Experimentally freezing quantum discord in a dissipative environment using dynamical decoupling Harpreet Singh Kavita Dorai Arvind Department of Physical Sciences Indian Institute of Science Education Mohali Young Quantum-2017 Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 1 / 28 decoupling Outline 1 Introduction 2 Quantum correlations for two-qubit BD state 3 Experimental demonstration of time inveriant quantum discord 4 Decoupling strategies Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 2 / 28 decoupling Introduction The interaction of a quantum system with its environment causes the rapid destruction of crucial quantum properties, such as quantum superpositions and of quantum correlations in composite systems. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 3 / 28 decoupling Quantum correlations for two-qubit BD state Consider a two-qubit system with Hilbert space and computational H base 00 ; 01 ; 10 ; 11 . fj i j i j i j ig A generic two-qubit state 1 A B P3 A B ρAB = I + ~u~σ I + I ~v~σ + wjk σ σ 4 ⊗ ⊗ j;k=1 j ⊗ k We are interested in system undergoing phase damping, initially in the state 1 P3 A B ρAB = IAB + ci σ σ 4 i=1 i ⊗ i With c1 = 1 and c2 = c3. − Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 4 / 28 decoupling Dynamics of phase damping are goverened by the master equation: i X y 1 y ρ_AB (t) = [ HAB ; ρ] + (Li z ρL L Li z ; ρ ) (1) − ; i;z − 2f i;z ; g } i,α (i) where the Lindblad operator Li;z = pγi σz acts on the ith qubit and (i) describes decoherence and σz denotes the Pauli matrix of the ith qubit. The constant γi is approximately equal to the inverse of decoherence time. Solving above equation c1(t) = c1(0)exp( 2γt), c2(t) = c2(0)exp( 2γt), and c3(t) = c3(0) = c3. − − Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 5 / 28 decoupling Total mutual information of system is given by (ρAB ) = S(ρA) + S(ρB ) S(ρAB ) I − where ρA(B) is the reduced density matrix of subsystem and S(ρ) = Tr ρlog2ρ is the Von Neumann entropy. For BD− typef of statesg the mutual information is: P3 (ρAB ) = 2 + λl log2λl I l=0 where λl are the eigenvalues of ρAB given by λ0 = [1 + c1 c2 + c3]=4, λ1 = [1 c1 + c2 + c3]=4, − − λ2 = [1 + c1 c2 c3]=4, and λ3 = [1 c1 + c2 c3]=4 − − − − Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 6 / 28 decoupling 1 Classical correlations (ρAB ) are given by: C (ρAB ) = max[S(ρA) S(ρAB Πk )]; (2) C fΠk g − jf g where the maximum is taken over the set of projective measurements P Πk and S(ρAB Πk ) = k pk S(ρk ) is the conditional entropy of Af , giveng the knowledgejf g of the state of B, with ρk = TrB (Πk ρAB Πk )=pk and pk = TrAB (ρAB Πk ). For BD state P2 1+(−1)j χ j (ρAB ) = log [1 + ( 1) χ] C j=1 2 2 − where χ = max c1 ; c2 ; c3 . fj j j j j jg 1S. Luo, Phys. Rev. A 77, 022301 (2008). Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 7 / 28 decoupling 2 X 1 + ( 1)j χ(t) (ρ(t)) = − log [1 + ( 1)j χ(t)] C 2 2 − j=1 2 j X 1 + ( 1) c1(t) j (ρ(t)) = − log [1 + ( 1) c1(t)] I 2 2 − j=1 2 j X 1 + ( 1) c3 j + − log [1 + ( 1) c3] 2 2 − j=1 (ρ) (ρ) (ρ) (3) D ≡ I − C where χ(t) = max c1(t) ; c2(t) ; c3(t) . fj j j j j jg For a time t < ¯t = 1 ln( c1(0) ), 2γ c3(0) j P2 1+(−1) c3 j (ρ) = log [1 + ( 1) c3] D j=1 2 2 − Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 8 / 28 decoupling Simulation 1.4 Quantum corr. Classical corr. 1.2 Total corr. 1 0.8 t¯= 0.05s 0.6 0.4 Correlation 0.2 t¯ 0 0 0.1 0.2 0.3 Time(s) Figure: Time evolution of total correlations (green triangles), classical correlations (blue squares) and quantum discord (red circles) of the BD state. With intial c1 = 1, c2 = 0:7, c3 = 0:7, and 2γ = 7:5. − Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 9 / 28 decoupling Nuclear magnetic interaction Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 10 / 28 decoupling NMR Qubits 1 A molecule contains spin I = 2 nuclei placed in B0 represents a single qubit, with internal Hamiltonian: H0 = µ.B = µz B0 = γn~B0Iz = ~!LIz (4) − − − − ∆E = ¯hωL 1 A molecule containing N-coupled spin I = 2 nuclei placed in B0 represents a N-qubit, with internal Hamiltonian N N X i X i j H0 = !i I + 2π Jij I :Iz (5) − z z i=1 i<j th where Jij is the scalar coupling and !i is the Larmor frequency of i spin. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 11 / 28 decoupling NMR States At temperature T, NMR qubits are at thermal equilibrium state e−H0=kB T ρeq = (6) Tr[e−H0=KB T ] For high temperature, Ei << KB T I H0=kB T I H0=kB T I H0 ρeq − − N N (7) ≈ Tr[I H0=kB T ] ≈ Tr[I ] ≈ 2 − 2 kB T − 2N −1 X ∆ρ I i (8) ≈ − z i=0 ~!i −5 where = N 10 2 kB T ≈ Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 12 / 28 decoupling Pseudo-Pure State Figure: At thermal equilibrium for Figure: Pseudo-Pure ∆ = I + I + 2I I ρ = 105I + ∆ρ, for two-qubit ρ z1 z2 z1: z2 Homo-nuclear system ∆ρ = Iz1 + Iz2 Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 13 / 28 decoupling NMR System Details (a)1H (b) Qubit1 Qubit2 13 1 0 1 0 C | i | i | i | i νC =11815.9Hz νH =4792.4Hz J12=215.1Hz 0 0 C (c) | i | i T1 =16.6s H T1 =7.9s C T2 =0.3s T H =2.9s 2 8.2 8.0 7.8 79.5 78.5 77.5 ωH (in ppm) ωC (in ppm) Figure: (a) Chloroform-13C molecular structure with 1H and 13C labeling the first and second qubits, respectively. Tabulated system parameters are: chemical shifts νi , the scalar coupling interaction strength J12 (in Hz) and relaxation times T1 and T2 (in seconds). NMR spectrum of (b) the thermal equilibrium state after a π=2 readout pulse and (c) the pseudopure 00 state. j i Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 14 / 28 decoupling We aim to prepare an initial BD state with the parameters c1(0) = 1, c2(0) = 0:7,c3(0) = 0:7. − 0 0:07 0:00 0:00 0:07 1 B 0:00 0:43 0:43 0:00 C ρth = B C BD @ 0:00 0:43 0:43 0:00 A 0:07 0:00 0:00 0:07 The two-qubit system was initialized into the pseudopure state 00 using the spatial averaging technique, with the density operatorj giveni by 1 ρ00 = − I + 00 00 (9) 4 j ih j with a thermal polarization 10−5 and I being a 4 4 identity operator. ≈ × Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 15 / 28 decoupling Figure: (a) Quantum circuit for the initial pseudopure state preparation, followed ✭ ✝✮ ◆ by the block for BD state ♥ ✶ ♦ preparation. The next block ✄ ❍ ❥✵✐ ✁ t ❡ ✁ ❛ t ❤ s ☎②✆ ✝♠✞❝✝✟ ❞ ✠❝✡✉♣✟✞✆ ❣ ✚ depicts the DD scheme used ✄ ❛ ❇ ❉ ✂ ❧ t ☛✠ ✉✠✆ ❝✠ ❛ ✁ ❙ ✄ ✶ ✸ ✁ t to preserve quantum ❈ ❥✵✐ ✄ ♥ ■ discord. The entire DD sequence is repeated N ✙ ✙ ✙ ✙ times before measurement. ✭ ❜ ✮ ✌ ✍ ✍ ✍ ✍ ✹ ✹ ✷ ✷ ◆ (b) NMR pulse sequence ❚ ❳ ✔ ✕ ✖ ✭ ✮ ❖ corresponding to the ▼ ① ① ✲ ① ✲ ✏ ① ① ✲ ① ✲ ① ① quantum circuit. The rf ❖ ❳ ✔ ✖ ✎ ✽✭ ✮ ● pulse flip angles are set to ❘ ◦ ◦ ❆ α = 46 and β = 60 , while ❳ ✔ ✗✻ ✖ ✭ ✮ P ① ✲ ① ① ✏ ① ✲ ① ✏ ☞ all other pulses are labeled ✜ ✜ ✑ ✒ ✑ ✒ ❨ ❑ ✘ ✘ ✛✢ with their respective angles ✓ and phases. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 16 / 28 decoupling E The experimentally achieved ρBD (reconstructed using the maximum 2 likelihood method ) had parameters c1(0) = 1:0, c2(0) = 0:68 and c3(0) = 0:68 and a computed fidelity of 0:99. exp − ρBD = 0 0:080 0:003 + 0:000i 0:003 + 0:000i 0:080 + 0:000i 1 B 0:003 0:000i 0:42 0:420 + 0:001i 0:003 + 0:000i C B − C @ 0:003 0:000i 0:420 0:001i 0:420 0:003 + 0:000i A − − 0:080 0:000i 0:003 0:000i 0:003 0:000i 0:080 − − − All experimental density matrices were reconstructed using a reduced tomographic protocol, with the set of operations given by II ; IX ; IY ; XX being sufficient to determine all 15 variables for the two-qubitf system.g q 2 F = Tr pρtheoryρexptpρtheory (10) where ρtheory and ρexpt denote the theoretical and experimental density matrices respectively.