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 . {| i | i | i | i} 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 † 1 † ρ˙AB (t) = [ HAB , ρ] + (Li z ρL L Li z , ρ ) (1) − , i,z − 2{ i,z , } } i,α
(i) where the Lindblad operator Li,z = √γi 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− type{ of states} 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 {Πk } − |{ }
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 A{ , given} the knowledge|{ } 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 . {| | | | | |}
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) . {| | | | | |} 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 = ~ωLIz (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 = ωi I + 2π Jij I .Iz (5) − z z i=1 i th where Jij is the scalar coupling and ωi 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 ~ωi −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. | 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. − 0.07 0.00 0.00 0.07 0.00 0.43 0.43 0.00 ρth = BD 0.00 0.43 0.43 0.00 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 operator| giveni by 1 ρ00 = − I + 00 00 (9) 4 | ih | 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 ✄ ❍ ❥✵✐ ✁ ❡ ✁ ❛ ❤ ☎②✆ ✝♠✞❝✝✟ ❞ ✠❝✡✉♣✟✞✆ ❣ ✚ depicts the DD scheme used ✄ ❛ ❇ ❉ ✂ ❧ ☛✠ ✉✠✆ ❝✠ ❛ ✁ ❙ ✄ ✶ ✸ ✁ 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 ❳ ✔ ✗✻ ✖ ✭ ✮ 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 = 0.080 0.003 + 0.000i 0.003 + 0.000i 0.080 + 0.000i 0.003 0.000i 0.42 0.420 + 0.001i 0.003 + 0.000i − 0.003 0.000i 0.420 0.001i 0.420 0.003 + 0.000i − − 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-qubit system.} q 2 F = Tr √ρtheoryρexpt√ρtheory (10) where ρtheory and ρexpt denote the theoretical and experimental density matrices respectively. 2H. Singh, Arvind, and K. Dorai, Phys. Lett. A, 380, 3051 (2016) Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 17 / 28 decoupling (a) Simulation (b) Experiment 1.4 Quantum corr. 1.4 Quantum corr. Classical corr. Classical corr. 1.2 1.2 Total corr. Total corr. 1 1 0.8 0.8 t¯= 0.05s t¯= 0.05s 0.6 0.6 0.4 0.4 Correlation Correlation 0.2 t¯ 0.2 t¯ 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 Time(s) Time(s) Figure: Time evolution of total correlations (green triangles), classical correlations (blue squares) and quantum discord (red circles) of the BD state.(a)Simulated with γzq 6.9/s γdq 9.8/s. (b) experimental results. ≈ → Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 18 / 28 decoupling (a) Initial State (T = 0s) 00 00 11 11 01 01 10 10 10 10 01 01 11 00 11 00 Simulated Experimental (b) T = 0.06s 00 00 00 00 01 11 11 01 11 11 10 01 10 10 01 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 (c) T = 0.12s 00 00 00 00 01 11 11 01 11 11 10 01 10 10 01 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 (d) T = 0.17s 00 00 00 00 01 11 11 01 11 11 10 01 10 10 01 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 (e) T = 0.23s 00 00 00 00 01 11 11 01 11 11 10 01 10 10 01 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 Figure: Real (left) and imaginary (right) parts of the experimental tomographs of the (a) Bell Diagonal (BD) state, with a computed fidelity of 0.99. (b)-(e) depict the state at T = 0.06, 0.12, 0.17, 0.23s, with the tomographs on the left and the right representing the simulated and experimental state, respectively. The rows and columns are labeled in the computational basis ordered from 00 to 11 . | i | i Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 19 / 28 decoupling Spin echo Dynamical Decoupling strategies When the state is known. When the subspace is known. Interested in protection coherence only. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 20 / 28 decoupling (a) XY 4(s) We implemented the τ4 τ4 τ4 τ4 τ4 2 2 symmetrized XY4 DD scheme x y x y for τ4 = 0.58ms and an experimental time for one run of (b) XY 8(s) 2.43 ms. We implemented the XY8 DD τ τ 8 τ8 τ8 τ8 τ8 τ8 τ8 τ8 8 2 2 scheme a ( providing a better xyxyyxyx compensation for pulse errors), for τ8 = 0.29ms and an Figure: NMR pulse sequence corresponding experimental time for one run of to DD schemes (a) XY4(s), and (b) XY8(s). 2.52 ms. All the pulses are of flip angle π and are labeled with their respective phases. The aPhys.Rev. A, 87, 042309 (2013) pulses act on both qubits simultaneously. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 21 / 28 decoupling (a) XY4(s) (b) XY8(s) 1.4 Quantum corr. 1.4 Quantum corr. Classical corr. Classical corr. 1.2 1.2 Total corr. Total corr. 1 1 t¯= 0.10s t¯= 0.13s 0.8 0.8 0.6 0.6 0.4 0.4 Correlation Correlation 0.2 t¯ 0.2 t¯ 0 0 0 0.1 0.2 0.3 0 0.1 0.2 0.3 Time(s) Time(s) Figure: Time evolution of total correlations (green triangles), classical correlations (blue squares) and quantum discord (red circles) of the BD state. With dynamical decoupling sequence (a)XY4(s) (b) XY8(s) Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 22 / 28 decoupling XY4 XY8 Real Imaginary Real Imaginary (a) N = 20 00 00 00 00 11 01 01 11 01 11 11 10 01 10 10 10 10 01 10 10 10 01 01 11 01 11 00 11 00 11 00 00 (b) N = 40 00 00 00 00 01 11 01 11 01 11 01 11 10 10 10 10 10 10 10 10 01 01 01 11 01 11 00 11 00 11 00 00 (c) N = 60 00 00 00 00 11 01 11 01 11 01 11 01 10 10 10 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 (d) N = 80 00 00 00 00 11 11 11 01 01 10 01 11 01 10 10 10 10 10 01 10 01 10 01 01 11 00 11 00 11 00 11 00 (e) N = 100 00 00 00 00 11 01 11 01 01 11 01 11 10 10 10 10 10 10 10 10 01 01 01 01 11 00 11 00 11 00 11 00 Figure: Real (left) and imaginary (right) parts of the experimental tomographs of the (a)-(e) depict the BD state at N = 20, 40, 60, 80, 100, with the tomographs on the left and the right representing the BD state after applying the XY4 and XY8 scheme, respectively. The rows and columns are labeled in the computational basis ordered from 00 to 11 . | i | i Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 23 / 28 decoupling (a) XY 16(s) τ τττττττττττττττ τ 2 2 We implemented the x y x y y x y x-x-y-x-y-y-x-y-x symmetrized XY16 DD scheme for τ = 0.145ms and an (b) KDDxy experimental time for one run of 2 2.75ms. τ τ k τk τk τk τk τk τk τk τk τk k 2 2 We implemented the KDDxy a π π π 2π π π 2π DD scheme , τ = 0.116 ms 6 0 2 0 6 3 2 π 2 3 k and an experimental time for Figure: NMR pulse sequence corresponding one run of 2.86 ms. to DD schemes (a) XY16(s), and (b) aPhys. Rev.Lett., 106, 240501 (2011). KDDxy . All the pulses are of flip angle π and are labeled with their respective phases. The pulses act on both qubits simultaneously. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 24 / 28 decoupling (a) XY16(s) (b) KDDXY 1.4 Quantum corr. 1.4 Quantum corr. Classical corr. Classical corr. 1.2 1.2 Total corr. Total corr. 1 1 t¯= 0.24s t¯= 0.22s 0.8 0.8 0.6 0.6 0.4 0.4 Correlation Correlation 0.2 t¯ 0.2 t¯ 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 Time(s) Time(s) Figure: Time evolution of total correlations (green triangles), classical correlations (blue squares) and quantum discord (red circles) of the BD state.With dynamical decoupling sequence (a)XY16(s) (b) KDDxy Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 25 / 28 decoupling XY16 KDDxy Real Imaginary Real Imaginary (a) N = 20 00 00 00 00 11 01 01 11 01 11 11 10 01 10 10 10 10 01 10 10 10 01 01 11 01 11 00 11 00 11 00 00 (b) N = 40 00 00 00 00 01 11 01 11 01 11 01 11 10 10 10 10 10 10 10 10 01 01 01 11 01 11 00 11 00 11 00 00 (c) N = 60 00 00 00 00 11 01 11 01 11 01 11 01 10 10 10 10 10 01 10 01 10 01 10 01 11 00 11 00 11 00 11 00 (d) N = 80 00 00 00 00 11 11 11 01 01 10 01 11 01 10 10 10 10 10 01 10 01 10 01 01 11 00 11 00 11 00 11 00 (e) N = 100 00 00 00 00 11 01 11 01 01 11 01 11 10 10 10 10 10 10 10 10 01 01 01 01 11 00 11 00 11 00 11 00 Figure: Real (left) and imaginary (right) parts of the experimental tomographs in (a)-(e) depict the Bell Diagonal (BD) state at N = 20, 40, 60, 80, 100, with the tomographs on the left and the right representing the BD state after applying the XY16 and KDDxy preserving DD schemes, respectively. The rows and columns are labeled in the computational basis ordered from 00 to 11 . | i | i Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 26 / 28 decoupling Conclusion Our results show that quantum discord, which remains unaffected under certain decoherence regimes, can be preserved for very long times using dynamical decoupling schemes. Our experiments have important implications in situations where persistent quantum correlations have to be maintained in order to carry out quantum information processing tasks. Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 27 / 28 decoupling Thanks Harpreet Singh, Kavita Dorai, Arvind Experimentally freezing quantum discord in a dissipativeYoung environment Quantum-2017 using dynamical 28 / 28 decoupling