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Entanglement Measures in Quantum Computing
About distinguishable and indistinguishable particles, entanglement, exchange and
correlation
Szilvia NagyDepartment of Telecommunications,Széchenyi István University, Győr
Péter Lévay, János Pipek, Péter Vrana, Szilárd Szalay,
Department of Theoretical Physics, Budapest University of Technology and Economics, Budapest
Contents
• Motivation• Realization of entangled states• Distinguishable and indistinguishable particles
propertiesentanglement’s two facemeasures for entanglement
Schmidt and Slater ranksConcurrence and Slater correlation
measureentropies
• Generalization, entanglement types in three or more particle systems
Motivation
• Entanglement plays an essential role in paradoxes and counter-intuitive consequences of quantum mechanics.
• Characterization of entanglement is one of the fundamental open problems of quantum mechanics.
• Related to characterization and classification of positive maps on C* algebras.
• Applications of quantum mechanics, like quantum computingquantum cryptographyquantum teleportation
is based on entanglement. “Entanglement lies in the heart of quantum computing.”
Physical systems
• Quantum dots: the charge carriers are confined /restricted/ in all three
dimensionsit is possible to control the
number of electrons in the dotsthe qubits can be of orbital or
spin degrees of freedomtwo qubit gates can be e.g. magnetic field
• Neutral atoms in magnetic or optical microtraps
Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)
Distinguishable and indistinguishable particles
• Not identical particles• Large distance or
energy barrier• No exchange effects
arise
• Identical particles• Small distance and
barrier• Exchange properties
are essential
Distinguishable particles
Small overlap between and The exchange contributions are small in the
Slater determinants
BAABi
For two particles and two states A B
Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)
Indistinguishable particles
Large overlap between and The exchange contributions are significant in the
Slater determinants
21211
2
1 t
If the energy barrier is lowered
Indistinguishable particles
A mixed state of two Slater determinants arises
Suppose, that after time evaluation
2121
21212
2
1
t
Distinguishable particles
We get one of the Bell states
Rising the barrier again – increasing the distance
BABAABi
2
1
What is entanglement?
Basic concept: two subsystems are not entangled if and only if both constituents possess a complete set of properties.→separability of wave functions in Hilbert space
Distinguishable particlesthe two subsets are not entangled, iff the system’s Schmidt rank r is 1, i.e. only one non-zero coefficient is in the Schmidt decomposition.
Indistinguishable particles the two subsets are not entangled, iff the system’s Slater rank is 1, i.e. only one non-zero coefficient is in the Slater decomposition.
1 ,0 2
1
i
ii
r
iiii zzbaz
1Tr2 , 0 T
PPPPccP
Distinguishable and indistinguishable particles
• Not identical particles• Large distance or
energy barrier• No exchange effects
arise• Schmidt
decomposition
→Schmidt rank
• Identical particles• Small distance and
barrier• Exchange properties
are essential• Slater decomposition
→Slater rank
i
iii baz
i
ccP 0
Distinguishable particles - concurrence
The state can be written as
The concurrence is
Concurrence can also be introduced for indistinguishable particles.
Magic basis for two particles
2322
21
121
0
imim
mm
mj
jj
2j
jC
Indistinguishable particles – η measure
Both C and are 0 if the states are not entangled and 1 if maximally entangled.
The definition of the Slater correlation measure
0 ccP
PP
if
Schliemann & al. Phys. Rev. A 64 022303 (2001)
Distinguishable and indistinguishable particles
• Not identical particles• Large distance or
energy barrier• No exchange effects
arise• Schmidt
decomposition
→Schmidt rank• concurrence
• Identical particles• Small distance and
barrier• Exchange properties
are essential• Slater decomposition
→Slater rank measure
i
iii baz
i
ccP 0
Von Neumann and Rényi entropies
In our case
Good correlation measures for fermions. The von Neumann entropy is
And the th Rényi entropies are
0 Trlog1
1
logTr
2
21
ρS
ρρS
2
21
2
221
11 with
0 1log1
11
1log1log1
x
xxS
xxxxS
The minimum of the entropy
According to Jensen’s inequality
thus the von Neumann entropy is
and S=1 iff =0, i.e., if the Slater rank is 1.
MSN
MSN
212
222
loglog
loglog
NS
SS
MS
N 21
22
22
12
2
logloglog
log0
It can be shown that
Distinguishable and indistinguishable particles -
summary• Not identical particles• Large distance or
energy barrier• No exchange effects
arise• Schmidt decomposition
→Schmidt rank• Concurrence
• Smin=0
• Identical particles• Small distance and
barrier• Exchange properties
are essential• Slater decomposition
→Slater rank• η measure
• Smin=1
i
iii baz
i
ccP 0
The measures of entanglementThe connection between the entropy and the concurrence for specially
parameterized two-electron states:
Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)
The measures of entanglementThe connection between the concurrence and for
specially parameterized two-electron states:
221 12C
Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)
The measures of entanglement
The connection between the entropy and for specially parameterized two-electron states:
2
21
221
11 with
1log1log1
x
xxxxS
Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)
Three fermions
With
There are at least two essentially different types of entanglement if three or more particles are present.
3 particles, 6 one-electron states
ijk
kliijlkiij
P
jlkPBPA
PP
DetBDetABTrATrABT
~
,),(),(
,
444
321123
##2123
And the “dual state”
Lévay& al. Phys. Rev. A 78, 022329 (2008)
3 particles, 6 one-electron states:
Non-entangled states (separable or biseparable):
Entangled state type 1
Entangle state type 2
Three fermions
321
123 .0~,0
eeeP
PT ijk
213132321321
123
21
.0
eeeeeeeeeeeeP
T
Lévay& al. Phys. Rev. A 78, 022329 (2008)
• Developing a series of measures useable for any particles with any (finite) one-fermion states
• Basis: Corr by Gottlieb&Mauser
• Generalization: the distance not only from the uncorrelated statistical density matrix, but from characteristic correlated ones.
Future plans
A.D. Gottlieb& al. Phys. Rev. Lett 95, 123003 (2005)
Recent publications by the group
•Lévay, P., Nagy, Sz. and Pipek, J.,Elementary Formula for Entanglement Entropies of Fermionic Systems,Phys. Rev. A, 72, 022302 (2005).
•Szalay, Sz., Lévay, P., Nagy, Sz., Pipek, J.,A study of two-qubit density matrices with fermionic purifications,J. Phys. A - Math. Theo., 41, 505304, (2008).
•Lévay, P., Vrana, P.,Three fermions with six single particle states can be entangled in two inequivalent ways,Phys.Rev. A, 78 022329, (2008).