PMC

The cyan vertical line at 0.7068 represents the fidelity between ρ_3qubits and pure three-qubit SC state. Green vertical line at 0.82274 represents fidelity between experiment and pure SC state. In the figure, “a” represents the fidelity between the estimation and experiment . “b” represents the fidelity between the estimation and target pure state. represents the distribution of fidelity when 499 copies of ρ_3qubits are projected into the bases of S_3qubits,1; 1 copy of ρ_3qubits is projected into the bases of S_3qubits,2, 1 copy of ρ_3qubits is projected into S_3qubits,3 and 499 copies of ρ_3qubits are projected into S_3qubits,4. The number of events of fidelities between estimations and target pure state in this case is denoted by pink triangle, pink dashed line is applied to connect them. The number of events of fidelities between the estimation and experiment in this case is denoted by blue triangle and blue dashed line is applied to connect them. represents fidelity distribution obtained from optimization distribution on four settings applied by 1000 copies of experimental three-qubit SC state () built by Pauli measurements. represents 363 copies of ρ_3qubits are projected into S_3qubits,1, 257 copies of ρ_3qubits is projected into S_3qubits,2, 267 copies of ρ_3qubits is projected into S_3qubits,3 and 113 copies of ρ_3qubits is projected into S_3qubits,4. Red circles are applied to denote the events of fidelities between the estimations and experiment. Red line is applied to connect them. Black squares represent the number of events of fidelity between the estimation and target pure state. Black line is applied to connect them. It is observed that performs better for fidelity estimation than . performs better than . Therefore, optimization distribution of copies of ρ_3qubits performs better than the performance of two settings.

a Continuum LDOS spatially averaged over a period of PDW (8) at low-temperatures (T=0 and T=0.04t) and pure PDW state at a high temperature (T=0.09t) obtained using parameter set doping p = 0.125, t = 400 meV, t’ = -0.3t and J = 0.3t. PDW+DSC state exhibits V-shaped nodal LDOS due to the presence of the DSC component. Pure PDW state has Bogoliubov-Fermi pockets (in contrast to nodes in PDW+DSC state), which leads to a large E = 0 LDOS. The LDOS is obtained after the effects of linear inelastic scattering is incorporated, where and α = 0.25 using the experimental fits in ref. . A non-zero LDOS at zero-bias in PDW+DSC state is a consequence of the finite artificial broadening . b q-space schematic showing the most prominent wavevectors = [(±1/8, 0); (0, ±1/8)]2π/a₀ and appearing in the Fourier transform of the mean-fields and other related quantities. c Spatial variation of hole density (δ) in PDW+DSC state (at T=0 and T=0.04t) and pure PDW state (at T=0.09t). Hole density modulates with a periodicity of 8 in PDW+DSC state due to presence of the DSC component and a periodicity of 4 in pure PDW state due to the absence of the DSC component, as expected from Ginzburg-Landau theories. d Spatial variation of d-wave gap order parameter in PDW+DSC state (at T=0 and T=0.04t) and pure PDW state (at T = 0.09t) exhibiting 8-periodic modulations corresponding to the PDW component of the gap. e Temperature evolution of the uniform () and PDW () components of the d-wave gap order parameter in PDW+DSC state (0<T<0.085t) and pure PDW state (0.085t<T<0.11t). The uniform component of the gap decreases sharply with temperature, becoming negligibly small but finite compared to the PDW component for 0.05t<T<0.085t. This ‘fragile PDW+DSC’ state is shown in white background. For T>0.085t PDW+DSC state becomes unstable and only pure PDW state (shown in pink background) exists as a stable solution of the RMFT equations. f Temperature evolution of the and components of hole density (δ) in PDW+DSC and pure PDW state in the same temperature range as in (e). The component mirrors the temperature evolution of the uniform component of the gap in panel (e), as expected from Ginzburg-Landau theories. component is the dominant component at all temperature leading to 4–periodic charge density wave.

a Predicted QPI signature of pure 8 PDW state at T = 0.09t. b Predicted of pure 4 CDW state at T = 0.09t. The CDW states show very different features compared to the PDW state. c Measured of Bi₂Sr₂CaDyCu₂O₈ (p ≈ 0.08) for the pseudogap phase at T = 1.25T_c. d Measured of Bi₂Sr₂CaDyCu₂O₈ (p ≈ 0.08) for the pseudogap phase at T = 1.5T_c. e The measurements of the pseudogap phase agree much better with the pure PDW scenario (a) than with the pure CDW (b).

(a) Density matrix of the initial state of the form . (b) Density matrix of the metamaterial distilled state. Note that the weights of the components in the distilled state are more balanced than the starting state. (c) Entanglement distillation performance of different metamaterials for a fixed non-maximally entangled pure state. Entanglement of formation (EOF) (red), fidelity (green) and purity (blue). The EOF and the fidelity of the distilled states with respect to the maximally entangled state are higher than the initial state (no array case) for all tested metamaterial nanoantenna arrays. The antenna arrays do not affect the purity of the state. (d) Entanglement distillation performance of a fixed metamaterial nanoantenna array for various non-maximally entangled pure states of the form (blue) and (red). The inset shows the fidelity of the distilled state to the maximally entangled state and respectively.