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Transcript of 1 Combinatorics and Quantum Information Simone Severini Department of Combinatorics & Optimization,...
1
Combinatorics and
Quantum Information Simone Severini
Department of Combinatorics & Optimization, and
Institute for Quantum Computing,
University of Waterloo, Canada
www.iqc.ca/~sseverin
Isaac Newton Institute, Cambridge, March 2008
2
Quantum Information
Quantum information may be defined as the study of the achievable limits to information processing possible within quantum mechanics.
Goals of quantum information:
• Determine limits on the class of information processing tasks which are possible in quantum mechanics (e.g., limitations on the class of measurements that may be performed on a quantum system).
• Provide constructive means for achieving information processing tasks (e.g., development of unbreakable schemes for doing cryptography based upon the principles of quantum mechanics, study of the power of quantum computational devices).
(M. Nielsen)
3
• Quantum teleportation (applications: quantum memories, quantum repeaters). • Superdense coding (primitive in quantum protocols).• Secure key distribution in cryptography.• Quantum computing (e.g., discrete Log, search in databases).• Simulation of quantum mechanical evolutions (e.g., many-body systems, applications in condensed matter physics).• Secret sharing (e.g., secure distributed computation).• Quantum Games (e.g., lower bounds in algorithms, study of phase transitions).• Quantum Lithography and Imaging (e.g., increased signal intensities, extensions of optical microscopy).• Quantum Sensors (improve the sensitivity of measurements across the electromagnetic spectrum, measuring microwaves).
Phenomena and Applications
4
Entanglement Theory
What is entanglement?
• Entanglement is a property of certain states of quantum systems.• Entanglement is a physical resource (like energy or time).• It is one of the key ingredients for a quantum computer, allowing to solve certain tasks more quickly. • It is an NP hard problem to decide whether a given state is entangled or not.
Goals of entanglement theory:
• Find easily implementable criteria for detecting and classifying entanglement.
• Isolate the properties that characterize entanglement.
• Determine how useful entanglement is.
I would not call entanglement one but rather the characteristic trait of quantum mechanics
(E. Schrödinger)
5
• Reconstruction of quantum states: Young diagrams, Marginal inequalities (Christandl et al. 2005, Klyachko, 2004)• Quantum state transformations: Schubert calculus and varieties (Hayden et al. 2004, Knutson 2004)• Classification of entangled states: Hyperdeterminants (Miyake et al., 2003), Combinatorics of permutations (Clarisse et al., 2005), Knots invariants (Kauffman et al., 2004)• Toy states and entanglement detection: graph laplacians (Braunstein et al., 2005)• One-way quantum computation: local operations on graphs (Briegel et al., 2002)• Quantum-error correction: codes over Z₄ (Calderbank et al., 2003)• Bell's inequalities: cut polytopes (Avis et al., 2005)• Non-locality, CHSH games: extremal set theory (Cleve et al., 2005)• Optimal measurements: finite geometry, projective designs (Wootters et al., 2005)• Randomization of quantum states: quantum expanders (Ta-Shma et al., 2006)• Information transfer and spin network control: spectral graphtheory (Cambridge people, Bose >2000), power domination in graphs (Aazam et al., 2008)• Topological quantum computation: Hopf algebras (Friedman, Kitaev, 1998)• Complexity parameters and simulation: (Valiant 2000, Shi, Josza 2006)
Combinatorics
6
• Classification of entangled states: combinatorics of permutations
• Toy states and entanglement detection: graph laplacians
• Quantum-inspired algorithm for graph matching
• Information transfer and spin network control: spectral graphtheory, power domination in graphs
• Non-locality, CHSH games: extremal combinatorics
Plan of this talks
7
Clarisse, Ghosh, Sudbery, S, Phys. Rev. A 72, 012314 (2005)
quant-ph/0502040
Clarisse, Ghosh, Sudbery, S, Phys. Lett. A, 365 (2007)
quant-ph/0611075
Classification of entangled states and combinatorics of
permutations
8
What is a quantum state?
†
:
Def. (
0
T r
S
(
ta t
1
e)
)
d
2 †
:
o r s.t
Def.
. 1 and
is ide
(Pure st
n tifi
a
ed w ith
te) d
d
9
What is entangled?
0
:
there are no : ,
Def. (Entang led sta te)
: and
s.t.
p qA B
A p B qi A i B i
A Bi i ii
r
r
there are no
Def. (Ent
a
an
nd
gled pure s
s.t.
ta te) p qA B
p qA B
10
How to measure entanglement?
2
:
S : (1 T r( ))1
is the reduced density m atrix w .r
Def. (L
.
inear entrop
t
)
y
.
d dA B
A
dA B
d
d
12
What is required to be a good entangler ?
2
, 1 , 1
(
2
)
,
( ) :
, norm alize
Def
d p
. (Entang ling po
rob. m easures on un it spheres
w er o f ( )
m x ( )1
)
a
d dA B
U d
e U S U d d
d d
de U
U U d
d
15
Mutually Orthogonal Latin Squares (MOLS)
1 2 3
2 3 13 1 2
1 3 22 1 33 2 1
1,1 2,3 3,22,2 3,1 1,33,3 1,2 2,1
16
MOLS permutation
1,1 2,3 3,22,2 3,1 1,33,3 1,2 2,1
1 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 0
17
What are the best entanglers ?
2
2
Let 6.
Then ( )1
if is a MOLS permutation of size .
This is max. in (
Theorem.
).
d
de U
d
U d
U d
18
Open problemsCombinatorial problems: • What about dimension 6?• Classify permutations according to the their entangling power.• What about multipartite systems (i.e., many subsystems)?
Notice: The max. # of MOLS of side l is • For a larger # of subsystems can we do it with other unitaries? • If we can not? 1. e(U) is not plausible; 2. entanglement for a few subsystems is special(or MOLS are objects from a larger class?).
1l
20
Braunstein, Ghosh, S, Ann. Comb. 10 (2006), 291-327 quant-ph/0406165
Braunstein, Ghosh, Mansour, S, Wilson, Phys. Rev. A 73, 012320 (2006) quant-ph/0508020
Hildebrand, Mancini, S, Math. Structures Comput. Sci. 18:1 (2008), 205-219cs.CC/0607036
Toy states and entanglement detection
21
Associating graphs to states
( , ), undirected graph, with no self-loops
, adjacency matrix
, degree matrix
, combinatorial la
Def. (Laplacian
placian
0,
1 ( ) is a quantum state
Tr( )
state)
T
G V E
M
L M
L L L
G L
(Spin networks, bosonic graphs, graph states, etc.)
22
3/
How hard is to detect entanglement?
Give
Decision p
n: :
Question: is entangled?
This is NP-comple
ro .
te
blemp qA B
uuuuuuuuuuuuuuuuuuux
£ £
23
11 1
1
11 1
1
:
is
Def. (Partia l transpose)
B
p qA B
p
p pp
ii
T Tp
T Tp pp
M M
M M
M q q
M M
M M
Partial transpose
24
Positivity of Partial Transpose (PPT) Criterion
2 2 2 3
:
If is separab le then 0
The converse is not necessarily true
: or ,
Theore
i
m .
Theor
s separab le iff 0
em .
B
B
p qA B
A B A B
25
Associating a lattice to the graph
( ) :
| ( ) | .
Le t { } and { } standard bases for and .
Labe l each vertex by a ordered pa ir ( , ).
D raw on the { } X { } la ttice ,
P r
accord ing to the chosen la
oce
be l
dure .
ing .
p qA B
p qA B
G
V G pq
i j
i j
G i j
29
Degree Criterion and entanglement
2 Let ( ) :
Then ( ) ( ) iff ( ) is separable.
Then ( ) 0 iff
Theore
( ) is separable
m.
.
B
B
qA BG
G G G
G G
£ £
30
• The concurrence of entangled laplacian states in dimension four is exactly fractional: 1/(# edges).• Characterize graphs whose laplacian state is entangled independently of the labelling.• Consider multipartite systems: Wang, Wang, Elec. J. Comb. 14 (2007) Hassan, Joag, J. Phys. A: Math. Theor. 40 (2007); J. Math. Phys. 49 (2008)
Open problems
Let be a connected graph.
Then ( ) is entangl
Conjec
ed
ture.
iff 0.B
G
G
32
Emms, Hancock, S, Wilson, Elec. J. Comb. 13 (2006), #R34. quant-ph/0505026See this morning talk by E. Hancock, for applications in pattern recognition: Emms, Hancock, Wilson, ICPR, IEEE Computer Society (2006).
For a different approach seeAudenaert, Godsil, Royle, Rudolph, J. Comb. Theory, Ser. B 97(1): 74-90 (2007).math.CO/0507251
A quantum-inspired graph matching algorithm
34
Graph isomorphism
Problem.
• Are graphs G and H actually the “same graph”?
(Pictures - Courtesy of D. Bacon)
35
Let ( , )
Then D ( ( ), ), where ( , )
Def. (Bi-or
,( , ) ( ) iff
ientation)
{ , } ( )G G
G V E
V G A i j j i A D i j E G
( ) #{
Def. (D
:
egree)
( , ) ( )}Gd i j i j A D
,
( , ),( , )
For all ( , ),( ,
Def. (Orthogonal matrix represe
) ( )
2 / ( ) , if ; ( )
0, ot
ntation
herwise.
)
G
i li j k l
i j k l A D
d j j kU G
( ) (
Lemm
)
a.TU G U G I
Representing a graph with an orthogonal matrix
36
,
Let ( , )
1, if { , } ( );
Def. (Adjacency matr
( )0, otherwis
)
.
ix
ei j
G V E
i j E GM G
GraphsDef. (Isomor and
iff : ( ) ( )
phic graphs)T
G H
G H P PM G P M H
Graphs and
If then Sp( ( )) Sp( ( )).
The converse is not nec
Lemma.
essarily true.
G H
G H M G M H
Graphs and
Prob
?
lem. (Graph Isomo
So far the prob
rp
l
hism Pro
em is in
b
lem
NP A
)
co M
G H
G H
Graph Isomorphism Problem
37
is a srg( , , , )
| ( ) | ;
the degree of each vertex is ;
, , if { , } ( ) then , have exactly common neigh;
Def. (Strongly
, , if { , } ( ) then , have exactly co
regular graph) G n m a b
V G n
m
i j i j E G i j a
i j i j E G i j b mmon neigh.
Let and be srg( , , , )
Then Sp( ( )) Sp( ( )), even if
Lemma.
non-isomorphic.
G H n m a b
M G M H
1/3( log )
and are srg( , , , ).
Problem.
?
So
(SRG I
far the
somorphism
problem is .
Problem)
O n n
G H m n a b
G H
n
Strongly regular graphs
38
,
,
Let be a real matrix.
1, if 0;
Def. (Support of a matr
0, o
ix
therwise
)
.i j
i j
M
MM
3 3
Let and be srg( , , , ), where 64.
Then iff Sp( ( ) ) Sp( ( )
Fact. (Theorem proved by inspecti
)
on)
G H n m a b n
G H U G U H
3 3
Let and be srg( , , , ), where 64.
Then iff Sp
Conjec
( ( ) )
ture.
Sp( ( ) ).
G H n m a b n
G H U G U H
Result
40
Aazami, S, A covering problem with a propagation rule: formulations and algorithms, submitted to SWAT08. Important reference:Burgardt, Giovannetti, Phys. Rev. Lett. 99 (2007).
A game for controlling spin systems
Perfect state transfer Saxena, S, Shparlinski, Int. J. Quantum Inf. (2007), 417-430.quant-ph/0703236Important reference:Christandl, Datta, Ekert, Landahl, Phys. Rev. Lett. 92 (2004).
52
Nondiscriminatory propagation problem
Given a graph .
Find a minimum cardinality set of initially colored vertices
required to propagate the color in the entire .
G
G
57
Result
We can solve NONDISCRIMINATORY PROPAGATION
optimally in polynomial time on graphs of bounded tree-width.
(This is done by reformulating the problem as an orientation
one, and by making u
Theo
se
rem.
of a dynamic-programming algorithm.)
1
Determine complexity. (It is NP-hard for weigh
Open
ted graphs.)
Opt(HC ) 2 ?
Good expanders have
pr
la
oblems
rge t
.
Op .
dd
58
D
A graph with quant
ef. (Spin system on a
um particles on the ve
gra
rt
ph
.
)
icesG
r
r
¡0
( )0
Assign each ( ) to a stand
Def. (Rule of th
ard basis vector v.
v : .
e dynamics
, where .
)
iA G tt
v V G
e t
r r( )
( ,
Def. (F
,
idelity
) | | .
)T iA G t
Gf v w t w e v
r r( )
Def. (Perfect state transfer, PST
( , , ) |
)
| 1.T iA G tGf v w t w e v
For the purpose of studying state transfer (in the XY model), we can introduce the following:
Perfect state transfer
59
¤
Given :
if s.t. ( , , ) 1
then { , , , } Sp( ( ), .
The converse is not necessarily true.
However
Lemma
, s.t. , ( , ) .
.
,
1
G
i ji j k l
k l
G
G
t f v w t
M G
t v f v v t
Construct graphs:
Small degree;
Large number of vertices;
Gives PST between very distant vertices;
( . ., -cubes give PST between antipodal
Proble
vertic
m.
es).e g d
60
If a circ. gives PST then it has integral spectrum.
(We characterize those graphs; small diam
Lemma
)
.
eter.
G
Result: circulant graphs
¢ ¢
The Cayley graphs, ( , ), where .
Equivalently, the graphs with circulant adjacency matrix.
(Many uses: VLSI design
Def. (Circulan
, computer net
t graph)
works, .)
n nX S S
etc
Circ. graphs on 2 1 vertices do not
Theore
give T.
m.
PSn k
62
Parallel repetitions of Clauser-Horne-Shimoni-Holt
(CHSH) Games
R. Cleve, W. Slofstra, F. Unger, S. Upadhyayquant-ph/0608146
70
Best quantum strategy
2
Pr[Alice & Bob win*]
cos ( / 8) 0.85
-* when they share |
Optimal by the Tsirelson’s Inequality
76
A classical strategy
*Pr[Alice & Bob win ]
(3 / 4)n
* concatenating strategies
on the single bits
78
For n = 2 there is better
00 01 10 11
00 00 00 00 00
01 00 01 00 01
10 00 00 10 10
11 00 01 10 11
Input Output
79
For n = 2 there is better
00 01 10 11
00 00 00 00 00
01 00 01 00 01
10 00 00 10 10
11 00 01 10 11
00 00 00 10
00 00 00 00 00
00 00 01 00 01
00 00 00 10 10
01 00 01 10 11
Input Output
80
For n = 2 there is better
00 01 10 11
00 00 00 00 00
01 00 01 00 01
10 00 00 10 10
11 00 01 10 11
00 00 00 10
00 00 00 00 10
00 00 00 00 10
00 00 00 00 10
01 01 01 01 11
Input Output
85
Best classical strategy for playing the game for an arbitrary
number of times
Pr[Alice & Bob win] ?
86
Upper bound
• Semidefinite programming (Feige-Lovasz relaxation)
• Density of squares in (0,1)-matrices (Peleg)