entanglement in bipartite and tripartite quantum systems...section 3.3.2 shows that the ppt...

Gradu Amaierako Lana/Trabajo Fin de Grado

Fisikako Gradua/Grado en Fısica

Entanglement in bipartite

and tripartitequantum systems

Hodei Eneriz

Director:

Prof. Enrique Solano

Codirector:

Dr. Mikel Sanz

Department of Physical ChemistryFaculty of Science and Technology

University of the Basque Country UPV/EHU

Leioa, February 2015

Acknowledgements

I would like to thank Prof. Enrique Solano for giving me the opportunity tojoin the QUTIS group and work in such an inspiring atmosphere. This hasdefinitely enhanced my interest in physics and contributed in fading awaymany doubts about pursuing a scientific career.

I am especially grateful to Dr. Mikel Sanz who always offered his timeand patience, no matter how busy he was.

Thanks also to all QUTIS members, who revived my curiosity with everydiscussion held in the office.

I finally thank my friends and family for their support and encourage-ment.

Contents

Contents 3

1 Introduction and Objectives 4

2 Mathematical background 62.1 Density operator . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Von Neumann measurements . . . . . . . . . . . . . . . . . . 92.4 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Bipartite entanglement 123.1 EPR paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Bell states and Schmidt decomposition . . . . . . . . . . . . . 133.3 Mixed entangled states and PPT criterion . . . . . . . . . . . 14

3.3.1 Entanglement under unitary evolution . . . . . . . . . 173.3.2 PPT for 3⊗ 3 pure states . . . . . . . . . . . . . . . . 18

4 Manipulation and classification of entanglement 214.1 Introduction to different kinds of entanglement . . . . . . . . 214.2 LOCC tasks and entanglement classification . . . . . . . . . . 234.3 Stochastic LOCC . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 SLOCC classification for non-genuine

tripartite entanglement . . . . . . . . . . . . . . . . . . . . . . 324.5 SLOCC classification for symmetric 3-qubit

states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Conclusions 36

Bibliography 37

3

Chapter 1

Introduction and Objectives

The counterintuitive properties of entanglement were first discussed by Al-bert Einstein in 1935, in a joint paper with Boris Podolsky and NathanRosen [1]. It was Erwin Schrodinger, who shortly thereafter coined theword entanglement and described it as “not one but rather the character-istic trait of quantum mechanics”. Although these first studies criticizedquantum mechanics, by arguing that quantum description of physical real-ity is not complete, repeated experiments have verified that photons, ionsand more recently, solid-state systems such us quantum dots or SQUIDs(superconducting quantum interference devices), can show this behavior.

This behavior implies the existence of global states of composite systemswhich cannot be written as a product of the states of individual subsystems.Schrodinger concluded that “best possible knowledge of a whole does notinclude best possible knowledge of its parts and this is what keeps comingback to haunt us”.

In 1964 John Bell accepted the incompleteness conclusion as a workinghypothesis and proposed the local hidden variable model (LHVM) [2], sum-marizing the deterministic world idea by the following assumptions [3]: (i)measurement results are determined by properties the particles carry priorto and independent of the measurement (“realism”), (ii) results obtained atone location are independent of any actions performed at space-like separa-tion (“locality”) and (iii) the setting of local apparatus are independent ofthe hidden variables which determine the local results (“free will”). He thenproved that these assumptions impose constraints, in the form of inequali-ties, on correlations in experiments. The outcomes obtained when suitablymeasuring some entangled quantum state violate the Bell inequalities.

But it was not until the 80s that a convincing test of violation of the Bellinequalities was performed. However the experiments carried out by Aspectet al. [4] and [5] and many others since, still suffer from locality and/ordetection loopholes. The ones of the first kind are due to the fact that the twodetections are separated by a time-like interval, which means that the first

4

detection might influence the second one by some kind of signal, whereas thedetection loopholes are consequence of particles not always being detected atboth wings of the detector. Advancements in technology during the last 30years have led to significant elimination of loopholes as well as a vast varietyof methods to test the Bell inequalities. Therefore, for most physicist, it isby this time unrealistic to hold to a local realistic view.

A main problem is, however, that it is not easy to give a definition ofentanglement other than it is a property of entangled states. That is why theproblem has been shifted: nowadays the fundamental question in quantumentanglement theory is which states are entangled and which are not [6].Being a property of correlations between quantum systems, entanglementdefies a classical description and, unfortunately, the structure that natureconceals seems to be, in general, very complex.

Other than entanglement detection, much efforts have been devoted tothe concept of entanglement manipulation. As well as the interest in testingquantum phenomena, with the ability of manipulating individual quantumsystems, the possibility of using quantum correlations as a resource to per-form tasks that are inefficient or even impossible by classical means, hasbecome stronger. The development of this concept is a central element inmodern quantum information science.

This work is essentially a review of the entanglement phenomenon inbipartite and tripartite quantum systems. However, original calculationsboth analytically and numerically have been presented: in subsections 3.3.1the evolution of entanglement under a Ising Hamiltonian is considered; sub-section 3.3.2 shows that the PPT criterion identifies entanglement of everypure 3⊗ 3 quantum state, and finally in section 4.5 we present an ingeniousmethod to verify whether GHZ and W states belong to the same SLOCCclass or not.

Our approach is going to be that of defining entangled states, but before,we shall define states that are not entangled, which is actually simpler.We shall see later on that manipulation of entanglement plays in fact afundamental role in entanglement theory.

5

Chapter 2

Mathematical background

In this section we are going to introduce some mathematical concepts thatare of great importance for the subject we are treating here.

2.1 Density operator

The state of a quantum system may not be completely known. Mathemat-ically, a quantum state is represented by a state vector |ψ〉 in a Hilbertspace, which we call a pure state. A mixed quantum state corresponds to aprobabilistic mixture of pure states and it applies when there is not enoughinformation to specify the normalized state |ψ〉. If pn are the probabilitiesthat the system is in a normalized state |ψn〉 then the expectation value ofan operator A is

〈A〉 =∑n

pn〈ψn|A|ψn〉. (2.1)

In this context of lack of information, it appears to be convenient to intro-duce the density operator ρ which is the hermitian operator

ρ =∑n

pn|ψn〉〈ψn|. (2.2)

Density operators of this form represent a statistical mixture of states. Whenρ = |ψi〉〈ψi| we have a pure state and ρ2 = ρ. Given that tr(ρ) = 1, itfollows that tr(ρ2) = 1 for pure states. For mixed states on the other hand

tr(ρ2) =∑n

p2n < 1.

Density operators are also positive, which means that for any state |φ〉

〈φ|ρ|φ〉 =∑n

pn|〈φ|ψn〉|2 ≥ 0, (2.3)

because every term in the sum is positive or zero.

6

It is also convenient to introduce a measure of the uncertainty of thestate vector in the form of entropy, the von Neumann entropy

S(ρ) = −tr(ρ log ρ), (2.4)

which multiplying it by Boltzmann’s constant gives the thermodynamic en-tropy.

In 1930, Paul Dirac introduced the idea of the reduced density matrix.In the quantum description of physics, systems are composed by more ele-mentary subsystems. If the system is composed of two subsystems A andB then the possible states of the total system form the total space that isgiven by the tensor product

Htot = HA ⊗HB, (2.5)

of Hilbert spaces.The term subsystem has a large meaning here. It may refer to different

particles or to degrees of freedom such as the spin. Let us consider as anexample two particles of spin 1/2. Htot = C2

1⊗C22 if only the spins are taken

into account. The state of the total system can be:

|ψ1〉 = |+〉 ⊗ |+〉. (2.6)

Moving to the density matrix formalism this reads:

ρ = |ψ〉〈ψ| = |+ +〉〈+ + | =

1 0 0 00 0 0 00 0 0 00 0 0 0

(2.7)

in the {|++〉, |+−〉, |−+〉, |−−〉} basis representation. The reduced densitymatrices read now as follows:

ρa = trb(|+ +〉〈+ + |) = |+〉〈+|(〈+|+〉)2 = |+〉〈+| =(

1 00 0

)(2.8)

and equally

ρb = |+〉〈+| =(

1 00 0

). (2.9)

More generally,

ρa = trb(ρ) =∑n

b〈ψn|ρ|ψn〉b, (2.10)

where {|ψn〉b} forms the basis that spans the second subsystem and similarlyfor ρb.

7

2.2 Unitary operators

An operator U is said to be unitary if U †U = UU † = I. Unitary operatorspreserve the inner products between vectors:

(U |v〉, U |w〉) = 〈v|U †U |w〉 = 〈v|w〉. (2.11)

Defining |wi〉 ≡ U |vi〉, where |vi〉 is an orthonormal basis set and therefore

so is |wi〉, we can write U =∑i

|wi〉〈vi|.

Unitary operators are very useful in quantum mechanics because theyare generated by Hermitian operators as we are going to prove next. Firstnote that any operator can be written in the form A+ iB, where A and Bare hermitian, and therefore iB is skew-Hermitian, so that

U = A+ iB, (2.12)

U † = A− iB (2.13)

and then the unitarity implies that

U †U = A2 +B2 + i[A,B] = I, (2.14)

UU † = A2 +B2 − i[A,B] = I. (2.15)

It follows now that A and B must commute and then A2 + B2 = I, whichallows us to write A = cosC and B = sinC, so that U = exp(iC).

In particular, the unitary time evolution operator, U(t) = exp(−iHt/~),is generated by the Hamiltonian H. Considering a initial density operatorwritten in the diagonalized form

ρ(0) =∑n

pn|ψn(0)〉〈ψn(0)|, (2.16)

the formal solution of the Schrodinger equation is |ψn(t)〉 = U(t)|ψn(0)〉 andalong with the bra equivalent we are able to write

ρ(t) =∑n

pnU(t)|ψn(0)〉〈ψn(0)|U †(t) = U(t)ρ(0)U †(t). (2.17)

It is a unitary evolution of ρ.

8

2.3 Von Neumann measurements

Von Neumann or projective measurements are the kind of measurementsthat are introduced in introductory quantum theory courses within thescheme of the postulates of quantum mechanics. Here this process is go-ing to be defined using the density operator formalism.

If a quantum system is described by a density operator ρ and if A =∑λn|λn〉〈λn| is an operator associated to an observable, then after the

measurement of the magnitude A, the probability of getting the result λn is

p(λn) = 〈λn|ρ|λn〉 = tr(ρ|λn〉〈λn|) = tr(ρPn). (2.18)

Here the operator Pn is called the projector over the space of the eigenvalueλn of A. A von Neumann measurement is one in which the probability for agiven measure is given in this form. If the eigenstates of A are degenerate,then the probability for the corresponding eigenvalue will read

p(λn) =∑i

tr(ρ|λin〉〈λin|). (2.19)

Now, the reduction of the wave-packet postulate tells us that the measure-ment is accompanied by a change in the density operator of the form

ρ→ ρ′n =PnρPn

tr(PnρPn)=PnρPnp(λn)

, (2.20)

that is, the projection of ρ onto the space of eigenstates associated with themeasurement result divided by the probability for the observed measurementoutcome, so that normalization is ensured. The last step follows from thecyclicity of the trace, tr(AB) = tr(BA), and from the fact that projectorsare orthonormal, which means that PiPj = Piδij .

2.4 Qubit

A qubit is a quantum system having two orthogonal states. It is the quan-tum analogue to the term bit in information theory and it can hold one bitby preparing it either in the state |0〉 or |1〉. However, due to the superpo-sition principle in quantum mechanics, a qubit can be also prepared in anysuperposition state of the form

|Qubit〉 = α|0〉+ β|1〉, (2.21)

where α and β are complex numbers.A physical implementation of a qubit can be provided in principle by

any quantum system with two states such as the two orthogonal polariza-tion states of a photon or the orientation of a spin-half particle, associatingconventionally |+〉 to |0〉 and |−〉 to |1〉.

9

It is helpful to represent the qubit states as points on the surface ofa sphere, the Bloch sphere depicted in figure 2.1. North and south polescorrespond respectively to |0〉 and |1〉 and more generally, opposite pointsrepresent mutually orthogonal states. Finally the x and y axes hold theeingenstates of σx and σy respectively.

Figure 2.1: Bloch sphere.

Other than the generic qubit pure state

|ψ〉 = cos

(θ

2

)|0〉+ eiϕ sin

(θ

2

)|1〉, (2.22)

mixed states can also be visualized in this representation. Thanks to itshermiticity, the density operator ρ can be written as a combination of thefour Pauli operators

I =

(1 00 1

), (2.23)

σx =

(0 11 0

), (2.24)

σy =

(0 −ii 0

), (2.25)

σz =

(1 00 −1

), (2.26)

10

with real coefficients a, b and c:

ρ =1

2(I + aσx + bσy + cσz). (2.27)

This permits as to associate a, b and c with the x, y and z components of theBloch vector and the eigenvectors |λ〉 and |φ〉 of ρ are also eigenvectors ofaσx+bσy+cσz corresponding to the eigenvalues ±

√a2 + b2 + c2. Therefore,

we can write the diagonalized density operator ρ as follows:

ρ =1

2(1 +

√a2 + b2 + c2)|λ〉〈λ|+ 1

2(1−

√a2 + b2 + c2)|φ〉〈φ|. (2.28)

This reduces to the pure state |λ〉〈λ| for a2 + b2 + c2 = 1, that is, for thestates lying on the surface of the Bloch sphere. For a2 + b2 + c2 < 1, onthe other hand, we get mixed states and the Bloch vector describes a pointinside the sphere.

Furthermore, unitary operators on single qubits are naturally visualizedin the Bloch representation. One way of seeing this is writing them as acomposition of the rotation operators about the x, y and z axes

Rx(ε) = exp(−i ε

2σx

)=∞∑n=0

1

n!

(−i ε

2σx

)n= cos

( ε2

)I− i sin

( ε2

)σx,

(2.29)

Ry(ε) = exp(−i ε

2σy

)=∞∑n=0

1

n!

(−i ε

2σy

)n= cos

( ε2

)I− i sin

( ε2

)σy,

(2.30)

Rz(ε) = exp(−i ε

2σz

)=

∞∑n=0

1

n!

(−i ε

2σz

)n= cos

( ε2

)I− i sin

( ε2

)σz.

(2.31)

These operators are themselves unitary and so is

U =eiχRz(δ)Ry(µ)Rz(ν)

=

(ei(χ−δ/2−ν/2) cos µ2 −ei(χ−δ/2+ν/2) sin µ

2

ei(χ+δ/2−ν/2) sin µ2 ei(χ+δ/2+ν/2) cos µ2

). (2.32)

Here, δ, µ and ν are the Euler angles that describe an orientation in 3-dimensional euclidean space and χ simply acts to change the global arbitraryphase of the state vector. By combining such real numbers we can constructany unitary operation on single qubits.

11

Chapter 3

Bipartite entanglement

3.1 EPR paradox

The EPR paradox [1], named after Albert Einstein, Boris Podolsky andNathan Rosen, was a thought experiment which revealed what later wouldbe called entanglement. Bohm presented the EPR paradox in terms of apair of spin 1/2 particles prepared in a zero total angular momentum state[7]:

|ψ〉AB =1√2

(|+〉A|−〉B − |−〉A|+〉B). (3.1)

Two distant parties, called usually Alice and Bob in computer and quantuminformation sciences, each have one of the pair of the quantum states, A andB respectively. Translating it now to the qubit notation this reads

|ψ〉AB =1√2

(|0〉A|1〉B − |1〉A|0〉B). (3.2)

If Alice chooses to measure σz then she immediately establishes that thestate of Bob’s qubit is |1〉B or |0〉B, corresponding, respectively, to her results+1 and -1. If Alice measures σx, however, then what she establishes is thatthe state of Bob’s qubit is |0′〉B = 2−1/2(|0〉B+ |1〉B) or |1′〉B = 2−1/2(|0〉B−|1〉B), corresponding again to her results -1 and +1 respectively, as

|ψ〉AB =1

2√

2((|0′〉A + |1′〉A)⊗ (|0′〉B − |1′〉B)

− (|0′〉A − |1′〉A)⊗ (|0′〉B + |1′〉B))

=1√2

(|1′〉A|0′〉B − |0′〉A|1′〉B). (3.3)

σz and σx have no common eigenstates and therefore there is no quantumstate having well-defined values for both observables. Then by the choice

12

of measuring one or the other, Alice establishes either of two incompati-ble properties of Bob’s qubit that should not be possible to establish atthe source where the qubits were created. Alice’s measure seems to changeinstantaneously Bob’s qubit and this conflicts with what is called local re-alism. According to locality, physical influences should not propagate fromone party to the other at a speed greater than that of light. Realism lies onthe belief that the properties of Bob’s qubit exist prior to and independentlyof the measurement.

3.2 Bell states and Schmidt decomposition

The Bell states are usually presented as the most simple examples of en-tangled states. Furthermore they have been realized in a number of diverseexperiments. They are conventionally written in the following form:

|Ψ+〉 =1√2

(|0〉|1〉+ |1〉|0〉), (3.4)

|Ψ−〉 =1√2

(|0〉|1〉 − |1〉|0〉), (3.5)

|Φ+〉 =1√2

(|0〉|0〉+ |1〉|1〉), (3.6)

|Φ−〉 =1√2

(|0〉|0〉 − |1〉|1〉). (3.7)

They are also known as the maximally entangled two-qubit states. Let uswrite for example the antisymmetric state used in the Bohm’s version of theEPR experiment:

ρ = |Ψ−〉〈Ψ−| = 1

2

0 0 0 00 1 −1 00 −1 1 00 0 0 0

. (3.8)

Evaluating the reduced density operator either for the a or the b subsystemwe get

ρa = trb(ρ) =1

2(|0〉〈0|(〈1|1〉)2 + |1〉〈1|(〈0|0〉)2) =

1

2

(1 00 1

)= ρb. (3.9)

So the reduced density matrix corresponds to a mixed state. This is some-what surprising as the total system is pure. It means that it is not possibleto specify the exact state of a single qubit when this is entangled to anotherone. In other words, our entangled state is not separable. Thus, tr(ρ2) < 1

13

for a subsystem of a bipartite pure state is a signature of entanglement.Quantum superposition leads to a kind of correlations that cannot be ex-plained by classical means and it is by the word entanglement that thisphenomena is known.

Before giving a more precise definition of entanglement, which in fact it isgoing to be that of separability, let us define the Schmidt decomposition [8].It is always possible to write an entangled pure state as summation of theorthonormal sets |λn〉 and |φn〉, by a suitable choice of the local unitaryoperators UA and UB:

|ψ〉 = UA ⊗ UB∑ij

aij |i〉a|j〉b =∑n

an|λn〉a|φn〉b, (3.10)

where an are non-negative real numbers and aij the elements of a diagonal-izable matrix A → UAAU

TB that completely represents the state provided

that the basis has been specified. This form is known as the Schmidt de-composition and the orthonormal states are the eigenstates of the reduceddensity operators

ρa =∑n

a2n|λn〉〈λn|, (3.11)

ρb =∑n

a2n|φn〉〈φn|. (3.12)

Both density operators have the same eigenvalues a2n and if the states |λn〉and |φn〉 are the eigenstates of a pair of operators

A =∑n

λn|λn〉〈λn|, (3.13)

B =∑n

φn|φn〉〈φn| (3.14)

and if the eigenvalues are distinct then it follows that the outcome of ameasurement of B is uniquely determined by a measurement of A and viceversa. Observables A and B are perfectly correlated.

Pure states that are not entangled have a corresponding Schmidt decom-position which has one and only one Schmidt coefficient.

3.3 Mixed entangled states and PPT criterion

The existence of correlated properties, of course, is not something particularto entangled states. Indeed, unlike pure states, all correlated mixed statesare not entangled. Let us first introduce an uncorrelated mixed state of twosystems A and B defined on HAB = HA ⊗HB:

14

ρ = ρA ⊗ ρB. (3.15)

A correlated but not entangled state will be one formed by a mixture of statesof these kind and it will not display the intrinsically quantum correlationsassociated with entanglement. It is a separable state:

ρ =∑i

pi ρiA ⊗ ρiB. (3.16)

An entangled mixed state is the one that cannot be represented nor approx-imated by this form [6].

It would be useful to have a universal method to tell whether a givenstate ρ is entangled or not but in general the problem of separability ofmixed states appears to be extremely complex and finding one such methodis still an important research problem in quantum information theory. Thereare however some operational criteria for some cases.

For 2 ⊗ 2 and 2 ⊗ 3 cases there is a sufficient and necessary conditionfor a mixed state to be entangled known as the positive partial transpose(PPT) criterion [9]. Let us write a given state in the following form:

ρ =∑mn

ρmn |m〉〈n|, (3.17)

for a single qubit the desity matrix will be:

ρ =

(ρ00 ρ01ρ∗01 ρ11

)(3.18)

as ρ is an hermitian operator. The transpose of the density operator reads

ρT =

(ρ00 ρ∗01ρ01 ρ11

), (3.19)

which itself represents a possible density operator for a quantum system,given that the transpose operation does not change the eigenvalues of amatrix.

The partial transpose operation performs the transpose on one of thesubsystems so that an unentangled mixed state will become

ρPTB =∑ij

pij ρiA ⊗ (ρjB)T , (3.20)

if the transpose is operated on subsystem B. This represents an allowedstate for the whole system given that so does the substate (ρiB)T for the Bsubsystem. The same thing happens if we take the transpose on subsystemA, in fact, it turns out that ρPTA = (ρPTB )T , what permits us to use eitherof them indistinctively.

15

Now, if the partial transpose is performed on an entangled state we often(always for 2⊗2 and 2⊗3 entangled cases) find that the ρPT has one or morenegative eigenvalues and as this is not permitted for a density operator, weconclude that the state is indeed an entangled one. In other words, thePPT is necessary but not sufficient to guarantee that a state is separable forstates bigger than 2⊗ 3.

We have developed, using MATLAB computing language, a functionthat computes the partial transpose of a matrix and a little program thatapplies it for the PPT criterion:

For example the antisymmetric |Ψ−〉 = 1√2(|0〉|1〉−|1〉|0〉) Bell state used

in the EPR experiment has an associated partial transpose that reads

ρPT =1

2

0 0 0 −10 1 0 00 0 1 0−1 0 0 0

. (3.21)

One of the eigenvalues of this operator is negative and hence the state isentangled.

16

3.3.1 Entanglement under unitary evolution

A more interesting example is to consider the evolution of a state under acertain Hamiltonian and see how the entanglement properties change. Wepresent a one-dimensional Ising model representing two spins-1/2. Eachspin is allowed to interact with its neighbor and there is no external fieldinteracting with the lattice. Then, the Hamiltonian reads as follows:

H = −J(σ+1 σ−2 + σ−1 σ

+2 ), (3.22)

where J characterizes the interaction and

σ+1 σ−2 = (|0〉〈1|)⊗ (|1〉〈0|) = |01〉〈10| =

0 0 0 00 0 1 00 0 0 00 0 0 0

, (3.23)

σ−1 σ+2 = (|1〉〈0|)⊗ (|0〉〈1|) = |10〉〈01| =

0 0 0 00 0 0 00 1 0 00 0 0 0

. (3.24)

The diagonalization of the Hamiltonian leads to

Hd = J

0 0 0 00 −1 0 00 0 1 00 0 0 0

, (3.25)

{|00〉, |Ψ+〉 = 1√2(|01〉+ |10〉), |Ψ−〉 = 1√

2(|01〉 − |10〉), |11〉} being the eigen-

vectors. Now that the Hamiltonian is diagonalized, let us look at the timeevolution of a state that is not stationary, for example |01〉:

U(t)|01〉 = exp(−iHt/~)1√2

(|Ψ+〉+ |Ψ−〉)

=1

2((|01〉+ |10〉) exp(iJt/~) + (|01〉 − |10〉) exp(−iJt/~))

= cos(Jt/~)|01〉+ i sin(Jt/~)|10〉. (3.26)

We have used our MATLAB function to see how the entanglement evolves.The evolution of the negative eigenvalue of the partially transposed densitymatrix, along with the von Neumann entropy of the reduced density matrixρA, are depicted in figure 3.1. In figure 3.2, on the other hand, all the eigen-values have been plotted using the Mathematica piece of software, in orderto compare with our results. In both figures J/~ = 1 applies. When the

17

Figure 3.1: numerical calculation of the minimal eigenvalue of the partiallytransposed matrix and the von Neumann entropy of ρA, in blue and violetrespectively.

value of the entropy is 1 for the reduced density operator, the total state isa Bell state. This agrees with the reasoning in the previous section of thereduced density matrix being in its most mixed form when the total stateis maximally entangled.

It is clear after this example that, unlike what happens under the effectof local unitaries, entanglement can change under global unitary evolution.

3.3.2 PPT for 3⊗ 3 pure states

Finally we have also tested states of higher dimension in order to investigatewhether, not only the necessity but also, the sufficiency of the PPT criterionholds al least for pure states. We tried with a system made of two qutrits,that is, a 3 ⊗ 3 quantum system. In this case, 3 Schmidt coefficients areenough to write down any entangled pure state, for example:

|abc〉 = a|02〉+ b|10〉+ c|21〉. (3.27)

Then the density matrix has a maximum of 9 positive terms that dependon our three Schmidt coefficients a,b and c:

18

Figure 3.2: evolution of the eigenvalues of the partially transposed densitymatrix given by Mathematica.

|abc〉〈abc| =

0 0 0 0 0 ab 0 0 00 0 0 0 0 0 0 0 ac0 0 a2 0 0 0 0 0 00 0 0 b2 0 0 0 0 00 0 0 0 0 0 bc 0 0ab 0 0 0 0 0 0 0 00 0 0 0 bc 0 0 0 00 0 0 0 0 0 0 c2 00 ac 0 0 0 0 0 0 0

. (3.28)

Having the 9×9 density matrix parametrized in such way, we calculated thepartial transpose and then solved the eigenvalue problem with Mathematica.As much as three eigenvalues turn out to be negative, namely, −ab,−acand −bc. Any of the 5 other possible Schmidt decompositions made of thesame three coefficients a, b and c, can be reached using local unitaries fromexpression 3.27. One simply needs to apply the local identity U1 = I for thefirst qutrit and local unitaries of the form U2 = |i〉〈1|+ |j〉〈2|+ |k〉〈0|, wherei, j, k ∈ {0, 1, 2} and i 6= j 6= k 6= i, for the second one.

19

For the states |ab〉 = a|02〉 + b|10〉, |ac〉 = a|02〉 + c|21〉 and |bc〉 =b|10〉 + c|21〉 we also find a negative eigenvalue in each of the respectivepartially transposed density matrices.

Therefore, we conclude that every entangled 2-qutrit pure state is iden-tified by the PPT criterion. In fact, a certain degree of mixture is neededin order an entangled state not to be identified by the PPT criterion [10].

20

Chapter 4

Manipulation andclassification of entanglement

4.1 Introduction to different kinds of entanglement

For states composed of more than two subsystems, the variety of entangledstates is much richer.

Let us consider first the simplest case in which only two of the threequbits are entangled with each other. One such state could be

|Ψ〉 =1

2(|010〉+ |100〉+ |011〉+ |101〉)

=1√2

(|01〉+ |10〉)⊗ 1√2

(|0〉+ |1〉). (4.1)

It is a state that can be separated in a Bell state and in a single qubit andconsequently it is said to be partially separable. This state is in fact a bisep-arable state, a concept that will be introduced more formally in subsequentsubsections, where entanglement classes are going to be defined.

However, three qubits can be fully entangled, that is to say, the propertiesof any qubit are correlated with both of the others. An example of a suchstate is the Greenberg-Horne-Zeilinger (GHZ) state [11]:

|GHZ〉 =1√2

(|000〉+ |111〉). (4.2)

We can evaluate the reduced density operator for each subsystem to showthat no pure state can be associated to a single qubit in the same way wedid for bipartite pure cases. The density operator of the GHZ state takesthe following form in the {|000〉, |001〉, |010〉, |011〉, |100〉, |101〉, |110〉, |111〉}basis:

21

|GHZ〉〈GHZ| = 1

2

1 0 0 0 0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 01 0 0 0 0 0 0 1

. (4.3)

Indexing our three subsystems a, b and c, it turns out that we get a mixedstate tracing here over the qubit c:

ρab = trc(|GHZ〉〈GHZ|) =1

2(|00〉〈00|〈0|0〉+ |11〉〈11|〈1|1〉)

=1

2(|00〉〈00|+ |11〉〈11|) =

1

2

1 0 0 00 0 0 00 0 0 00 0 0 1

. (4.4)

This remaining mixed state has no entanglement at all. We would havegotten the same result, of course, had we traced out any of the other twoqubits.

Tracing over either of the remaining qubits gives the same reduced den-sity matrix we got in the bipartite pure example:

ρa = ρb = ρc =1

2

(1 00 1

). (4.5)

Again, these mixed states tell us that the |GHZ〉 state is fully entangled.Another example of an entangled three qubit pure state is the W state [12]

|W 〉 =1√3

(|001〉+ |010〉+ |100〉), (4.6)

which gives

|W 〉〈W | = 1

3

0 0 0 0 0 0 0 00 1 1 0 1 0 0 00 1 1 0 1 0 0 00 0 0 0 0 0 0 00 1 1 0 1 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

(4.7)

and

22

ρab = ρac = ρbc =1

3

1 0 0 00 1 1 00 1 1 00 0 0 0

, (4.8)

so that

ρa = ρb = ρc =1

3

(2 00 1

). (4.9)

An important difference with respect to the GHZ state is the form of thereduced density matrix ρab. Taking its partial transpose we find that alleigenvalues of the resulting matrix are not positive or zero and therefore,according to the PPT criterion, ρab is entangled. Unlike the entanglement ofthe GHZ state, the one of the W can be said to be robust when disposing oneof the three qubits because the remaining state retains some entanglement.

In the following subsections we are going to introduce entanglementclasses and a kind of operations that are necessary in order to understandthem.

4.2 LOCC tasks and entanglement classification

Local quantum operations and classical communications (LOCC) [13] is amethod where, as the name tells us, operations are performed on part of thetotal system and where the result of the measure is communicated classicallyto the other parts. These parts can then perform another local operationon their subsystems, depending on measurements of other parties. Let usillustrate this with an example using the following two Bell states:

|Ψ+〉 =1√2

(|01〉+ |10〉) (4.10)

|Φ+〉 =1√2

(|00〉+ |11〉). (4.11)

For both states, Alice is in possession of the first qubit while Bob owns thesecond one, as usual. They can choose measuring one of the states, althoughthey do not know which state |Ψ+〉 or |Φ+〉 they are measuring exactly.Assuming that Alice measures σz on her qubit and that communicates herresult, which is either 1 or -1, to Bob using a classical communicationschannel, then Bob will measure either -1 or 1 respectively in case they areusing the |Ψ+〉 state. However, if they are using the |Φ+〉 state, then Bobwill measure either 1 or -1 respectively. So after receiving Alice’s messagewith the outcome of her measurement and performing his own measurement,Bob is able to distinguish which of the two states they have been using.

23

Figure 4.1: Scheme of a possible LOCC arrangement.

But what is the precise meaning of a local operation? Local operationsare operations that can be achieved by the composition of the followingfundamental steps [14]:

• Unilocal unitary transformations. Operations of the form

U ⊗ I⊗ ...⊗ I, (4.12)

that is, the identity for all parties except for one, for which a uni-tary operator acts. For instance, in a bipartite system this kind ofoperations are either of the form UA ⊗ IB or IA ⊗ UB.

• Unilocal von Neumann measurements. The identity acts for allthe parties except for one, on which a von Neumann measurement iscarried out:

Pn ⊗ I⊗ ...⊗ I, (4.13)

Pn being the corresponding projector.

• Addition or subtraction of an uncorrelated ancilla. One of theparties either couples or decouples an additional, or ancillary, quantumsystem to its subsystem, so that the density operator of the wholesystem transforms in the following ways:

ρ→ ρ′ = ρ⊗ ρanc, (4.14)

ρ→ ρ′′ = tranc(ρ), (4.15)

24

respectively.

Classical communication, on the other hand, means that either zero orone is sent from Alice to Bob and not a superposition of both. This merelyallows local operations by one party to be conditioned on the outcome ofmeasurements performed earlier by other parties.

In the spirit of non-local properties so closely related to entanglement, itmay appear quite natural to view states that differ only by local operationsas equivalently entangled. However LOCC-based classification of entangle-ment appears to be extremely complicated as we shall see.

Let us first drag our attention to the most intuitive case of separablestates. States of the form

ρABC... =∑i

pi ρiA ⊗ ρiB ⊗ ρiC ⊗ ... (4.16)

of many parties A, B, C, etc. can be created from scratch by means ofLOCC. First Alice acts on her subsystem in order to sample from a pre-viously known probability distribution pi. Now by telling all other partiesher outcome, each one acts on his or her party so that every one gets hisor her ρiX and then discards the information about the probability of thatoutcome. As these states satisfy a local hidden variable model (LHVM),their correlations do not defy a classical description. We conclude, as well,that LOCC cannot produce entanglement from an unentangled state.

We are now in position to justify the concept of maximally entangledtwo-qubit states we introduced as synonym of Bell states. The name comesfrom the fact that these state are more entangled than all others and, as weshall see now, all other states can be created from the maximally entangledones by means of LOCC alone.

Limiting ourselves first to only using local unitaries (LU), we can try tocharacterize bipartite states using the Schmidt normal form

|SNF 〉 =

min(d1,d2)∑n

√pn |λnλn〉, (4.17)

where d1 and d2 are the dimension of the subsystems’ Hilbert spaces and pnare probabilities.

From the definition U †U = UU † = I of unitary operators follows thatlocal unitaries are invertible, that is, a local unitary can be inverted by a localunitary to retrieve the departure state. Hence, and as LOCC cannot produceentanglement, states related by local unitaries have the same amount ofentanglement. Therefore, any bipartite quantum state can be written in theSchmidt normal form without affecting the entanglement properties.

Two quantum states are equivalent under local unitary transformationsif and only if their normal forms coincide and therefore, it follows that allclasses can be parametrized by the angle θ in the two qubits case:

25

cos θ |00〉+ sin θ |11〉. (4.18)

This is possible thanks to the ability of diagonalizing each of the reduceddensity matrices, something that is achieved with the adequate local unitary.If θ = 0, π

2 , π, 3π2 , then we have a product state whereas when cos θ and

sin θ are equal in magnitude, then the state is going to be most stronglyentangled.

We conclude that a continuous parameter such as θ, appears to be nec-essary in order to label all equivalence classes under local unitaries.

Now let us see how different the situation looks like using LOCC. Wewill start from the maximally entangled

|Φ+〉 =1√2

(|00〉+ |11〉) (4.19)

state and apply the following LOCC recipe, that solves an exercise proposedin [15]:

Alice begins by adding an ancilla in the state |0〉 that gives

1√2

(|00〉A|0〉B + |01〉A|1〉B). (4.20)

Then, she applies the local unitary operation that takes |00〉 to cos θ|00〉 +sin θ|11〉 and |01〉 to sin θ|01〉+ cos θ|10〉, namely

cos θ 0 0 − sin θ0 sin θ − cos θ 00 cos θ sin θ 0

sin θ 0 0 cos θ

⊗ IB

1√2

10010000

=

cos θ 0 0 0 0 0 − sin θ 00 cos θ 0 0 0 0 0 − sin θ0 0 sin θ 0 − cos θ 0 0 00 0 0 sin θ 0 − cos θ 0 00 0 cos θ 0 sin θ 0 0 00 0 0 cos θ 0 sin θ 0 0

sin θ 0 0 0 0 0 cos θ 00 sin θ 0 0 0 0 0 cos θ

1√2

001√2

0000

26

then, this is also true for mixed states.More generally, any bipartite state of n-dimensional subsystems can be

prepared with certainty from the maximally entangled

|Φn+〉 =

1√n

(|0, 0〉+ |1, 1〉+ ...+ |n− 1, n− 1〉) (4.27)

state, by means of LOCC alone.However, it is easy to see that there exist pairs of states that cannot

be converted one into the other with certainty. For example, considering abipartite system formed by two 3-level subsystems or qutrits, there is totalcertainty of success for the transformation

1√2

(|00〉+ |11〉)→ 1

2|00〉+

√3

2|11〉, (4.28)

using a certain LOCC protocol.Now for a system formed by a pair of qutrits we can also consider the

following transformation:

1√2

(|00〉+ |11〉)→ 1√1 + ε2

(1

2|00〉+

√3

2|11〉+ ε|22〉

). (4.29)

No matter how small ε is, this transformation has zero probability of successfor any LOCC protocol, because the number of Schmidt coefficients, alsoknown as the Schmidt number or the Schmidt rank, has been increased.

Clearly LOCC operations are in general non-invertible because they canproject out Schmidt terms diminishing the Schmidt number of the statewhereas the increasing is not possible. Therefore a state |ψ〉 can be con-verted into |φ〉 by means of LOCC with some probability if and only if thecorresponding Schmidt numbers satisfy the relation nψ ≥ nφ.

We mentioned when we introduced the Schmidt decomposition thatstates that can be written using only one Schmidt coefficient, that is, theones that correspond to unity Schmidt number, are clearly not entangledat all. It is reasonable to assert now that the entanglement of a state char-acterized by a given Schmidt number is less powerful than that of a statewhich has a bigger Schmidt number.

So far this LOCC approach has provided us with a tool for classifica-tion of entanglement rooted in some physically meaningful criterion. Inthe following subsection we will introduce a more general scheme with themotivation of presenting a classification of entanglement that works for mul-tipartite systems.

28

4.3 Stochastic LOCC

It is a natural generalization of LOCC to consider stochastic local quantumoperations assisted by classical communication (SLOCC) [16]. By letting theprotocol be successful just stochastically, instead of requiring to be successfulat each instance as the LOCC protocol does, it is not imposed that the finalstate has to be achieved with certainty. This loosened version of LOCC isalso known as filtering operation [15].

This method consists of several rounds of LOCCs, which are dependenton previous measurement results. Success is possible if and only if one ofthe branches does the job of getting the final state |φ〉 from the original one|ψ〉 and then it is said that |ψ〉 and |φ〉 are equivalent under SLOCC. Inother words, interconversion is not asked to be deterministic, in the sensethat probability of succeeding is not required to be 1.

SLOCCs are best understood within the formalism of quantum operatorsdescribed by Karl Kraus in [17]. These operators permit us to describe quan-tum measurements in a more general way than the von Neumann schemedoes. We begin by considering two systems: the target, that is, the systemwe want to measure, and a second ancillary system or probe [18]. Thesesystems have dimensions M and N respectively and the interaction betweenthem is described by a unitary operator acting on both that can be writtenas

U =∑

nn′mm′

unm,n′m′ |nm〉〈n′m′| =∑nn′

|n〉〈n′| ⊗Ann′

=

A11 A12 ... A1N

A21 A22 ... A2N

... ... ... ...AN1 AN2 ... ANN

, (4.30)

where |m〉 corresponds to the target and |n〉 to the probe, and the sub-blocks

Ann′ =∑mm′

unm,n′m′ |m〉〈m′| are M ×M matrices. Due to unitarity∑n

A†niAni = Bii = I (4.31)

for every sub-block Bii in U †U = I and we can then write An ≡ Ani andA†n ≡ A†ni = A∗in, so that the previous restriction reads∑

n

A†nAn = I. (4.32)

Given that the unitary operator that governs the interaction between thetarget and the prove can be any one that acts in the joint system, the probe

29

may start in the state |0〉 without loss of generality:

ρtot = |0〉〈0| ⊗ ρ. (4.33)

The action of the unitary leads to

ρU =UρtotU† =

(∑n

|n〉〈0| ⊗An

)(|0〉〈0| ⊗ ρ)

(∑n

|0〉〈n| ⊗A†n

)=∑n

|n〉〈n| ⊗AnρA†n. (4.34)

By allowing the target and the probe to interact, we have let them to getcorrelated, and now, a von Neumann measurement on the probe will provideinformation about the target. The action of the projector Pn = |n〉〈n| ⊗ Ifor example, after tracing out the probe, gives

ρ′n =trprobe(PnρUPn)

C=

AnρA†n

tr(AnρA†n)

=AnρA

†n

tr(|n〉〈n| ⊗AnρA†n)

=AnρA

†n

tr(PnρUPn)=AnρA

†n

pn, (4.35)

where C is a normalization constant that turns out to be the probability pnof finding the probe in the state |n〉.

If we do not know the outcome of the measurement, however, all we cansay is that the state of the target is going to be

ρ′ =∑n

pnρ′n =

∑n

AnρA†n, (4.36)

that is, an averaging over all the possible results of a von Neumann measure-ment in the probe. This last form is called the operator-sum representationor the Kraus representation of ρ′, and the operators An are known as Krausoperators.

Now that we have introduced this very useful formalism of measurementoperators, we are ready to tackle the SLOCC scheme and its close relationto invertible local operators (ILOs) [19]. We shall consider first the Schmidtnormal form of a general state |ψ〉 in a bipartite system made of a qutritand a qubit:

(UA ⊗ UB)|ψ〉 = cos θ|00〉+ sin θ|11〉. (4.37)

If we apply now the operation given by

A⊗ IB =

(2∑i=0

|λi〉〈i|

)⊗

(1∑i=0

|i〉〈i|

), (4.38)

30

where A acts on the qutrit and IB on the qubit, then the reduced densitymatrices will be changed in the following way:

ρψA =

cos2 θ 0 00 sin2 θ 00 0 0

→ AρψaA† = cos2 θ|λ0〉〈λ0|+ sin2 θ|λ1〉〈λ1|,

(4.39)

ρψB =

(cos2 θ 0

0 sin2 θ

)→ IBρψb I

†B =

(cos2 θ 0

0 sin2 θ

). (4.40)

So it follows that for the states |ψ〉 and |φ〉 such that

|φ〉 = (A⊗ IB)(UA ⊗ UB)|ψ〉, (4.41)

the ranks of the corresponding reduced density matrices satisfy

r(ρφA) = r(ρφB) ≤ r(ρψA) = r(ρψB). (4.42)

This expression is in fact a general one for arbitrarily big bipartite systems,and for the multipartite case, with parties A,B, ..., N , the generalizationreads as follows:

r(ρφk) ≤ r(ρψk ), (4.43)

where |φ〉 = (A ⊗ B ⊗ ... ⊗N)|ψ〉 and k = A,B, ..., N . This is because theoperation can be seen as a composition of the local operators A⊗ IB...N andIA ⊗ (B ⊗ ... ⊗ N) and similarly for the other parties. The probability ofsuccess of the composition will have the form of a product of probabilitiesfor each step pApB...pN .

Now, if the operators A,B, ..., N are invertible, then |ψ〉 = (A−1⊗B−1⊗... ⊗ N−1)|φ〉, that is, the operation can be reversed locally and then bothpure states can be reached from each other using SLOCC. We say that |ψ〉and |φ〉 are equivalent under SLOCC. Also as r(ρφk) ≤ r(ρψk ) and r(ρφk) ≥r(ρψk ), it follows that r(ρφk) = r(ρψk ). This is an important result. It meansthat SLOCC protocols, unlike LOCC, do conserve the local ranks of purestates.

Let us consider the simplest transformation of this kind, that is, A⊗IB...Nwith A invertible. We can write then the following Schmidt decompositionsfor the initial and final states:

|ψ〉 =n∑i=1

aψi |i〉A|µi〉B...N , (4.44)

31

|φ〉 =

n∑i=1

aφi (UA|i〉A)|µi〉B...N , (4.45)

where the local unitary UA relates the local basis for both states in Alice’sm-dimensional part. It is clear then that the operator A should be of theform

A = UA

(n∑i=1

aφi

aψi|i〉〈i|+

m∑i=n+1

|λi〉〈i|

). (4.46)

In this expression the vectors |λi〉 play no role and therefore we can write|i〉 instead:

A = UA

(n∑i=1

aφi

aψi|i〉〈i|+

m∑i=n+1

|i〉〈i|

). (4.47)

This is a great step, because it makes A diagonal and therefore invertible,because all the diagonal elements are nonzero. The general case, again,corresponds to composing this operation with IA⊗B⊗ IC...N , for which thesame argumentation can be applied, and so on and so forth. So this is theprove that if |ψ〉 and |φ〉 are SLOCC equivalent, then the operator relatingthem can always be chosen to be invertible.

Summarizing, two pure multipartite states are SLOCC equivalent if andonly if they are related by an invertible local operator (ILO).

The consideration of the value of the ranks of the reduced density matri-ces, or local ranks, has proved vital for an understanding of SLOCC proto-cols. Next we shall see their usefulness for the classification of multipartiteentanglement. Let us present the case of non-entangled and bipartitely en-tangled or biseparable three qubit states first.

4.4 SLOCC classification for non-genuinetripartite entanglement

Using the fact that local ranks of multipartite states are invariant underILOs we can distinguish four classes or families that are inequivalent underSLOCC, when talking about states that do not have genuine three partiteentanglement.

First of all, we present the non-entangled or separable state, that is, theone that can be taken into

|ψa−b−c〉 = |000〉, (4.48)

using some convenient local unitaries. We get that

32

ρa = ρb = ρc =

(1 00 0

), (4.49)

so that r(ρa) = r(ρb) = r(ρc) = 1.The representatives of the other three classes are the bipartitely maxi-

mally entangled

|ψab−c〉 =1√2

(|00〉+ |11〉)|0〉, (4.50)

|ψac−b〉 =1√2

(|00〉+ |11〉)|0〉, (4.51)

|ψbc−a〉 =1√2

(|00〉+ |11〉)|0〉, (4.52)

states, for which respectively r(ρc) = 1, r(ρb) = 1 and r(ρa) = 1, being theother local ranks equal to 2, as they correspond to the maximally entangledqubits in each of the cases.

Two, out of the three, local ranks are different from one state to anotherand therefore we conclude that the four states belong to four inequivalentclasses under SLOCC.

Note that other states within the biseparable classes can be obtainedfrom one of the three representatives by means of LOCC with total certainty,in the same manner we did for bipartite states, and the separable state canbe obtained from any of them.

4.5 SLOCC classification for symmetric 3-qubitstates

The |GHZ〉 and |W 〉 states we introduced at the beginning of this sectionwere both symmetric with respect to the permutation of the qubits. On theother hand, the symmetric subspace has the advantage of a lower increasingof its dimension with the number of parties: if we consider a 1/2 spin com-pound system formed of n spins, the |S,M〉 or Dicke states are simultaneouseigenstates of the collective spin operators S2 and Sz. The states having thehighest value of the total angular momentum quantum number S = n/2 aresymmetric with respect to permutation, and form a subset of all 2n Dickestates [20]. As there are 2S + 1 possible values of M for every value of S, itfollows that the dimension of the symmetric subspace is n+ 1.

Finally, and most importantly, limiting ourselves to the symmetric sub-space permits us to act with the same invertible local operator A on each ofthe qubits, because then it is sufficient to look for a symmetric ILO [21]:

|φS〉 = (A⊗ ...⊗A)|ψS〉, (4.53)

33

where |φS〉 and |ψS〉 are symmetric states.For 3 qubits, the 4 linearly independent symmetric Dicke states are usu-

ally written as follows:

∣∣∣∣32 , 3

2

⟩= |000〉,∣∣∣∣32 , 1

2

⟩=

1√3

(|001〉+ |010〉+ |100〉),∣∣∣∣32 , −1

2

⟩=

1√3

(|110〉+ |101〉+ |011〉),∣∣∣∣32 , −3

2

⟩= |111〉, (4.54)

where in the right hand side we already substituted the individual angularmomentum projection values by their qubit notation counterparts. Thesestates form a basis on the 3-qubit symmetric subspace and therefore anystate belonging to it can be expressed as a linear combination of them:

|φS〉 = w

∣∣∣∣32 , 3

2

⟩+ x

∣∣∣∣32 , 1

2

⟩+ y

∣∣∣∣32 , −1

2

⟩+ z

∣∣∣∣32 , −3

2

⟩, (4.55)

where w, x, y and z are complex coefficients.This scheme suggested us a straightforward technique to find different

SLOCC families. Indeed, using the expression 4.53, where we parametrizethe invertible operator as

A =

(a bc d

), (4.56)

so that the inverse is given by

A−1 =1

ad− bc

(d −b−c a

), (4.57)

we get a system of equations relating w, x, y and z to a, b, c and d for agiven |ψS〉. Taking |ψS〉1 = |GHZ〉 = 1√

2(|000〉+ |111〉) and |ψS〉2 = |W 〉 =

1√3(|001〉+ |010〉+ |100〉), we have found that the two systems of equations

we get are incompatible, that is, there do not exist w, x, y and z coefficientssuch that they can be written in terms of the a1, b1, c1 and d1 as well as thea2, b2, c2 and d2, at the same time:

34

w =1√2

(a31 + c31),

x =1√2

(a21b1 + c21d1),

y =1√2

(a1b21 + c1d

21),

z =1√2

(b31 + d31), (4.58)

w′ =√

3a22c2,

x′ =1√3

(a22d2 + 2a2b2c2),

y′ =1√3

(b22c2 + 2a2b2d2),

z′ =√

3b22d2. (4.59)

=⇒ w 6= w′, x 6= x′, y 6= y′, z 6= z′. (4.60)

We performed the same test with |ψS〉1 = |000〉 and |ψS〉2 = |111〉 and thistime we got a relation between a1, b1, c1 and d1, and a2, b2, c2 and d2. Thisis how it should be, of course, because |000〉 and |111〉 are local unitarilyequivalent and local unitaries are a special case of SLOCC operations, there-fore belonging to the same SLOCC family, namely, the family of separablestates.

35

Chapter 5

Conclusions

In this final chapter, we would like to summarize some important aspects ofentanglement we learned during this theoretical study.

• The Schrodinger equation not only admits product states, but alsosuperpositions of them, leading to correlations between observablesthat defy a classical reasoning. This does not mean, however, thatall superposition states are entangled. Some turn out to be separa-ble. This concept of separability is the one on which the definition ofentanglement lies.

• To tell whether or not a state is separable, there does not exist ageneral procedure. However different criteria exist for some particularcases. For pure states it is enough to look for the reduced densitymatrices and see if they represent a mixed state. Then the state isentangled. For mixed states of two qubits or a qubit and a qutrit, anecessary and sufficient condition exists, which is given by the PPTcriterion.

• For three qubits the variety of entangled states grows. If the LOCCscheme confirms us that there are different degrees of entanglement inthe bipartite case, being the Bell states the maximally entangled ones,SLOCC shows that in the tripartite case, although no such thing asthe maximally entangled state exists, a classification based on the localranks can be envisaged: separable states have local ranks which equalall to 1; non-totally separable but non-genuinely entangled tripartitestates have one local rank equalling to 1; genuinely entangled tripartitestates such as the W or the GHZ have local ranks all equalling to 2.

• However, W and GHZ states are not reachable from each other bymeans of ILOs and therefore are representatives of distinct SLOCCclasses.

36

Bibliography

[1] Einstein, Podolsky and Rosen, Phys. Rev. 47, 777 (1935).

[2] Bell, J. S., Physics. 1, 195 (1964).

[3] Yehuda B. Band and Yshai Avishai, Quantum Mechanics with Appli-cations to Nanotechnology and Information Science, Academic Press,Waltham, U.S. (2012).

[4] Aspect, A., P. Grangier and G. Roger, Phys. Rev. Lett. 47, 460 (1981).

[5] Aspect, A., J. Dalibard and G. Roger, Phys. Rev. Lett. 49, 1804 (1982).

[6] R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod.Phys. 81, 865 (2009).

[7] D. Bohm, Quantum Theory, Prentice-Hall, Upper Saddle River, U.S.(1951).

[8] M. A. Nielsen and I. L. Chuang, Quantum Computation and QuantumInformation, Cambridge University Press, Cambridge, U.K. (2000).

[9] Peres, A., Phys. Rev. Lett. 77, 1413 (1996).

[10] P. Horodecki, M. Lewenstein, G. Vidal and I. Cirac, Phys. Rev. A 62,032310 (2000).

[11] D. M. Greenberger, M. A. Horne and A. Zeilinger, Bell’s Theorem,Quantum Theory, and Conceptions of the Universe, Kluwer Academic,Dordrecht, Netherlands (1989).

[12] A. Zeilinger, M. A. Horne and D. M. Greenberger, NASA Conf. Publ.,3135 (1992).

[13] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolinand W. K. Wootters, Phys. Rev. Lett. 78, 2031 (1996).

[14] G. Vidal, Journ. of Mod. Opt. 47, 355 (2000).

37

[15] D. Bruß and G. Leuchs, Lectures on Quantum Information, Wiley-VCH, Weinheim, Germany (2006).

[16] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin and A. V. Tapliyal,Phys. Rev. A 63, 012307 (2000).

[17] K. Kraus, States, Effects, and Operations, Springer-Verlag, Berlin, Ger-many (1983).

[18] K. Jacobs, Quantum Measurement Theory and its Applications, Cam-bridge University Press, Cambridge, U.K. (2014).

[19] W. Dur, G. Vidal and I. Cirac, Phys. Rev. A 62, 062314 (2000).

[20] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cam-bridge University Press, Cambridge, U.K. (1995).

[21] P. Mathonet, S. Krins, M. Godefroid, L. Lamata, E. Solano and T.Bastin, Phys. Rev. A 81, 052315 (2010).

38

entanglement in bipartite and tripartite quantum systems...section 3.3.2 shows that the ppt...

Documents