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Entanglement in Quantum Information Processing
25 April, 2004
Les Houches
Samuel L. Braunstein
University of York
Classical/Quantum State Representation
Bit has two values only: 0, 1
Information is physical
BITS  QUBITS
Superposition
between two rays
in Hilbert space
 0  1
Entanglement
between (distant)
objects
 00   11
Many qubits leads to ...
Fast Quantum Computation


(Shor)
(Grover)
(slide with permission D.DiVincenzo)
Computational Complexity
Computational complexity: how the `time’ to complete
an algorithm scales with the size of the input.
For machines that can
simulate each other in
polynomial time.
PSPACE
NP
BQP
BPP
factoring*
P
primality
testing
Quantum computers add a new complexity class: BQP†
*Shor, 35th Proc. FOCS, ed. Goldwasser (1994) p.124
†Bernstein
& Vazirani, SIAM J.Comput. 25, 1411 (1997).
Picturing Entanglement
Pure states are entangled if
  A

B
Alice

AB
Bob
e.g., Bell state
00  11
(picture from Physics World cover)
Computation as Unitary Evolution
 

j1 jn 0,1
Evolves via
j1    jn
j1 jn
U :  U 
Any unitary operator U may be simulated
by a set of 1-qubit and 2-qubit gates.*
e.g., for a 1-qubit gate:
U
(i )
1
:  j1 ji  jn 

M
ki 0,1
ji ki
 j k  j
1
i
n
*Barenco, P. Roy. Soc. Lond. A 449, 679 (1995).
Entanglement as a Resource
 

j1 jn 0,1
j1 jn
j1    jn
State unentangled if
j
1 j2  jn
 a j1 b j2  d jn
“Can a quantum system be probabilistically
simulated by a classical universal computer?
… the answer is certainly, No!”
“Hilbert space is a big place.”
“Size matters.”
Richard Feynman (1982)
Carlton Caves 1990s
Anonymous
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).
Entanglement as a Prerequisite for Speed-up
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
Gottesman-Knill theorem*
• The Pauli group Pn is generated by the n-fold tensor product
of
 x ,  y,  z , 12
and factors ±1 and ±i.
• Subgroups of Pn have compact descriptions.
•
[ ,, 
(1)
z
( n)
z
]  Pn
[ x   x ,  z   z ]
• Gates:
stabilizes
stabilizes
1 0
1 1 1 

 , H 

 ,
2 1  1
0 i 
0   0
0  0 1 1
 x ,  y ,
map subgroups of Pn to subgroups of Pn.
1

0
,
z 0

0

.
0 0 0

1 0 0
0 0 1

0 1 0 
 any computation restricted to these gates may be
simulated efficiently within the stabilizer formalism.
*Gottesman, PhD thesis, Caltech (1997).
Entanglement as a Prerequisite for Speed-up
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
Mixed-State Entanglement
Since
A j   j A j
mixture
   pj j
write
j   j  j
A  trA 
so
j
For
 AB
on
unentangled if:
p
j
A
j
Η A  HB
 AB   p j  Aj   Bj
j
otherwise entangled.
, pj  0
j
Test for Mixed-State Entanglement
Consider a positive map
 AB
that is not a CPM

 AB
( A )  0
s.t.
 AB
( 1)(  AB )  0
entangled
 negative eigenvalues in

 A
( 1)(  AB )
entangled.
For  1 = partial transpose, this is necessary & sufficient
on 2x2 and 2x3 dimensional Hilbert spaces.
But positive maps do not fully classify entanglement ...
Peres, Phys.Rev.Lett. 77, 1413 (1996).
Horodecki3, Phys.Lett.A 223, 1 (1996).
Liquid-State NMR Quantum Computation
Utilizes so-called pseudo-pure states
1 
  n 1    
2
which occur in NMR experiments with small

For any unitary transformation U
U U
†
is pseudo-pure with

replaced by
U
The algorithm unfolds as usual on pure state perturbation
for traceless observables
A, A    A 
Each molecule is a little quantum computer.
(figure from Nature 2002)
NMR Quantum Computation (1997 - )
Selected publications:
Nature (1997), Gershenfeld et al.,
Nature (1998), Jones et al.,
Nature (1998), Chuang et al.,
Science (1998), Knill et al.,
Nature (1998), Nielsen et al.,
Nature (2000), Knill et al.,
Nature (2001), Lieven et al.,
NMR scheme
Grover’s algorithm
Deutsch-Jozsa alg.
Decoherence
Teleportation
Algorithm benchmarking
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?


   d  w(n1 ,, nn ) Pn1    Pnn
 
 
1
1
Pn  (12  n   ) 
d j (12  3n  n j ) Pn j

2
4
n


w(n1 ,, nn ) 
1
(4 ) n
 
 
tr[  (12  3n   )    (12  3n   )]
most negative eigenvalue 4n-1(-2) = -22n-1


 22 n1
 w(n1 , , nn ) 
(4 ) n


whereas for w(n , , n )  0,  is unentangled
1
n
1 
1   
For NMR states  
n
2




1 
so w (n1 , , nn ) 
  w1 (n1 ,, nn )
n
(4 )
1   (1  22 n 1 )

n
(4 )


if w (n1 , , nn )  0
   unentangled
1
  
2 n 1
1 2


unentangled
In current liquid-state NMR experiments  ~ 10-5, n < 10 qubits
 no entangled states accessed to-date …or is there?
Braunstein et al, Phys.Rev.Lett. 83, 1054 (1999).
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.
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.
*Linden & Popescu, Phys.Rev.Lett. 87, 047901 (2001).
Entanglement as a Prerequisite for Speed-up
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
Grover’s Search Algorithm*
Suppose we seek a marked number from
x  0, , N  1 satisfying:
1,
f (x)  
0,
x  x0
otherwise
Classically, finding x0 takes O(N) queries of
Grover’s searching algorithm* on a quantum
f (x.)
computer only requires O(N) queries.
1
sin θ 0 
N
0
target_sta te  x0
1
2
20
initial_st ate 
1
N
x
x
*Grover, Phys.Rev.Lett. 79, 4709 (1997).
Can there be Speed-up without Entanglement?
At step k
k
In Schmidt basis
Project
cos  k

x  sin  k x0

N  1 x  x0
N  2n
 k  (2k  1)0
 1 (k ) g e  2 (k ) e g 
1 
k 
1N    k  k onto { g g  , g e , e g  , e e. }
N
N
1 

( 4)
k 
14    k  k 

4   ( N  4)  N

1
  k 
is entangled when
.
1  N 1 (k )2 (k )
Since projection cannot create entanglement,
if  k unentangled     k .
At step k, the probability
1
p
²of· success
² kk ´
 x220(k)
k
1 + Np(k¸)11 (k)¸
x0  1
p(k) = hxor
must be amplified through repetition
parallelism
(many molecules).
0 j½
k jx 0
0i < 1
k+ 1
Each repetition involves k+1 function evaluations.
N N M R = min
k
p(k)
(min)
`Unentangled’ query complexity(N N
2)(N¡ min
1)
+ NMR
N cl ass =
n
1
2
3
4
5
6
7
8
N
2
4
8
16
32
64
128
256
kopt
0
1
1
2
0
0
0
0
( m i n)
( m i n)
NNM R
2
2
5.48
12.89
32
64
128
256
N
N cl ass 
1
2.25
4.38
8.44
16.47
32.48
64.49
128.50
k
k 1
p(k )
(using
  k )
( N  2)( N  1)
2N
We find that entanglement is necessary for obtaining
speed-up for Grover’s algorithm in liquid-state NMR.
Braunstein & Pati, Quant.Inf.Commun. 2, 399 (2002).
Entanglement as a Prerequisite for Speed-up
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 in Communication: Teleportation
in
Alice
  
In the absence of entanglement,
the fidelity of the output state
F =  out  is bounded.
Bob
out
Entanglement
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 teleportation
experiments (DiMartini et al, 1998 for qubits; Kimble et al 1998 for
coherent states). Entanglement matters!
Absence of entanglement precludes better-than-classical fidelity (NMR).
NB Teleportation only uses operations covered by G-K (or generalization
to infinite-dimensional Hilbert space†). Simulation is not everything ...
*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 todate, 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