Entanglement in Quantum Information Processing

Samuel L. Braunstein

University of York

25 April, 2004Les Houches

Classical/Quantum State Representation

Bit has two values only: 0, 1

Information is physical

BITS QUBITS

10 Superposition between two raysin Hilbert space

1100 Entanglement between (distant) objects

Many qubits leads to ...

(slide with permission D.DiVincenzo)

Fast Quantum Computation

(Shor)

(Grover)

Computational complexity: how the `time’ to complete an algorithm scales with the size of the input.

Quantum computers add a new complexity class: BQP†

†Bernstein & Vazirani, SIAM J.Comput. 25, 1411 (1997).

Computational Complexity

*Shor, 35th Proc. FOCS, ed. Goldwasser (1994) p.124

For machines that cansimulate each other inpolynomial time.

P

NP

primalitytesting

factoring*

BPP

BQP

PSPACE

Pure states are entangled if

Picturing Entanglement

(picture from Physics World cover)

BobAliceBA AB

e.g., Bell state

1100

Computation as Unitary Evolution

Any unitary operator U may be simulatedby a set of 1-qubit and 2-qubit gates.*

e.g., for a 1-qubit gate:

*Barenco, P. Roy. Soc. Lond. A 449, 679 (1995).

Evolves via

njj

jj jjn

n

11,01

1

UU :

ni

i

iini jkjk

kjjjji MU 11

1,0

)(1 :

Entanglement as a Resource

“Can a quantum system be probabilistically simulated by a classical universal computer? … the answer is certainly, No!”

Richard Feynman (1982)

“Size matters.”Anonymous

“Hilbert space is a big place.”Carlton Caves 1990s

Theorem: Pure-state quantum algorithms may be efficiently simulated classically, provided there is a bounded amount of global entanglement.

Jozsa & Linden, P. Roy. Soc. Lond. A 459, 2011 (2003). Vidal, Phys. Rev. Lett. 91, 147902 (2003).

njj

jj jjn

n

11,01

1

State unentangled if

nn jjjjjj dba 2121

Naively, to get an exponential speed-up, the

entanglement must grow with the size of the input.

Caveats:

• Converse isn’t true, e.g., Gottesman-Knill theorem

• Doesn’t apply to mixed-state computation, e.g., NMR

• Doesn’t apply to query complexity, e.g., Grover

• Not meaningful for communication, e.g., teleportation

Entanglement as a Prerequisite for Speed-up

stabilizes .

Gottesman-Knill theorem*

• Subgroups of PPn have compact descriptions.

• Gates: , , , , ,

any computation restricted to these gates may be simulated efficiently within the stabilizer formalism.

i0

01

11

11

2

1H x y

0100

1000

0010

0001

map subgroups of PPn to subgroups of PPn.

z

*Gottesman, PhD thesis, Caltech (1997).

stabilizes• PPn],,[ )()1( nzz 00

],[ zzxx 1100

• The Pauli group PPn is generated by the n-fold tensor product

of , , , and factors ±1 and ±i.x y z 21

Naively, to get an exponential speed-up, the entanglement must grow with the size of the input.

Caveats:

• Converse isn’t true, e.g., Gottesman-Knill theorem*

• Doesn’t apply to mixed-state computation, e.g., NMR

• Doesn’t apply to query complexity, e.g., Grover

• Not meaningful for communication, e.g., teleportation

Entanglement as a Prerequisite for Speed-up

Mixed-State Entanglement

mixture so

Since writejjjAA jjj

jj

jp jj

j ApAA tr

For onAB

unentangled if: ,j

BjA

jjAB p 0jp

otherwise entangled.

BA HΗ

Test for Mixed-State Entanglement

s.t.

Consider a positive map

that is not a CPM

0)( A A

AB 0))(1( AB

entangledAB

negative eigenvalues in

entangled.

))(1( ABAB

Peres, Phys.Rev.Lett. 77, 1413 (1996).Horodecki3, Phys.Lett.A 223, 1 (1996).

For = partial transpose, this is necessary & sufficient

on 2x2 and 2x3 dimensional Hilbert spaces.

But positive maps do not fully classify entanglement ...

1

Liquid-State NMR Quantum Computation

(figure from Nature 2002)

The algorithm unfolds as usual on pure state perturbation

for traceless observables ,

For any unitary transformation

Utilizes so-called pseudo-pure states

Each molecule is a little quantum computer.

12

1n

which occur in NMR experiments with small U

is pseudo-pure with replaced by†UU U

A AA

NMR Quantum Computation (1997 - )

Selected publications:

Nature (1997), Gershenfeld et al., NMR schemeNature (1998), Jones et al., Grover’s algorithmNature (1998), Chuang et al., Deutsch-Jozsa alg.Science (1998), Knill et al., DecoherenceNature (1998), Nielsen et al., TeleportationNature (2000), Knill et al., Algorithm benchmarkingNature (2001), Lieven et al., Shor’s algorithm

But mixed-state entanglement and hence computation is elusive.

Physics Today (Jan. 2000), first community-wide debates ...

Does NMR Computation involve Entanglement?

)]31()3(1tr[)4(

1),,( 221

nnnnwnn

most negative eigenvalue 4n-1(-2) = -22n-1

ndnnnn PPnnw

1),,( 1

jnjjn PnndnP

)31(4

1)1(

2

122

whereas for , is unentangled

n

n

nnnw)4(

2),,(

12

1

0),,( 1 nnnw

Braunstein et al, Phys.Rev.Lett. 83, 1054 (1999).

In current liquid-state NMR experiments ~ 10-5, n < 10 qubits

For NMR states

so

unentangled

unentangledif

no entangled states accessed to-date …or is there?

12

1n

nnnnw

)4(

1),,( 1

),,( 11 nnnw

n

n

)4(

)21(1 12

0),,( 1 nnnw

1221

1

n

Can there be Speed-Up in NMR QC?

For Shor’s factoring algorithm, Linden and Popescu*showed that in the absence of entanglement,

no speed-up is possible with pseudo-pure states.

*Linden & Popescu, Phys.Rev.Lett. 87, 047901 (2001).

Caveat:

Result is asymptotic in the number of qubits (current NMR experiments involve < 10 qubits).

For a non-asymptotic result, we must move away from computational complexity,say to query complexity.

Naively, to get an exponential speed-up, the entanglement must grow with the size of the input.

Caveats:

• Converse isn’t true, e.g., Gottesman-Knill theorem*

• Doesn’t apply to mixed-state computation, e.g., NMR

• Doesn’t apply to query complexity, e.g., Grover

• Not meaningful for communication, e.g., teleportation

Entanglement as a Prerequisite for Speed-up

Grover’s Search Algorithm*

Suppose we seek a marked number from

satisfying:

*Grover, Phys.Rev.Lett. 79, 4709 (1997).

1,,0 Nx

Classically, finding x0 takes O(N) queries of .

Grover’s searching algorithm* on a quantum

computer only requires O(N) queries.

)(xf

,0

,1)(xf 0xx

otherwise

0

1

20

2

0tetarget_sta x

x

xN

1ateinitial_st

N

1θsin 0

Can there be Speed-up without Entanglement?

Project onto .

Since projection cannot create entanglement,

if unentangled .

At step k

In Schmidt basis

is entangled when .

0sin1

cos

0

xxN

kxx

kk

nN 2

0)12( kk

gekegk )()( 21

kkNk N

1

1},,,{ eegeeggg

)()(1

1

21 kkNk

k k

kkk NN

N

4)4( 1

1

)4(4

Braunstein & Pati, Quant.Inf.Commun. 2, 399 (2002).

² · ²k ´1

1+Np¸1(k)¸2(k)

p(k) = hx0j½k jx0i < 1

NNMR =mink

k+1p(k)

Nclass =(N +2)(N ¡ 1)

N

n N kopt N (min)NMR Nclass

1 2 0 2 12 4 1 2 2.253 8 1 5.48 4.384 16 2 12.89 8.445 32 0 32 16.476 64 0 64 32.487 128 0 128 64.498 256 0 256 128.50

² · ²k ´1

1+Np¸1(k)¸2(k)

p(k) = hx0j½k jx0i < 1

NNMR =mink

k+1p(k)

Nclass =(N +2)(N ¡ 1)

N

n N kopt N (min)NMR Nclass

1 2 0 2 12 4 1 2 2.253 8 1 5.48 4.384 16 2 12.89 8.445 32 0 32 16.476 64 0 64 32.487 128 0 128 64.498 256 0 256 128.50

² · ²k ´1

1+Np¸1(k)¸2(k)

p(k) = hx0j½k jx0i < 1

NNMR =mink

k+1p(k)

Nclass =(N +2)(N ¡ 1)

N

n N kopt N (min)NMR Nclass

1 2 0 2 12 4 1 2 2.253 8 1 5.48 4.384 16 2 12.89 8.445 32 0 32 16.476 64 0 64 32.487 128 0 128 64.498 256 0 256 128.50

² · ²k ´1

1+Np¸1(k)¸2(k)

p(k) = hx0j½k jx0i < 1

NNMR =mink

k+1p(k)

Nclass =(N +2)(N ¡ 1)

N

n N kopt N (min)NMR Nclass

1 2 0 2 12 4 1 2 2.253 8 1 5.48 4.384 16 2 12.89 8.445 32 0 32 16.476 64 0 64 32.487 128 0 128 64.498 256 0 256 128.50

We find that entanglement is necessary for obtaining speed-up for Grover’s algorithm in liquid-state NMR.

At step k, the probability of success

must be amplified through repetition or parallelism (many molecules).

Each repetition involves k+1 function evaluations.

`Unentangled’ query complexity (using ))(

1minN(min)

NMR kp

kk

N

NN

2

)1)(2(

k

1)( 00 xxkp k

Naively, to get an exponential speed-up, the entanglement must grow with the size of the input.

Caveats:

• Converse isn’t true, e.g., Gottesman-Knill theorem*

• Doesn’t apply to mixed-state computation, e.g., NMR

• Doesn’t apply to query complexity, e.g., Grover

• Not meaningful for communication, e.g., teleportation

Entanglement as a Prerequisite for Speed-up

Entanglement in Communication: TeleportationAlice Bob

Entanglement

outin

In the absence of entanglement, the fidelity of the output stateF = is bounded.

e.g., for teleporting qubits, F 2/3 whereas for the teleportation of coherent states in an infinite-dimensional Hilbert space F 1/2.*

Fidelities above these bounds were achieved in teleportationexperiments (DiMartini et al, 1998 for qubits; Kimble et al 1998 forcoherent states). Entanglement matters!

Absence of entanglement precludes better-than-classical fidelity (NMR).

NB Teleportation only uses operations covered by G-K (or generalizationto infinite-dimensional Hilbert space†). Simulation is not everything ...

out

*Braunstein et al, J.Mod.Opt. 47, 267 (2000)†Braunstein et al, Phys.Rev.Lett. 88, 097904 (2002)

Summary

The role of entanglement in quantum information processing is not yet well understood.

For pure states unbounded amounts of entanglement are a rough measure of the complexity of the underlying quantum state. However, there are exceptions …

For mixed states, even the unentangled state description is already complex. Nonetheless, entanglement seems to play the same role (for speed-up) in all examples examined to-date, an intuition which extends to few-qubit systems.

In communication entanglement is much better understood,but there are still important open questions.

Entanglement in communication

The role of entanglement is much better understood, but there are still important open questions …

Theorem:*

additivity of the Holevo capacity of a quantum channel.

additivity of the entanglement of formation.

strong super-additivity of the entanglement of formation.

If true, then we would say that

wholesale is unnecessary!

We can buy entanglement or Holevo capacity retail.

*Shor, quant-ph/0305035 some key steps by: Hayden, Horodecki & Terhal, J. Phys. A 34, 6891 (2001). Matsumoto, Shimono & Winter, quant-ph/0206148. Audenaert & Braunstein, quant-ph/030345