Pure-state tomography with the expectation value of Pauli operators

We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary $n$-qubit pure state among all quantum states. We show that only 11 Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with 16 measurements, and only 31 Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with 64 measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure-state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.

Figure 1

Average fidelity of reconstructed density matrices compared to the ideal state using an optimum Pauli measurement set for 3 qubits. The error bars are given by standard deviation of the said fidelity over 100 instances. For small noise $\eta$, the state is very close to pure, and the protocol returns a high-fidelity density-matrix reconstruction. As noise $\eta$ increases, the pure-state assumption becomes less useful, which yields a low-fidelity estimation.

Figure 2

Molecular structure of (a) two-qubit sample ${}^{13}$ C-labeled chloroform and (b) three-qubit sample diethyl-fluoromalonate. The corresponding tables on the right side summarize the relevant NMR parameters at room temperature, including the Larmor frequencies (diagonal, in hertz), the $J$-coupling constant (off-diagonal, in hertz), and the relaxation time scales $T_1$ and $T_2$.

Figure 3

Reconstruction of density matrix for state number one. The upper two figures are real and imaginary part of density matrix of state reconstruction using all 16 Pauli measurements. The bottom two figures are real and imaginary part of density matrix of state reconstruction using 11 optimum Pauli measurements described earlier. The fidelity between the two density matrices is 0.992.

Figure 4

Performance of two-qubit protocol using selected Pauli measurements against randomly Pauli measurements. The blue diamond dots are the fidelity between density matrix reconstructed from all 16 Pauli measurements with density matrix reconstructed from random 11 Pauli measurements. The red line represents fidelity of reconstruction using our protocol, and the green dashed line shows the purity of density matrix reconstructed from all Pauli measurements.

Figure 5

(a) General scheme to create the GHZ state via global controls. $\mathrm{X}(\theta$) and $\mathrm{Y}(\theta )$ are, respectively, the global rotations with $\theta$ angle along $x$ and $y$ directions, and $\mathrm{ZZ}(\tau$) denotes a free evolution with the $\tau$ time under the model Ising Hamiltonian. (b) NMR sequence to realize the GHZ state creation from the PPS. Blue and red rectangles represent $\pi /2$ and $\pi$ rotations, respectively. The evolution times are $t_1=6.76$ ms, $t_2=6.49$ ms, and $t_3=2.84$ ms with our sample.

Figure 6

Reconstruction of density matrix for GHZ state. The upper two figures are real and imaginary part of density matrix of state reconstruction using all 64 Pauli measurements. The bottom two figures are real and imaginary part of density matrix of state reconstruction using 31 optimum Pauli measurements described in Eq. (6). The fidelity between the two density matrices is 0.960.

Figure 7

Performance of three-qubit protocol using selected Pauli measurements against randomly Pauli measurements. Blue dots represent fidelity between density matrix reconstructed from all 64 Pauli measurements and density matrix reconstructed from random 31 Pauli measurements. The red square represents fidelity of reconstruction using our protocol.

Figure 8

Measurement scheme for a polarization-encoded $n$-photon state. The $n$-qubit state is encoded in the polarizations of the $n$ photons. Each photon is measured using a quarter-waveplate (QWP), half-waveplate (HWP), and a polarizing beamsplitter (PBS) with a single-photon counting detector (SPD) at each of its output ports. The quarter- and half-waveplates are rotated to choose the measurement basis for each photon. Separable projective measurements are performed by counting coincident detection events between all $n$ photons.

Figure 9

Fidelity of the different reconstructions against the state intended to be prepared. The green dashed line show the fidelity between the aim pure state and the reconstructed state by 16 Pauli measurements, and the blue dash-dotted line shows the fidelity between the aim pure state and the reconstructed state by 11 Pauli measurements in our protocol. Both of them decrease when the purity is low, as our truly prepared state becomes more and more mixed along with the long evolution times. For low purities, our protocol is also worse than full-state tomography.