Entanglement measures in quantum computing
Download
1 / 25

Entanglement Measures in Quantum Computing - PowerPoint PPT Presentation


  • 113 Views
  • Uploaded on

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,

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Entanglement Measures in Quantum Computing' - gloria


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Entanglement measures in quantum computing

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
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
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
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)


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 and indistinguishable particles


Distinguishable particles

Small overlap between j and c

The exchange contributions are small in the Slater determinants

Distinguishable particles

For two particles and two states

A B

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


Indistinguishable particles

Large overlap between j and c

The exchange contributions are significant in the Slater determinants

Indistinguishable particles

If the energy barrier is lowered


Indistinguishable particles1

A mixed state of two Slater determinants arises

Indistinguishable particles

Suppose, that after time evaluation


Distinguishable particles1

We get one of theBell states

Distinguishable particles

Rising the barrier again – increasing the distance


What is entanglement
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.


Distinguishable and indistinguishable particles1

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

Distinguishable and indistinguishable particles


Distinguishable particles concurrence

The state can be written as

The concurrence is

Concurrence can also be introduced for indistinguishable particles.

Distinguishable particles - concurrence

Magic basis for two particles


Indistinguishable particles measure

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

Indistinguishable particles – η measure

The definitionof the Slater correlation measure

if

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


Distinguishable and indistinguishable particles2

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

h measure

Distinguishable and indistinguishable particles


Von neumann and r ny i entropies

In our case

Von Neumann and Rényi entropies

Good correlation measures for fermions. The von Neumann entropy is

And the ath Rényi entropies are


The minimum of the entropy

According to Jensen’s inequality

The minimum of the entropy

It can be shown that

thus the von Neumann entropy is

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


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

Distinguishable and indistinguishable particles - summary


The measures of entanglement

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

The measures of entanglement

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)


The measures of entanglement1

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

The measures of entanglement

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)


The measures of entanglement2

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

The measures of entanglement

Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)


Three fermions

With specially parameterized two-electron states:

Three fermions

There 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 fermions1
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
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
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).


ad