13/02/12 Role of complementarity on Entanglement detection Ryo - PowerPoint PPT Presentation

13/02/12 Role of complementarity on Entanglement detection Ryo Namiki Quantum optics group, Kyoto University 求職中

● Quantum entanglement A B ● Inseparability (entangled state) Form of density operators  AB ≠ ∑ i p i  Ai ⊗ Bi LOCC: e.g. , ∣  〉 AB =∑ i a i ∣ i 〉 A ∣ i 〉 B 〈 i ∣ j 〉 = i , j Local quantum Operation & ● Entanglement measure (LOCC monotone) Classical Communication Quantify the strength of quantum correlation A B time M  AB ≥ M  L LOCC  AB  A i 1 † B i 2 † A i 3 † B i 4 B i 2  i 1  † L LOCC  AB =∑ i B i 4 A i 3 B i 2 A i 1  AB A i 1 =∑ i  K Ai ⊗ L Bi  AB  K Ai ⊗ L Bi  † A i 3  i 2, i 1  B i 4  i 3, i 2, i 1  Guhne&Toth, Phys. Rep. 474, 1 (2009) Horodeckis, Rev. Mod. Phys. 81, 865, (2009)

Basic concepts on Quantum mechanics  p  2 V C : =∥[ x , p ]∥/ 2 4 ● Canonical uncertainty relation  x  2  p  2  C 2 3 Trade off 2 〈  2  x 〉〈  2  p 〉 ≥ 1 / 4 ∥[ x , p ]∥ 1 unphysical  x  2 4 U ● Quantum entanglement 0 1 2 3  AB ≠ ∑ i p i  Ai ⊗ Bi - Inseparability A B 〈 i ∣ j 〉 = i , j e.g. , ∣  〉 AB =∑ i a i ∣ i 〉 A ∣ i 〉 B - Entanglement measure M  AB ≥ M  L LOCC  AB 

Einstein-Podolsky-Rosen (EPR) state A B ∣  〉 AB : =∫ dx ∣ x 〉 A ∣ x 〉 B /  2  ∣ x 1 〉 ........... ∣ x 1 〉 ∣ x 2 〉 ........... ∣ x 2 〉 Simultaneous eigenstate of ∣ x 3 〉 ........... ∣ x 3 〉 p A    p B ,  x A −  x B ⋮ Positions are correlated and ∣ p 1 〉 ........... ∣ − p 1 〉 Momentums are anti-correlated ∣ p 2 〉 ........... ∣ − p 2 〉 〈  2   p B  〉 ~ 0 ; 〈  2   x B  〉 ~ 0 ∣ p 3 〉 ........... ∣ − p 3 〉 p A   x A −  ⋮  p  2 V V 4 4 EPR paradox! A B Uncertainty relation! 3 3 UV  1  x  2  p  2  C 2 u 2  v 2 = 1 2 2 U = 〈  2  u  x B  〉 / C x A − v  V = 〈  2  u  p B  〉 / C p A  v  1 1  x  2 4 U 4 U 0 1 2 3 0 1 2 3

Entanglement detection via EPR paradox Product criterion for entanglement C : =∣[ x , p ]∣/ 2 〈  2  u  x B  〉〈  2  u  p B  〉  C 2 x A − v  p A  v  〈  2  u  x B  〉〈  2  u  p B  〉  C 2 x A − v  p A − v  Uncertainty relation! V. Giovannetti et al., Phys. Rev. A 67, 022320 (2003) Stronger Correlation beyond the uncertainty limit V EPR paradox! 4 ● Quantum entanglement UV  1 3  AB ≠ ∑ i p i  Ai ⊗ Bi u 2  v 2 = 1 2 U = 〈  2  u  x B  〉 / C x A − v  V = 〈  2  u  p B  〉 / C p A  v  Give a constraint on the form 1 Of the density operator 4 U 0 1 2 3

Complementary correlations for entanglement Z ∣ 0 〉 = ∣ 0 〉 X ∣  0 〉 = ∣  0 〉 ● Maximally entangled state (of two qubits) Z ∣ 1 〉 =− ∣ 1 〉 X ∣  1 〉 =− ∣  1 〉 ∣  0 〉 = ∣ 00 〉  ∣ 11 〉 /  2 0 〉 = ∣ 0 〉  ∣ 1 〉 /  2 ∣  = ∣  0 〉  ∣  1 〉 /  2 0  1  ∣  1 〉 = ∣ 0 〉 − ∣ 1 〉 /  2 Simultaneous eigenstate of product Pauli operators: Z : = [ 0 − 1 ] 1 0 X A   Z A    X B , Z B Strong correlations on the conjugate variables X : = [ 0 ] 0 1  〈  X B 〉 = 0 X A −  〈  Z B 〉 = 0 1 Z A −  An average correlation of the Z-basis bits and X-basis bits exceeds 75% 〈  Z B 〉  1 X A  X B   Z A  the state is entangled:  AB ≠ ∑ i p i  Ai ⊗ Bi

Uncertainty relations and entanglement A B 〈  2  u  x B  〉〈  2  u  p B  〉  C 2 x A − v  p A  v  Continuous-variable systems Continuous-variable entanglement C : =∣[ x , p ]∣/ 2 〈  Z B 〉  1 X A  X B   Z A  Pair of two-levels systems Qubit-Qubit entanglement Pair of d-level systems Strength of measured correlations Uncertainty Qudit-Qudit entanglement relations? ● Fourier-based uncertainty relations ● Generalized Pauli-operators on d-level systems R. Namiki and Y. Tokunaga, Phys. Rev. Lett. 108, 230503 (2012)

Complementary elements and Uncertainty Relations Conjugate bases: on d- level system Two Fourier distributions Cannot have sharp peaks simultaneously Trade-off Never coexist on the unit circle (At least one is inside) Two (generalized) Pauli operators 〈 Z 〉 = ∑ j P  j  e i  j

Complementary elements and Uncertainty Relations Conjugate bases: on d- level system Discrete Fourier-based Uncertainty relations

Theorem. The state is entangled if it satisfies either of d × d level system R. Namiki and Y. Tokunaga, 2 qubits ( d =2) Phys. Rev. Lett. 108, 230503 (2012) 〈  Z B 〉  1 X A  X B   Z A  Discrete Fourier-based Uncertainty relations R. Namiki and Y. Tokunaga, Phys. Rev. Lett. 108, 230503 (2012)

Theorem. The state is entangled if it satisfies either of For d = 2,3 two conditions are equivalent. R. Namiki and Y. Tokunaga, For d 4 there are mutually exclusive subsets. ≧ Phys. Rev. Lett. 108, 230503 (2012) X: Verified to be entangled by the first condition 〇 . Y: Verified to be entangled by the first condition △ . ∣  l ,m 〉 =  l  m ∣  0,0 〉 X A Z B F  x  : Floor function

Basic concepts on Quantum mechanics  p  2 V C : =∥[ x , p ]∥/ 2 4 ● Canonical uncertainty relation  x  2  p  2  C 2 3 Trade off 2 〈  2  x 〉〈  2  p 〉 ≥ 1 / 4 ∥[ x , p ]∥ 1 unphysical  x  2 4 U ● Quantum entanglement 0 1 2 3  AB ≠ ∑ i p i  Ai ⊗ Bi - Inseparability A B 〈 i ∣ j 〉 = i , j e.g. , ∣  〉 AB =∑ i a i ∣ i 〉 A ∣ i 〉 B - Entanglement measure M  AB ≥ M  L LOCC  AB 

Theorem. Multi-level coherence  AB ≠ ∑ i p i ∣  i 〉〈  i ∣ k − 1 a i ∣ u i 〉 A ⊗ ∣ v i 〉 B ∣  i 〉 = ∑ i = 0  1 ≤ k ≤ d  ∣  〉 AB ∝ ∣ 0 〉 ∣ 0 〉  ∣ 1 〉 ∣ 1 〉  ∣ 2 〉 ∣ 2 〉  ..... Total correlations A B The state needs to include # of k+1 coherent superposition of the product states The figure k is called the Schmidt number which can quantify entanglement (entanglement monotone). For k = 1  AB ≠ ∑ i p i  Ai ⊗ Bi = ∑ i p i ' ∣  i 〉〈  i ∣ ⊗ ∣  i 〉〈  i ∣ R. Namiki and Y. Tokunaga, Phys. Rev. Lett. 108, 230503 (2012)

Theorem. Multi-level coherence of Quantum Gates  ' = E ≠ ∑ i A i  A i †  k − 1  = 1 2  1  k F  F d  rank  A i ≤ k Description by less-than rank -k Input-output correlation Kraus operators is not admissible! F = 1 2 d ∑ i  〈 i ∣ E  ∣ i 〉〈 i ∣  ∣ i 〉  〈  i ∣ E  ∣  i 〉 〈  i ∣  ∣  i 〉  ∃ A i s.t. ,rank  A i  k Z-basis X-basis   ' E Trace-preserving Tr  ' = Tr = 1 Ideal unitary gates † E ideal = U  U † U = 1 U rank  U = d rank  A ≤ k General physical maps k − 1 a i ∣ u i 〉 A ⊗ ∣ v i 〉 B † A i = 1  ' = E = ∑ i A i  A i ∑ i A i  A A ∣  〉 ∝ ∑ i = 0 † Degrade Schmidt number Less-than k R. Namiki and Y. Tokunaga, Phys. Rev. A 85, 010305(R) (2012).

Application for known experiments  ' = E ≠ ∑ i A i  A i †  k − 1  = 1 2  1  k F  F d  rank  A i ≤ k Input-output correlation (Average fidelity) F = 1 2 d ∑ i  〈 U i ∣ E  ∣ i 〉 〈 i ∣  ∣ U i 〉  〈 U  i 〉  i ∣ E  ∣  i 〉 〈  i ∣  ∣ U  A basic elements of quantum computer: CNOT gate Entanglement d = 4 Breaking F  = 1 k = 1 C-Not Gate Schmidt number k 2  F Z  F X  k = 2 (at least) k = 3 U C − NOT : ∣ i 〉  ∣ U i 〉 0.89 [24] 4 ∣ 0 〉 ∣ 0 〉  ∣ 0 〉 ∣ 0 〉  3  = 0.875 F ∣ 0 〉 ∣ 1 〉  ∣ 0 〉 ∣ 1 〉 3 0.86 [23] ∣ 1 〉 ∣ 0 〉  ∣ 1 〉 ∣ 1 〉  2  = 0.75 F ∣ 1 〉 ∣ 1 〉  ∣ 1 〉 ∣ 0 〉 2  1  = 0.625 F 0 〉 ∣  0 〉  ∣  0 〉 ∣  0 〉 ∣  ∣  0 〉 ∣  1 〉  ∣  1 〉 ∣  1 〉 ∣  1 〉 ∣  0 〉  ∣  1 〉 ∣  0 〉 ∣  1 〉 ∣ 1 〉  ∣  0 〉 ∣  1 〉

Summary Role of Complementary on Entanglement detection Quantum entanglement & Uncertainty relations Simultaneous correlations on complementary observables → Inseparability of two-body density operators → Strength of quantum correlations X X Z Z →(Coherence of Quantum Gates) ● Two measurement settings ● Multi-dimensional entanglement A B →Detection of Non-Gaussian entanglement uncertainty relations based on SU(2) and SU(1,1) generators R. Namiki and Y. Tokunaga, Phys. Rev. Lett. 108, 230503 (2012). (R. Namiki and Y. Tokunaga, Phys. Rev. A 85, 010305(R) (2012). ) R. Namiki, Phys. Rev. A 85, 062307 (2012).