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Entanglement Measures in Quantum Computing

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### Entanglement Measures in Quantum Computing

About distinguishable and indistinguishable particles, entanglement, exchange and correlation

Szilvia Nagy

Department 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
properties

entanglement’s two face

measures for entanglement

Schmidt and Slater ranks

Concurrence and Slater correlation measure

entropies

- Generalization, entanglement types in three or more particle systems

Motivation

- Entanglement plays an essential rolein 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 computing quantum cryptography quantum teleportationis based on entanglement. “Entanglement lies in the heart of quantum computing.”

Physical systems

- Quantum dots: the charge carriers are confined /restricted/ in all three dimensions it is possible to control the number of electrons in the dots the qubits can be of orbital or spin degrees of freedom two qubit gates can be e.g. magnetic field
- Neutral atoms in magnetic or optical microtraps

Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)

Large distance or energy barrier

No exchange effects arise

Identical particles

Small distance and barrier

Exchange properties are essential

Distinguishable and indistinguishable particlesSmall overlap between j and c

The exchange contributions are small in the Slater determinants

Distinguishable particlesFor two particles and two states

A B

Eckert & al. Ann. Phys. (NY) 299 p.88 (2002)

Large overlap between j and c

The exchange contributions are significant in the Slater determinants

Indistinguishable particlesIf the energy barrier is lowered

A mixed state of two Slater determinants arises

Indistinguishable particlesSuppose, that after time evaluation

We get one of theBell states

Distinguishable particlesRising the barrier again – increasing the distance

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.

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

Distinguishable and indistinguishable particlesThe concurrence is

Concurrence can also be introduced for indistinguishable particles.

Distinguishable particles - concurrenceMagic basis for two particles

Both C and h are 0 if the states are not entangled and 1 if maximally entangled.

Indistinguishable particles – η measureThe definitionof the Slater correlation measure

if

Schliemann & al. Phys. Rev. A 64 022303 (2001)

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

h measure

Distinguishable and indistinguishable particlesVon Neumann and Rényi entropies

Good correlation measures for fermions. The von Neumann entropy is

And the ath Rényi entropies are

According to Jensen’s inequality

The minimum of the entropyIt can be shown that

thus the von Neumann entropy is

and Sb=1 iff h=0, i.e., if the Slater rank is 1.

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

Distinguishable and indistinguishable particles - summaryThe connection between the entropy and the concurrence for specially parameterized two-electron states:

The measures of entanglementSzalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

The connection between the concurrence and specially parameterized two-electron states: hfor specially parameterized two-electron states:

The measures of entanglementSzalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

The connection between the entropy and specially parameterized two-electron states: h for specially parameterized two-electron states:

The measures of entanglementSzalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

With specially parameterized two-electron states:

Three fermionsThere are at least two essentially different types of entanglement if three or more particles are present.

3 particles, 6 one-electron states

And the “dual state”

Lévay& al. Phys. Rev. A 78, 022329 (2008)

Three fermions specially parameterized two-electron states:

3 particles, 6 one-electron states:

Non-entangled states (separable or biseparable):

Entangled state type 1

Entangle state type 2

Lévay& al. Phys. Rev. A 78, 022329 (2008)

Future plans specially parameterized two-electron states:

- 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.

A.D. Gottlieb& al. Phys. Rev. Lett 95, 123003 (2005)

Recent publications by the group specially parameterized two-electron states:

- 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).

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