Transcript Document

Characterization of
Two-qubit Operators
R. Sankaranarayanan
Department of Physics
National Institute of Technology
Tiruchirappalli – 620015
(www.nitt.edu)
[email protected]
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OUTLINE
•
Introduction
Model for computation
Some perspectives
Quantum States
Quantum Algorithms
Single qubit and Two-qubit operators
•
Characterization Tools
Geometry
Local Invariants
Entangling Power
Operator Entanglement
•
Conclusion
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INTRODUCTION
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Model for Computation
Fundamental unit: 0 and 1 – bit (classical)
Single Input gate: NOT (reversible)
Two Input gates:
OR, AND, XOR, NOR, NAND (irreversible)
General Logic Gate f :0, 1 0, 1
- sequence of n input bits converted to another
sequence of m output bits
n
NAND
m
Universal gate
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Some Perspectives
Logical gates are intrinsically irreversible (given the output of
the gate, input is not uniquely determined)
eg. Output of NAND gate is 1 for the inputs 00,01,10.
- information input to the gate is lost irretrievably when the gate
operates (information is erased)
Moore’s law:
computer power will double for constant cost roughly
once very two years.
(true for last 4 decades!)
- quantum effects will begin to interfere in the functioning of
electronic devices as they are made smaller and smaller!
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Quantum States
P1. Associated to any ‘isolated’ physical system is a complex vector
space with inner product (Hilbert Space) known as state space.
The system is completely described by a state vector in the
state space.
Simplest quantum mechanical system has two-dimensional
state space.   a 0  b 1 ; a 2  b 2  1 - Two-level system
P2. Evolution of ‘closed’ quantum system is described by a unitary
transformation.
  U  ; U U   I ; U 1    
(reversible)
If the state space is n-dimensional, U is n  n unitary matrix.
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P2’. Time evolution of the state of a closed quantum system is
described by the Schrodinger equation
d
i   Hˆ 
dt
; Hˆ – Hamiltonian operator
P3. Quantum measurements are described by a collection
of measurement operators, M i  i i ; i  1,2,n
- n dimensional Hilbert space
For two-level system,
M 0  0 0 ; M1  1 1
 M 0 M 0   a
such that
2

;  M1 M1   b
2
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State space of two-qubit (composite) system
H  H1  H 2
Four dimensional Hilbert space spanned by the linearly
independent vectors: 0  0 , 0  1 , 1  0 , 1  1
General state   a 00  b 01  c 10  d 11
with the normalization condition  
2
2
2
2
 1 or a  b  c  d  1 .
State space of n-qubit (composite) system
H  H1  H 2  H 3    H n
- 2 n Hilbert space
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State of a two-qubit (composite) system
  a 00  b 01  c 10  d 11  H  H1  H 2
Let  1  0   1  H1 and  2   0   1  H 2 .
If the state    1   2
- product (or) untangled state
condition: ad  bc  0
If the state    1   2
- entangled state
condition: ad  bc  0
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Product (unentangled) states
(i) 
1
 00  10   1  0  1   0
2
2

(ii)  
1
 00  i 01  10  i 11   1  0  1   1  0  i 1
2
2
2

Entangled states
(i)
1
   00  i 01  10  i 11
2
(ii)
 
1
 00  11
2


; ad  bc 
i
; ad  bc 
2
1
2
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For state of a two-qubit system:
  a 00  b 01  c 10  d 11
Concurrence, C( )  2 ad  bc
Range:
0  C ( )  1
For product state:
C ( )  0
For maximally entangled state: C ( )  1
eg. Bell states
Superposition and Entanglement
- New resource for computation
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Superdense coding
2 bits
Problem:
A
B
00,01,10,11
 
1
 00  11 
2
1. each qubit of a two-qubit state
shared by A & B
2. A applies I , X , Z , Y gates for 00,01,10,11 respectively
3. A sends her qubit to B
4. B applies CNOT followed by H gate to the 1st qubit
- two-qubit state 00 , 01 , 10 , 11
5. B makes measurement to retrieve 00,01,10,11
Transfer of one qubit = transferring two bits
C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992)
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Quantum search
Problem: Search an item in a random collection of N items.
eg. Search a name corresponds to a phone number
in telephone directory
Best case – 1 search, Worst case – N search
On average:
1
1  2    N  ~ N
N
2
search
Grover’s quantum logic – average no. of search ~
N
- quadratic improvement in random search
L.K. Grover, Phys. Rev. Lett. 79, 325 (1997)
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Prime factorization
Problem: factoring n-bit integer into prime factors
N  p q ~ 2n ; where p,q – primes
eg. 137703491 7919  17389
best classical algorithm – number field sieve
no. of operations – Oexpn1/ 3 log2 / 3 n
eg. 130 digit number – super computer (1012 flops)
- 42 days to factorize
400 digit number – 1010 years to factorize !
Quantum logic – no. of operations - On2 log n loglog n
- factorization is exponentially faster
400 digit number – 3 years to factorize !
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P.W. Shor, SIAM J. Comp. 26 (5), 1484 (1997)
Single qubit gates
Acts on single qubit system – SU(2) Group
Pauli-X
Pauli-Y
Pauli-Z
  X 
 Y
 Z
b  0 1 a
   
  
 a  1 0 b 
  b 0  i a
  
i    
 a   i 0 b 
 a  1 0  a
   
  

b
0

1
  
b 
0  1
0  i1
0 
1  0
1  i 0
1 1
0
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Hadamard
  H
  S 
1  a  b  1 1 1   a 

 

  
2 a  b
2  1  1  b 
0 
1 
1
2
1
2
0
0
Phase
1
1


 a  1 0 a 
   
  
 ib   0 i   b 
0  0
1 i1
Pi/8
 T 
 a  1 0
 i   
i
e 4 b 0 e 4

 
0 
1 
 a
 
  b 

0
i
e4
1
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Two-qubit gates
Acts on two-qubit system – SU(4) Group
SU (4)  SU (2)  SU (2)
- Local gates – do not produce entanglement
If
( A  B)  1   2  A  1  B  2   1'   2'
SU (4)  SU (2)  SU (2)
- Nonlocal gates – produce entanglement
U  1   2   1'   2'
Local gates
Nonlocal gates
Perfect Entanglers
(Bell state for some input product state)
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CNOT
SWAP
00  00
; 01  01
00  00
; 01  10
10  11
; 11  10
10  01
; 11  11
U CN
1

0

0

0

0
1
0
0
0
0
0
1
0

0
1

0 
U SWAP
1

0

0

0

0
0
1
0
0
1
0
0
0

0
0

1 
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CHARACTERIZING TOOLS
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Geometry
An arbitrary two-qubit gate
can be written as
- Pauli matrices,
Geometrical structure of
3-Torus with period
- real numbers
:
.
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Local Invariants
Two operators
are called locally equivalent if
they differ only by local operations:
Bell basis:
Transformation of from
standard basis to the basis:
Local gates are
real orthogonal matrices
in the Bell basis
where
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Theorem:
The complete set of local invariants of a two-qubit gate
with
is given by the set of eigenvalues
of the matrix
. The spectrum of the matrix
is completely described by a complex number
and a real number
.
Local invariant of two-qubits are defined as:
Y. Makhlin, Quantum Info. Process 1, 243 (2003)
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Eigenvalues of
:
Local invariants of a two-qubit gate:
Properties:
1. For every
2. For every
3. For every
,
and
,
and
,
,
Local invariants are unaffected by local operations.
Zhang J. et. al. Phys. Rev. A 67, 042313 (2003)
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Symmetry in
the base
L = CNOT
A3 = SWAP
Tetrahedron (Weyl Chamber):
Perfect entanglers in the Polyhedron:
(a)
and (b)
Zhang J. et. al., Phys. Rev. A 67, 042313 (2003)
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Local invariants of Weyl Chamber edges
O = Local gate
L = CNOT
A3 = SWAP
α
OA3 = SWAP
A1 A3= SWAP -α
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Local invariants of Polyhedron edges
Entangling Power
Average entanglement generated by the action of a two-qubit gate
on all product states
:
where
is the linear entropy of the state.
Properties:
1. For every
,
2. For every
,
3. For every
,
Entangling power is unaffected by local operations.
Zanardi P. et. al, Phys. Rev. A 62, 030301 (2000)
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Entangling power of Weyl Chamber edges
Entangling power of Polyhedron edges
Entangling power and local invariants
where
.
Since
If
, we have
,
.
. This is the case for the edges
QP, MN and PN.
Perfect entanglers are such that,
(c)
and (d)
Local invariants classify two-qubit gates as Perfect entanglers
and Non-perfect entanglers.
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Operator Entanglement
Nonlocal part of U can also be written as
where
Defining
are complex functions.
as Schmidt coefficients, we have
.
Measures of operator entanglement:
(i) Schmidt strength,
(ii) Linear Entropy,
Nielson M.A. et. al., Phys. Rev. A 67, 052301 (2003)
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Properties:
1. For every
,
,
2. For every
,
,
3. For every
,
,
Schmidt strength and Linear entropy are unaffected by local
operations.
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Schmidt coefficients of Weyl Chamber edges
Schmidt coefficients of Polyhedron edges
Schmidt coefficients of Polyhedron edges (continued)
Linear Entropy and Schmidt Strength
are not equivalent measures
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Linear Entropy and Local Invariants
Linear entropy of an operator U can also be written as
We can show that
, the maximum value
only for the gates on the edge A2A3 .
Further, we have
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O = Local gate
L = CNOT
A3 = SWAP
α
OA3 = SWAP
A1 A3= SWAP-α
A2 A3 – maximally entangled gates
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Linear entropy of Weyl Chamber edges
Linear entropy of Polyhedron edges
Local invariants of Polyhedron edges
Conclusion
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•
•
•
All the geometrical edges of two-qubit gates are one-parametric.
SWAP α and its inverse form two edges of the Weyl chamber.
Relation between entangling power and local invariant.
Local invariants classify gates as perfect and nonperfect
entanglers.
• Two measures of operator entanglement, (i) linear entropy and (ii)
Schmidt strength are not equivalent.
• Linear entropy of an operator in terms of geometrical points
and local invariants.
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References
• Balakrishnan, S. and R. Sankaranarayanan, Entangling characterization of
(SWAP)1/m and controlled unitary gates, Phys. Rev. A 78, 052305 (2008).
• Balakrishnan, S. and R. Sankaranarayanan, Characterizing the geometrical
edges of nonlocal two-qubit gates, Phys. Rev. A 79, 052339 (2009).
• Balakrishnan, S. and R. Sankaranarayanan, Entangling power and local
invariants of two-qubit gates, Phys. Rev. A 82, 034301 (2010).
• Balakrishnan, S. and R. Sankaranarayanan, Operator-Schmidt decomposition
and the geometrical edges of two-qubit gates, Quantum Inf. Process. 10(4), 449
– 461 (2011).
• Balakrishnan, S. and R. Sankaranarayanan, Measures of operator entanglement
of two-qubit gates, Phys. Rev. A 83, 062320 (2011).
Thank you for your attention!
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