Entanglement in multiqubit systems Maximally entangled states Antoni

Entanglement in multiqubit systems: Maximally entangled states Antoni Borras – Universitat Illes Balears with M. Casas – Universitat de les Illes Balears & IFISC J. Batle – Universitat Illes Balears A. R. Plastino, C. Zander – University of Pretoria (South Africa) A. Plastino – National University La Plata (Argentina) 1

Outline 1. Genuine entanglement in N qubit systems 2. Entanglement measures 3. Maximally search) entangled states (Numerical 4. Usefulness of maximally entangled states 2

1. Genuine entanglement in N qubit systems 3

Entanglement in bipartite systems § Maximum entanglement (2 qubits): n } Separable State: Otherwise: entangled state 4

Entanglement in 3 qubit systems 2 classes of genuine multipartite entanglement: SLOCC [ Dür, Vidal and Cirac PRA 62 062314 (2000) ] 5

Entanglement in 4 qubit systems n Genuine 4 qubit entanglement: [Verstraete et al PRA 65 052112 (2002) ] 6

Genuine N-qubit entanglement n n All reduced density matrices represent a mixed state. Maximally entangled state: All reduced density matrices are completely mixed 4 qubits: None state fulfills conditions. [ Higuchi and Sudbery PLA 272, 213 (2000) ] 7

2. Entanglement measures 8

Measures based on subsystems purity n Single qubit reduced density matrix: Meyer and Wallach J. Math. Phys. 43 4273 (2002) G. K. Brennen Quantum Inf. Comput 3 619 (2003) n Multipartite reduced density matrix: [ A. J. Scott PRA 69 052330 (2004) ] 9

Ex. 4 qubits Bipartion of a N qubit state Number on nonequivalent bipartitions? ? n ≠ N/2: n n = N/2: N AB CD AC BD AD BC BC AD B–D A-C C–D A-B 3 4 5 6 7 1 3 4 5 6 7 2 0 3 10 15 21 3 0 0 0 10 35 Ncuts 3 7 15 31 63 10

Sum over all possible bipartitions Bipartite measures used: • Linear entropy: • von Neumann entropy: • Renyi entropy: • Negativity: i are the negative eigenvalues of the partial transpose matrix associated with a given bipartition. 11

3. Maximally entangled states 12

Numerical search (Brown et al) Numerical maximization of entanglement: Initial mixed state (arbitrary) n ; n Entanglement measure: n Try to increase Neg: n Decrease pj and ck until convergence [ Brown et al. , JPA 38 1119 (2005) ] 13

Our algorithm Similar to the previous one. Main differences: Brown et al: Our algorithm: 1. Mixed States. 1. Pure States. 2. Negativity. 2. SL, SVN, SRe, Neg 3. If ck < 10 -2 then ck = 0 forever more. 3. We never kill ck 4. FORTRAN 4. Matlab. 14

Algorithm E(t=1) < E(t=0) . . E(t=0) E(t=1) > E(t=0) E(t=1) < E(t=0) … E(t=n) = Emax E(t=1) < E(t=0) ck = 0. 5 ck 0. 1 … ck 10 -9 15

Opt. 3 qubits Target Max. EL EVN ERe Neg EL EVN ERe Neg GHZ EL (1. 5) 1. 5 EVN (3) 3 ERe (2. 079441) 2. 079441 Neg (1. 5) 1. 5 16

4 qubits - I It doesn’t exist any 4 qubit states with all reduced density matrices completely mixed. [ Higuchi and Sudbery PLA 272, 213 (2000) ] § Proposed maximally entangled state: 17

4 qubits - II Target ? ? BSSB 4 Us EL (4. 25) 4. 0000 EVN (10) 9. 2018 9. 3773 ERe (6. 9315) 5. 8619 5. 9955 Neg (6. 5) 5. 9142 6. 0981 EL EVN ERe Neg EL X X X EVN X ERe X X X Neg X 18

5 qubits Target BSSB 5 EL (10) 10 EVN (25) 25 ERe (17. 3287) 17. 3267 Neg (17. 5) 17. 5 19

Target 6 qubits EL (23) EVN (66) 6 qb 23 66 ? ? ERe (45. 74771) Neg. (60. 5) 45. 74771 60. 5 20

7 qubits 7 qb Target ? ? EL EVN ERe Neg EL X X X EVN ERe X X EL EVN (49. 875) (154) 49. 5738 152. 6201 ERe (106. 74467) 91. 6518 Neg. (157. 5) 155. 8129 21

Normalized measures n N 3 4 5 6 7 ELmax EVNmax ERemax Negmax 1 3 4 5 6 7 0. 5 1 0. 6932 0. 5 2 0 3 10 15 21 0. 75 2 1. 3863 1. 5 3 0 0 0 10 35 0. 875 3 2. 0794 3. 2 Ncuts 3 7 15 31 63 E N 3 4 5 6 7 8 EL 1. 0 0. 9524 1. 0 0. 9942 0. 9932 EVN 1. 0 0. 9555 1. 0 0. 9913 0. 9833 ERe 1. 0 0. 8886 1. 0 0. 8898 0. 9469 Neg 1. 0 0. 9617 1. 0 0. 9901 0. 9894 22

4. Usefulness of maximally entangled states 23

Quantum Error Correction Codes n Encode state of n qubits in a state of N qubits to protect it against decoherence. (N > n) n A QECC protects information by encoding it nonlocally Entanglement n Usually, Quantum Error Correction Codes are highly entangled multiqubit states. 24

Task-Oriented Maximally Entangled [ P. Agrawal and B. Pradhan, quant-ph/0707. 4295 ] States n Perfect teleportation of 1 or 2 qubits using |BSSB 5> n Superdense coding using |BSSB 5> Transmission of 5 cbits sending 3 qubits [ S. Muralidharan and P. Panigrahi, quant-ph/0708. 3785 ] 25

Future Work n Increase dimension (N=9, 10, …) ? ? ? n Deeper insight in the 4 qubit case. n Multiqudit systems n Applications 26

Summary n Extension of numerical search up to 8 qubits. n All states with maximally mixed single qubit reduced density matrix. n Highly symmetric states 27