Multi-particle Entanglement in Quantum States and

Evolutions

Matthew Leifer - 1st Year Ph.D., Maths Dept., University of BristolSupervisor - Dr. Noah Linden

1. Background and Motivation

2. Physical Meaning of Entanglement

3. Quantum Mechanics

4. Entanglement in Quantum States

5. Entanglement in Quantum Evolutions

6. Further Investigations

1. Background and Motivation

• Quantum Mechanics is weird! – role of probability

– measurement problem (“collapse of wave-function”)

– non-local correlations (entanglement)

• Quantum Mechanics is Successful!– Atomic Physics and Chemistry

– Solid State Physics (semiconductors)

– Quantum Field Theory (Particle Physics)

– Anomalous magnetic moment of electron

• We do not have full control of the quantum degrees of freedom in these applications.

What happens if we can control quantum systems?

• Quantum Computers– Feynman (1982)

• Holy Grails of Information Theory– Polynomial time prime

factorisation - Shor (1994)

– Perfectly secure key distribution in cryptography

• Other discoveries– Teleportation - Bennett et al

(1992)

– Quantum error correction - Shor (1995)

• These procedures use entangled states! Peter Shor

Richard Feynman

3. Quantum MechanicsMeasurement

• Quantum states, |>, are vectors (rays) in a Hilbert space

• Usually we normalise s.t. < |> = 1

• Observables are represented by Hermitian operators (i.e Q s.t. Q† = Q)

• If we construct an orthonormal eigenbasis{|i>} of Q s.t. Q|i> = i |i> then |> = ai

|i> with |ai|2 = 1 and ai = <i|>

• The possible results of measurements of Q are its eigenvalues i

• The result of a measurement will be i with probability |ai|2

• After obtaining a value i, the state will become |i>

3. Quantum MechanicsQuantum Dynamics

• States can also evolve between measurements |> U |>

• Conservation of probability => states must remain normalised: <|U†U|> = <|> => U†U = 1

• Quantum evolutions are unitary!

• Can also see this from Schrödinger eqn.

dt

diH

In theory, can implement any unitary transformation by correct choice of H.

3. Quantum MechanicsSystems and Subsystems

• If we have 2 systems A and B, with Hilbert spaces HA and HB then the quantum state of the combined system is a vector in HAHB

• Example - 2 dimensional subsystems (spin-1/2 particles)

HA has basis {|0>A, |1>A}

HB has basis {|0>B, |1>B}

HAHB has basis {|0>A|0>B, |0>A|1>B, |1>A|0>B, |1>A|1>B}

or {|00>, |01>, |10>, |11>}

An example vector 10012

1 AB

4. Entanglement in Quantum States

• An entangled state is one that cannot be written as |AB> = |A>|B> for any choice of basis in HA and HB

• Specialise to n spin-1/2 particles.

• General unitary transformation

• Local unitary transformation

• Each copy of U(2) acts on corresponding particle

• Local unitaries do not change entanglement of state

n2C

NU 2U

222 UUU n U

4. Entanglement in Quantum States

# Non-Local Parameters

• In general

• Linear span of XTs = tangent space to orbit at v.

• No. linearly indep. XTs gives dimension of orbit.

• E.g. infinitesimal change under a trans. in 1 direction:

• Write ar=cr+idr (r = 0,1) and

• Then

• and f(c0,d0,c1,d1) f(c0-d1,d0+ c1,c1- d0,d1+ c0)

• so

• Similarly we can find u0,u2,u3. Only 3 are linearly indep. So we have 4-3 = 1 non-local parameter

0

vefvfX Ti

T

0

11 ai

aii

Tdcdc 1100

Tcdcd 0011

Tcdcdufuf

0011110

where,.

4. Entanglement in Quantum States

Polynomial Invariants

• Construct invariants by contracting with U(2) invariant tensors) (ij and ij)in all possible ways

• Example: for 1 particle

• For 2 particles

• General case: Contract a’s with a*’s using ’s in all possible ways until we have as many functionally indep. invariants as non-local params.

ji i

iijiij aaaa,

**

ji

ijijjjii

jjiijiij aaTraaaaI,

†*

,,,

*1

11

1111

2†****

2 11111111aaTraaaaaaaaI jkjlilikllkkjjiikjjlliik

4. Entanglement in Quantum States

Stability Groups

• Each orbit has a stability group < U(2)n.

• Certain states have larger stability groups than the generic case.

• States with maximal symmetry are especially interesting.

• Example: 3 particles

– Generic states have no stability group.

– Singletvector is invariant under SU(2)U(1)

– Direct products are invariant under U(1)3

– GHZ are invariant under U(1)2 and discrete symmetry Z2

2312

1 10010

8333232131

323222121313212111

1

T

5. Entanglement in Quantum Evolutions

• Consider UV1UV2, where UU(2n) and V1,V2 U(2)n

• Does orbit space make sense?

• Apply same ideas– No. invariant parameters

– Canonical points

– Polynomial invariants

• 1 particle case -– Lie Algebra elements can now work on both

sides.

• 2 particle canonical form -– How are j’s related to polynomial

invariants?

)sincos1( 2 jj

ieU

jjjie

n

n

U

U22

2

6. Future Work

• Density matrix formalism - Linden, Popescu and Sudberry 1998

• Find canonical forms, polynomial invariants and special orbits for n particle unitaries.

• Determine relation between non-locality in states and evolutions.

• Allow measurements. What is the significance of

– Carteret, Linden, Popescu and Sudberry (1998)

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