Optimal Entanglement Generation from

Quantum Operations

Matt Leifer, Leah Henderson and Noah Linden

Dept. of Mathematics, University of Bristol

UK

quant-ph/0205055

Outline

1) Processing entanglement in states

2) Non-local Operations

2.1) Characterising non-locality

Entanglement Generation

Classical communication

Simulation

2.2) Examples: CNOT, SWAP

2.3) General 2-qubit unitaries

3) Results on entanglement generation

3.1) Single-copy results

3.2) Additivity

4) Reversibility

1) Processing entanglement in quantum states

One way to characterise entanglement is to ask what you can do with it.

Bipartite states

Singlets

Singlets

Single or multiple copies of state. states which cannot be distilled with

< 2 copies. Non-additive

AsymptoticallyReversibility is achieved for pure states.

Multi-particleFew general results are known

LOCC

LOCC

2) Non-local operations2.1) Characterising non-locality

An operation is non-local if its implementation requires an interaction between 2 or more qubits.

What can we do with them?– Perform quantum algorithms– Generate entanglement– Classical communication– Simulate other operations

Example - the CNOT gate

Generating entanglement

Product state 1 e-bit

Reversible?

2) Non-local operations2.2) Examples: CNOT

(Collins, Linden + Popescu)

Generating a CNOT from an e-bit + LOCC

BABA

BA

011100

1010

1) Append e-bit

BababA

10110010

2) Alice measures

(i) BBbaAbaA 10111000

aAaAaAaA iii 01,10vs11,00

(ii) BBbaAbaA 10001110

Alice does a CNOT on Aa and discards a

In case (ii) Bob needs to flip b 1 c-bit AliceBob

2) Non-local operations 2.2) Examples: CNOT

3) Bob does a CNOT on bB

BBbABBbA 01111000

4) Bob measures x on b

bbBBABBA

bbBBABBA

ii

i

10011100

10011100

In (ii) Alice needs to perform a z 1 c-bit Bob Alice

Can we use a CNOT to perform classical communication?

BABABABA 1101,0000 Yes

With a CNOT and an e-bit we can send 1 c-bit AliceBob and 1 c-bit BobAlice

Reversible for e-bits and c-bits on their own, but not both together

2) Non-local operations 2.2) Examples: SWAP

Example - the SWAP gate

Generating entanglement - Ancillas are required

2 e-bits are generated. Reversible?

Any non-local 2 qubit operation can be performed with 2 e-bits if classical communication is free.

CNOT and SWAP are special cases

Can generate integer number of e-bits

No initial entanglement is required

2) Non-local operations 2.3) General bipartite unitaries

Non-Local unitary gates

Converting between resources with LOCC

Entanglement generation

Distillation and dilution

Simulation

2) Non-local operations 2.3) General bipartite unitaries

Local equivalence

Canonical form

3) Results on entanglement generation

How much entanglement can be generated by a single copy of a 2-qubit unitary?

Entangling capacity

What entanglement measure should we use?

Square of concurrence

Entropy of entanglement

3) Results on entanglement generation

3.1) Single copy: Product states (Kraus + Cirac)

How much entanglement can be generated by a 2-qubit unitary acting on a product state?

12 C

2122 2sin C

3222 2sin C

2C

03

3) Results on entanglement generation

3.1) Single copy: Entangled States

How much entanglement can be generated by a 2-qubit unitary acting on an entangled state?

12 C

212 2sin C

322 2sin C

2C

03

3) Results on entanglement generation

3.1) Single copy: Other measures

Can we use results for other entanglement measures?

All measures are monotonic functions of each other.

For product states the optimal protocol is the same.

For entangled states the difference causes a problem, initial entanglement may be different, but the Schmidt basis is the same.

Optimal initial state and its entanglement

Region 2

3) Results on entanglement generation

3.1) Single copy: Other measures

Starting state

Entropy of entanglement

• Initial entanglement required

3) Results on entanglement generation

3.1) Single copy: Ancillas

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4) Additivity

Two copies

2

101202 11

fifi EEEEEEEE

Reversibility?

5) Conclusions

• Additivity – provides bounds on all the other multi-

copy conversion protocols.

• Are they all reversible?

• Other related properties of non-local

unitaries– classical capacity

– quantum capacity

• Generalisations– higher dimensions

– n-parties

– non-unitary operations

Mixed states

Pure states can achieve the entangling capacity with minimal initial entanglement.

Let be an optimal decomp.