Quantum Computing with Para-hydrogen Muhammad Sabieh Anwar sabieh@lums.edu.pk International Conference on Quantum Information, Institute of Physics, Bhubaneswar, March 12, 28 Joint work with: J.A. Jones (Oxford) and S.B. Duckett (York)
Contents Entanglement in two qubit mixed states Quantum computing with mixed states? The problem of initialization Pure and entangled states with para-hydrogen Purity sharing
Entanglement in two qubit mixed states
Entanglement: Pure States Separable ψ AB = φ A χ B = ( 1 ) / 2 ( 1 ) / 2 ( = ( 1 ) / 2 i 1 ) / 2 = ( i 1 1 i 11 ) / 2 Entangled ψ AB φ A χ B ϕ = ( 11 ) / 2 ϕ = ( 11 ) / 2 Bell or EPR States ψ = ( 1 1 ) / 2 ψ = ( 1 1 ) / 2
Entanglement and Density Matrices 1 Density matrix ρ ρ AB = k λ k ψ AB ψ AB Pure and mixed states Density matrices of the Bell States 1/ 2 ( 11 ) / 2 ρ = 1/ 2 1/ 2 1/ 2 ( 1 1 ) / 2 ρ = 1/ 2 1/ 2 1/ 2 1/ 2
Entanglement and Density Matrices 2 Maximally mixed state Maximally mixed state 11 ) / 11 1 1 1 1 ( 1/ 1/ 1/ 1/ = = ρ 1 ) / 1 1 1 1 1 1 1 ( = Another mixed state Another mixed state 2 2) / 1 ) / )( 1 ( ( 1/ 1/ 1/ 3/ = = ρ
Entanglement and Density Matrices 3 Separable density matrix ρ η ( ρ ρ AB = k A B ) k k Convex sum of direct product states Entangled density matrix ρ η ( ρ ρ AB k A B ) k k
Detecting and Quantifying Entanglement 1 Can we find a separable decomposition of? ρ = 5 1 1 5 18 1 ρ 11 ) / 11 1 1 1 1 ( 1/ 1/ 1/ 1/ = = ρ ) / ( = ψ ψ ψ ψ ϕ ϕ ϕ ϕ Ensemble fallacy D. T. Pegg, J. Jeffers J. Mod. Opt. 52, 1835 (25)
Detecting and Quantifying Entanglement 2 2 ) / 1 1 1 1 ( 2 1/ 2 1/ = = ρ 2 ) / ( = ψ ψ ψ ψ
Test for Separability PPT Test ψ ρ = ψ T ρ = 1/ ψ 1/ 2 = 2( 1/ 2 = 1/ 2 1/ 2 11 1/ 2 ) 1/ 2 1/ 2 1/ 2 Eigenvalues of ρ T are {1/2,1/2,1/2,-1/2}. ρ is non-separable or entangled. A. Peres Phys. Rev. Lett. 77, 113 (1996)
Separability of the Werner State Eigenvalues Eigenvalues of PPT of of PPT of ρ are {1/(1-3ε),1/(1 ε), 1/(1 ε), 1/(1 ε)}. ρ is non is non-separable or entangled only if separable or entangled only if ε > 1/3 > 1/3. ρ = (1-ε)1/ / ε ψ - >< ψ - Werner or pseudopure state = = 1 2 1 1 2 1 1 1 2 2 1 1 ε ε ε ε ε ε ρ ε ε ε ε ε ε T
The problem of initialization
Requirements for Practical QC Existence of Qubits Initialization Universal Quantum Networks Read-out or Measure Decoherence D.P. DiVincenzo quant-ph/277 (2)
Initialization 1 NMR states are highly mixed. Pure Mixed > 1> = = 1 ρ Ideal initial state
Initialization 2 1> E > Thermal equilibrium state.251 ρ = ρ eq = exp(-βh)/z where β=ħ/(k B T) E = γħb z = ħω =2.8 1-25 J k B T= 1-21 J at 1T, 3K Population difference only ~1 -.25.25.2999 k B T >> E ={{1/B,1/,1/,1/-B}}= }}=1/B/(I/(I z S z ) where B = ħω / kt.
Initialization 3 B Net Magnetic Moment Thermal Equilibrium
Pseudopure States.251.25.25.2999 Thermal equilibrium state 1 Ideal initial state Pseudopure state ρ ps = (1-ε)1/ ε ψ>< ψ ε is polarization. E = Thermal, mixed Maximally mixed Pure
Spin Temperature 1> > COLD 1> > COLD 1> > HOT
Problems with Pseudopure States Scalability Is NMR quantum mechanical at all? S.L. Braunstein et al. Phys. Rev. Lett. 83, (1999) J.A. Jones, Fortsch. der Physik 8, 99 (22).
Climbing Mount Scalable
Scalability Maximum pure state that can be extracted from thermal state ρ ps =(1-ε) 1 / 2 n ε ψ>< ψ Warren proposed a theoretical maximum on ε: ε ~ nb / (2 n ) where n is the number of qubits. 11> 1> 2 1> > W.S.Warren Science 277, 1688 (1997).
Scalability and Separability 1/(12 n/2 ) 1/(12 2n-1 ) n B / (2 n ) E ES S Present day NMR implementations work in region S. We can enter ES if n > 11 (room temperature, 1T). We may never enter E with thermal states!
Methods for Increasing Polarization High fields and low temperature B = ħω / kt Algorithmic concentration of polarization Polarization transfer from nuclear spins Polarization transfer from electron spins Chemically induced dynamical nuclear polarization (CIDNP)
Is an NMR Device a Quantum Computer? No entanglement What makes a quantum computer quantum? States or Dynamics? Scaling of the Hilbert space dimensionality? Superposition? Other models of better than classical computing.
The para-hydrogen approach
Para-hydrogen Induced Polarization 1 Need for non-thermal distributions Symmetrization postulate ψ> t = ψ> tr ψ> vib ψ> e ψ> ns ψ> r ψ> ns T 1 >= > T >=1/2 1/2 ( 1> 1>) S >=1/2 1/2 ( 1>- 1>) T -1 >= 1> ψ> r j=1,3,5, odd j=,2,, even (Y j m (π-θ, πϕ)=( )=(-1) j Y j m (θ, ϕ))
Para-hydrogen Induced Polarization 2 j ψ rot > ψ ns > o/p s symmetrical a asymmetrical p 1 a s o 2 s a p 3 a s o
Making Parahydrogen Para-hydrogen once is para-hydrogen forever!
Para-hydrogen Quantum Computer Cl The PASADENA Experiment H 12 C pairwise addition assymetric environment enhanced polarizations
Example PASADENA spectrum Rh H Before After
Experimental Setup
Schematic setup for the PASADENA Experiment
Some photographs depicting various units in the experiement
Flash Photolysis D. Blazina et al. Magn. Reson. Chem. 3, 2 (25)
PHIP Spectra Spin Temperature = 6. mk Effective Field =.5 MT ε =.916 1/ 2 1 1 1 1 Tomography results Ideal singlet
The NMR Experiment M.S. Anwar et al. Phys. Rev. Lett. 93, 51 (2) M.S. Anwar et al. Phys. Rev. A 71, 32327 (25)
Grover s algorithm
Classical Deutsch s Algorithm: f(1)
Quantum Deutsch s Algorithm M.S. Anwar et al. Phys. Rev. A 7, 3232 (2)
Quantum Deutsch s Algorithm M.S. Anwar et al. Phys. Rev. A 7, 3232 (2)
Purity dilution
Purity Sharing 1 M.S. Anwar et al. Phys. Rev. A 73, 22322 (26)
Purity Sharing 2
Purity Sharing 3