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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