Introduction to Entanglement

Allan Solomon, Paris VI

Mathematics Prelude

Paradigm: Quantum mechanics & Maths

Quantum Field Theory

For example, Feynman Diagrams involve

AnalysisRiemann Zeta fns and extensions

AlgebraHopf Algebra Braid Groups

CombinatoricsGraph TheoryCounting

TopologyKnot Theory

Mathematics Prelude Entanglement (Fr.) Intrication

(Eng.) Intricate = Complexity

involves Analysis, Algebra , Topology,Combinatorics

Borromean Rings – example of an entangled system

Physics Prelude: EPR paradox

Source emits spin singlet

Measurement by Alice on |y> determines Bob’s measurement.

The electrons are entangled.

“Entanglement is the characteristic trait of quantum mechanics”

Erwin Schroedinger, 1935.

“Interference (in Classical and Quantum Mechanics) is just the fact that the sum of squares is not the square of the sum.”

Richard Feynman

1 Vectors, Vector Spaces

A basic operation for vectors is addition. For mathematicians therefore, vector addition presents no surprises. For physicists, vector addition is such a remarkable property that in quantum mechanics the phenomena it gives rise to it go by many names, superposition rule, interference, entanglement,…

Vector notation

|0><1|

2121 1

0

0

1eeee

Maths Notation Physics Notation

1|0|,1|,0|

1

001),(

1001

2*121

*2

*1

eeee

ee

1|0

|1|0

00

1010

0

1*21ee

Dirac

2 Bipartite Spaces

0

0

1

0

1

0

0

121 ee

21 VVV Maths Notation Physics Notation

1,0|1|0|

If V1 has basis {ei} and V2 has basis {fj} then

{ei Äfj, i=1..m,j=1..n} is a basis for V1ÄV2

Every vector in V1ÄV2 is a sum of products; but not every vector is a product. If it is a product, then it is said to be non-entangled.

Bipartite Spaces: Entanglement

Example:

1221

2212221

21

)(

1

0

0

1

eeee

eeeeeee

ee

Not entangled

Entangled

“entangled” means not factorizable

1

0

1

0

0

1

1

0

3 States, Pure and Mixed

(a) Pure StatesVectors correspond to Pure states:

example

1

0

0

1||

i i

We may equally represent a Pure State | > y by the Operator (Projector) | >< |y y which projects onto that state:

PPP

2

**

****

NOTE: trace P=1 (Normalization) andP is Hermitian with (semi-)positive eigenvalues (0 and 1 ).

States

(b) Mixed StatesWe define a (mixed) state r as a positive matrix of

trace 1

i iPi

n

n

ii i

n

UU

UU

UU

1

.

0

0

...

0

.

1

0

0

.

0

1

.

01.

21

2

1

2

1

Note: A mixed state is a mixture of pure states.

which is a (convex) sum of pure states.

Mixed state is not a unique sum of pure states

Example:

|11|21

|11|21

||1|

||1|

)1(||||

1

0

0

1

21

|

0

0

1

0

|

bbaa

bab

baa

bbaa

ba

4 Entropy of a State (Von Neumann Entropy)

)log(

)log()(

ii i

tr

00log01log1)(00

01

State r has eigenvalues li

usually log2 Example(a): Pure state

Example(b): Mixed state

1)1log()1(log)0

0(

1)(0)1log()1(log)(

loglog)(

0

0

21

21

21

21

21

21

2211

2

1

Every Pure State has Entropy Zero.

Maximum entropy 1 for maximally random state.

(E is entropy here!)

Claude Shannon

John Von Neumann

5 Measures of Entanglement

Intuitively we expect

(1) (Pure) state (1/Ö2)(|0,0> + |0,1>)

No 0

(2) Bell state (1/Ö2)(|0,0> + |1,1>)

Yes 1

(3) Ö l |0,0> + Ö (1- l) |1,1>) Yes 0 Ð.?. Ð1

Entangled? E Measure

It turns out that the (VN) Entropy gives a measure of entanglement for pure states; but not directly, as all pure states have entropy zero.

We must first take the Partial Trace over one subsystem of the bipartite system.

6 Partial Trace

If V = VA VB then trB(QA QB)=QA tr(QB)

Extend to sums by linearity.Pure States: QA =|u1><u2| QB =|v1><v2|

then trB (QA QB)= |u1><u2| <v2| v1>

Example (non-entangled state):

00

01|)00||00(|

2

1)(

0000

0000

002/12/1

002/12/1

|)1,0|0,0)(1,0|0,0(|2

1

)1,0|0,0(|2

1|

aB

a

Tr

a

Entanglement (Entropy of partially traced state) is 0.

Example: Bell state

2/10

02/1

|b>=(1/ 2)(|0,0>+|1,1>)Ö

rb=(1/2)(|0,0>+|1,1>) )(<0,0|+<1,1|)

TrB(rb)=(1/2)(|0><0| + |1><1|) =

1)log()log()(2

1

2

1

2

1

2

1 b

John Stewart Bell

Example (Entangled state):

3/13/1

3/13/2

|)11||01||00||10||00(|31

)(

|0,1|1,0|0,0)(0,1|1,0|0,0(|31

)0,1|1,0|0,0(|3

1|

bB

b

Tr

b

Entropy of partially traced state is non-zero (=.55)

Entangled state (intermediate)

( )cos 20 0 ( )sin ( )cos

0 0 0 0

0 0 0 0

( )sin ( )cos 0 0 ( )sin 2

Pure state cos(q)|0,0>+sin(q)|1,1> =

( )cos

0

0

( )sin

Þ

So this measure of entanglement gives an intuitively correct variation from 0 (non-entangled) to maximum of 1 (Bell state) for PURE States.

Recap: Definitions

A pure state may be represented by a vector or a positive matrix (Projection matrix) with eigenvalues 1, 0, … 0, 0, 0

A mixed state is a (convex) sum of pure states and may be represented by a positive matrix of trace 1.

A pure state is separable (non-entangled) if it can be written as a product of vectors (factorizable).

A mixed state is separable if it can be written as a (convex) sum of separable (factorizable) pure states.

The expression of a mixed state as a convex sum of pure states is not unique.

Measure of Entanglement

The entanglement (E ) y of a pure bipartite state yÎVAÄVB is given by the Entropy of the Partial Trace of y

The entanglement (E ) r of a mixed bipartite state rÎVAÄVB is given by

E (r)=min{SliE(yi) | = r Sliyi}

This definition of entanglement measure for Mixed States is very difficult to apply, requiring infinite tests.

Example (revisited)

|11||11|1

||1|

||1|

,

)1(||||

1,1|0,0(||1,0||

2

1

2

1

4/14/3

2

1

bbaa

bab

baa

bbaa

ba

E( )=3/4´0+ 1/4´1=0.25 r E( 1)=1/2´0.118+ 1/2´0.118=0.118r (this IS the min and therefore the entanglement )

7 Concurrence (Physics viewpoint)

Spin Flip Operation

~

|*|||*|

~|*|||*|

yy

y

qubit

2-qubit

Concurrence C = |~||

|||||

~|||flip

flipFactorizable state

Bell state

C=0

C=1

So Concurrence gives a measure of entanglement for PURE states

Concurrence (Maths viewpoint)

Partial trace of Pure state r gives 2X2 matrix so Entanglement determined by eigenvalue equation l2- + =0l D

=(1l ± (1-4 ))/2Ö D

Concurrence C2=4D

C varies from 0 to 1 so Concurrence gives a measure of entanglement for PURE states

Equivalently (for pure states) C2=tr

)1log()1(log)( Recall

~||~~

Wootters’ Concurrence

Wootters[1,2] has shown that the form for C

},0max{)( 4321 C

where the l’s are the square roots of the eigenvalues of in descending order

Gives the entanglement for mixed states; i.e. it gives the correct minimum over Pure States.(Note the formula coincides with the previous for PURE states.)

~

[1] Hill, S and Wootters, WK, PhysRevLett 78,26,5022(1997)[2] Wootters, WK, PhysRevLett 80,10,2245(1998)

Feynman Nobel Lecture

The Development of the Space-Time View of Quantum Electrodynamics

“We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on.”

Tripartite entanglement

“Naïve Solution” Extend Concurrence to 3-subspace

Pure states by summing over (3) Partial Traces

Example

|)100|010|001()100|010|001(|)3/1(

)100|010|001(|3/1

Þ

Three Partial Traces are equal,

1

30 0 0

01

3

1

30

01

3

1

30

0 0 0 0

each with Concurrence 2/3 , leading to a 3-concurrence of(1/3) (2/3+2/3+2/3)=2/3

Tripartite states

However ……….The entangled state (1/ 2)(Ö |000>+|

111>)has 3 equal partial traces

1

20 0 0

0 0 0 0

0 0 0 0

0 0 01

2

which is separable (concurrence =0).

Borromean Rings analogy –

every cut leaves a separable system

A funny “Resource”

Physicists do experiments on the principle that they can be replicated in other laboratories – invariance under transformations.

In Quantum mechanics, we expect our measurable quantities to be invariant under Unitary Transformations (or anti-unitary – Wigner)

This is NOT the case for Entanglement!

Eugene Wigner

Open Problems

What is the significance of Entanglement for Quantum Computing?

Find a measure of Entanglement for 3 (or more) qubits (tripartite spaces,..).

Interaction with the Environment (Dissipation of Entanglement)