Entanglement Measures in Quantum Computing

1. 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

2. 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

3. 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.”

4. 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)

5. 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

6. 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)

7. Large overlap between j and c The exchange contributions are significant in the Slater determinants Indistinguishable particles If the energy barrier is lowered

8. A mixed state of two Slater determinants arises Indistinguishable particles Suppose, that after time evaluation

9. We get one of theBell states Distinguishable particles Rising the barrier again – increasing the distance

10. 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.

11. 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

12. 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

13. 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)

14. 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

15. In our case Von Neumann and Rényi entropies Good correlation measures for fermions. The von Neumann entropy is And the ath Rényi entropies are

16. 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.

17. 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

18. 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)

19. The connection between the concurrence and hfor specially parameterized two-electron states: The measures of entanglement Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

20. The connection between the entropy and h for specially parameterized two-electron states: The measures of entanglement Szalay& al. J. Phys. A - Math. Theo., 41, 505304 (2008)

21. With 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)

22. Three fermions 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)

23. Future plans • 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)

25. 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).