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 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 computing quantum cryptography quantum 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 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 particles Small overlap between  and  The exchange contributions are small in the Slater determinants 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 If the energy barrier is lowered

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

Distinguishable particles We get one of the Bell states Rising 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 particles the 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 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

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

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 if Schliemann & al. Phys. Rev. A (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

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

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. 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 S min =0 Identical particles Small distance and barrier Exchange properties are essential Slater decomposition →Slater rank η measure S min =1

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

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

The measures of entanglement The connection between the entropy and  for specially parameterized two-electron states: Szalay& al. J. Phys. A - Math. Theo., 41, (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 And the “dual state” Lévay& al. Phys. Rev. A 78, (2008)

3 particles, 6 one-electron states: Non-entangled states (separable or biseparable): Entangled state type 1 Entangle state type 2 Three fermions Lévay& al. Phys. Rev. A 78, (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, (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, (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, , (2008). Lévay, P., Vrana, P., Three fermions with six single particle states can be entangled in two inequivalent ways, Phys.Rev. A, , (2008).