ = max (hρ0 , A) − hρent , Ai) = max min (hσ, A) − hρent , Ai) = B(ρent )
A
A
σ∈S
which completes the proof.
19
5 MULTIPARTITE STATES
5 Multipartite states
The classification and quantification of entanglement is in its early stage when it comes to multipartite
systems. Many measures are designed to function in specific domains where they perform successful
operations. At the same time the same measures might fail badly in other areas.
The mathematical treatment of such measures is a delicate matter. The next few sections provide
the definitions of more or less often stated entanglement measures. [1]
5.1 Some entanglement measures
5.1.1 Geometric measure of entanglement
An attempt to quantify the entanglement of a pure state is given by the minimal distance of the state
from the set of all pure product states Φ.
Definition 5.1. A geometric measure Eg (Ψ) for a pure multipartite state is defined as follows:
Eg (Ψ):= − log max |hΨ|Φi|2
(5.1)
Φ
As shown in [18] this measure can be extended to an entanglement monotone measure for mixed
states. It is zero for separable states and rises up to its maximum for e.g. the maximal entangled
N -particle GHZ state.
5.1.2 Measure of entanglement by Barnum
Another measure was provided by Barnum.[3]
Definition 5.2. A generalized entanglement measure for qubit systems is given by
N
2 X
Egl := 2 −
tr(ρ2j )
N
(5.2)
j=1
where N is the number of qubits. ρj is the reduced density matrix of the j-th qubit.
Remembering the consequence of the Schmidt decomposition for pure bipartite states (section 3.1),
which stated that tr((ρA )2 ) = 1 for non entangled states, eq. 5.2 looks like an average of such
evaluations. However, a deeper connection would remain to be shown.
5.2 An entanglement witness
The concept of entanglement witnesses can in principle be extended to general multipartite systems
since the proof done in section 4.4.1 is independent of the dimension of the system. Of course, in
practice it is quite hard to find useful quantities.
In [9] the hermitian operator W in an N -partite system is studied:
W =
1
2
X
b~k (Q~+ − Q~− )
k
(5.3)
k
~k∈{0,1}N
where Q~± are projectors to generalized GHZ states, i.e. Q~± = |G~± ihG~± | with
k
k
1 G~± = √ |~ki ± σx⊗N |~ki
k
2
and b~k are constrained coefficients.
20
k
k
(5.4)
5.2 An entanglement witness
5 MULTIPARTITE STATES
It turns out that under the constraint (5.5) the hermitian operator W witnesses entanglement.
X
|b~k | ≤ 2N
(5.5)
~k
By choosing different sets of b~k one can obtain specific W ’s for different purposes.
21
6 THE CHOICE OF FACTORIZATION
6 The choice of factorization
Until now when we spoke e.g. about a bipartite qubit system in a 4-dimensional Hilbert space H2 ⊗ H2
we silently ignored the specific structure of these tensor factors. Mostly it was implied that the local
measurement is a projective measurement to the |0i state, which has of course its meaning, because
that is how it’s mainly done in experiments. However, that is certainly a special case. In general it is
allowed to do measurements w.r.t. all states in the eigenspace of a particular Hamilton operator that
relates to the actual present physical problem.
We will lift this restriction now and study what would happen to states (respectively entanglement)
if we chose different factorization of the tensor algebra. Among other things we will find that for an
entangled state we can always find a factorization of its algebra in which this same state does not
appear to be entangled but not the other way around!
6.1 The choice of factorization for pure states
For pure states the situation is quite clear as can be seen in the following section.[15]
e with H
e being a d1 × d2 -dimensional Hilbert-Schmidt space
Theorem 6.1. For any pure state ρ ∈ H
e
e
one can find a factorization of H, H = A1 ⊗ A1 , such that ρ is separable w.r.t. this factorization as
e = B1 ⊗ B2 such that ρ is maximal entangled.
well as a factorization H
Proof. To see this it suffices to know that for any |Ψi ∈ H there exists a unitary operator U which
transforms |Ψi in the following way:
1 X
U |Ψi = √
|Ψ1,i i ⊗ |Ψ2,i i
(6.1)
d i
e = A1 ⊗ A1 . So with unitary
with d = min(d1 , d2 ). Let’s consider |Ψi to be separable w.r.t. H
transformations we can transform every separable state in a maximal entangled one. Since ρsep =
|ΨihΨ| the transformed state has the form ρent = U ρsep U † ∈ A1 ⊗ A1 . Instead of the state ρ we can as
well transform the underlying algebra: B ⊗ B2 = U (A1 ⊗ A1 )U † . Now it becomes clear that ρ which
e = A1 ⊗ A1 is maximal entangled w.r.t H
e = B1 ⊗ B2 .
appears to be separable w.r.t H
6.2 The choice of factorization for mixed states
For mixed states the situation is not that easy and in fact much more interesting. To proof theorem
6.2 we need a maximal entangled ONB for a d × d-dimensional Hilbert space. The state (6.2) provides
one which is proven in lemma 6.1.[15]
Lemma 6.1. The set of vectors
X 2πi
|χkl i =
e d jl |ϕj i ⊗ |ψj+k i,
j = 1, ..., d; k = 1, ..., d
j
is a maximal entangled ONB of the d × d-dimensional Hilbert space Hd ⊗ Hd .
Proof. Evaluating hχkl |χmn i yields to
X 2πi 2πi
hχkl |χmn i =
e− d jl e d in (hϕj | ⊗ hψj+k |) (hϕi | ⊗ hψi+m |)
i,j
=
X
e
2πi
(lj−ni)
d
i,j
= δkm
hϕj |ϕi i hψj+k |ψi+m i
| {z } |
{z
}
=δij
X
e
2πi
j(l−n)
d
j
22
=δ(j+k),(i+m)
= δkm δln
(6.2)
6.2 The choice of factorization for mixed states
6 THE CHOICE OF FACTORIZATION
which proves that the set of |χkl i’s is an ONB.
That these states are entangled can be seen by considering the Schmidt decomposition (see section
3.1). The Schmidt rank equals its maximum value.
e with H
e being a d1 × d2 -dimensional Hilbert-Schmidt space
Theorem 6.2. For any mixed state ρ ∈ H
e H
e = A1 ⊗ A1 , such that ρ is separable w.r.t. this factorization. A
one can find a factorization of H,
e
factorization H = B1 ⊗ B2 such that ρ is entangled exists only beyond a certain bound of mixedness.
P
Proof. To realize the first statement one has to recall that ρ can be written as α pα |χα ihχα | where
|χα i can be an arbitrary state. We’ve already mentioned that every pure state can be transformed into
a separable state with a unitary transformation U : U |χα i = |ϕi,α i ⊗ |ψj,α i. Transforming ρ yields to
X
U ρU † =
pα |ϕi,α ihϕi,α | ⊗ |ψj,α ihψj,α |
(6.3)
α
which is separable.
On the other hand we can choose a unitary transformation U which transforms every state in a
P 2πi
maximal entangled one: U |χα i = |χkl i where |χkl i = j e d jl |ϕj i ⊗ |ψj+k i (see Lemma 6.1). The
transformed state ρ is a so called Weyl state:
X
U ρU † =
pk |ϕj ihϕj | ⊗ |ψj+k ihψj+k |
(6.4)
k,j
This is the expansion of U ρU † into a set of maximal entangled states. But this set is not convex,
therefore the state has not necessarily to be entangled.
Knowing this it makes sense to introduce the following definition:
Definition 6.1. If a state ρ remains separable for any factorization of its algebra (that is for any
unitary transformation U of ρ) it is called a absolutely separable state.
In [15] this theorem is stated more precisely for e.g. Werner states ρw in d × d dimensions:
ρw = αP +
1−α
I2
d2 d
0≤ α ≤1
(6.5)
1
where P is a projector to a maximal entangled state. This state is entangled for α > d+1
. It is shown
3
that if the largest eigenvalue λ1 of ρw satisfies λ1 > d2 then there exists always an algebra such that
ρw appears entangled.
Furthermore the following theorem has been proven for general mixed states in 2 × 2 dimensions:
Theorem 6.3. Let ρ be an arbitrary mixed state in 2×2 dimensions with an ordered spectrum {λi }, i =
1, .., 4 with λ1 ≥ λ2 ≥ λ3 ≥ λ4 . If the eigenvalues satisfy
p
(6.6)
λ1 − λ3 − 2 λ2 λ4 ≤ 0
then ρ is absolutely separable.
23
7 CONCLUSION
7 Conclusion
Let’s summarize the previous steps.
We started studying the simplest non trivial case, a pure bipartite state.
For these states concrete methods
available to exactly identify entanglement. Looking at the
P are
√
B
Schmidt decomposition |ΨAB i = kn=1 pn |uA
n , wn i we derived that one just has to evaluate the trace
of the squared reduced density operator to detect entanglement. Also the Schmidt rank quantifies
entanglement in a certain sense.
It was shown that the Von-Neumann entropy S(ρ) = −tr(ρ log ρ) as generalization of the entropy Ŝ
from thermodynamics (and therefore leading to a measure of information) offers useful properties by
considering sub systems and was our first candidate for an entanglement measure.
In mixed bipartite states we generalized the idea of the entropy and defined a general entanglement
measure to not only detect entanglement but to quantify it. We have seen that there is no unique
measure and stated a few possibilities, each for different purposes.
Further we saw that pure systems cannot be entangled with other systems. In this sense the whole
is not the sum of its parts but more:
»Bestmögliches Wissen um ein Ganzes schließt nicht notwendig das Gleiche für seine
Teile ein.« - Erwin Schrödinger[13]
»The best possible knowledge of a whole does not necessarily include the best possible
knowledge of all its parts.«
From the Hahn-Banach Theorem of functional analysis it follows that there must exist an operator
which detects entanglement, namely an entanglement witness, which was discussed as well. An operational criterion, the PPT-criterion, was derived with the help of these witnesses.
It was shown that one is even able to construct an optimal entanglement witness that was used
to proof the Bertlmann-Narnhofer-Thirring Theorem which connects the concept of entanglement
witnesses with the approach of entanglement measures.
The quantification of multipartite states is even more of a challenge. Mathematically the most
measures are not fully understood in the sense that many properties are just proven for special cases
which are merely assumed to hold in general. Not only the quantification is a problem: the question,
if a state is actually entangled or not, cannot be answered in general. Initial attempts were made to
provide tools, but that hopefully should only be the beginning.
Finally the factorization of the algebra was considered. The results were quite interesting. We can
transform every entangled state to a non-entangled one while the reverse is not true in general but just
for certain bound of mixedness.
24
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