Imperfect measurement settings: Implications for quantum state tomography and entanglement witnesses

Reliable and well-characterized quantum resources are indispensable ingredients in quantum information processing. Typically, in a realistic characterization of these resources, apparatuses come with intrinsic uncertainties that can manifest themselves in the form of systematic errors. While systematic errors are generally accounted for through careful calibration, the effect of remaining imperfections on the characterization of quantum resources has been largely overlooked in the literature. In this paper, we investigate the effect of systematic errors that arise from imperfect alignment of measurement bases—an error that can conceivably take place due to the limited controllability of measurement devices. We show that characterization of quantum resources using quantum state tomography or entanglement witnesses can be undermined with an amount of such imprecision that is not uncommon in laboratories. Curiously, for quantum state tomography, we find that having entanglement can help to reduce the susceptibility to this kind of error. We also briefly discuss how a given entanglement witness can be modified to incorporate the effect of such errors.

Figure 1

A schematic representation of the procedures involved in the characterization of a quantum state and how misalignment error can affect the resulting characterization. (1) The characterization protocol specifies measurement in certain bases (intended measurements). (2) Due to imperfect measurement devices, misalignment errors creep in during the experiments and the actual measurements performed differ from the intended ones. (3) The resulting measurement statistics are used to compute a given figure of merit, such as the fidelity or the expectation value of an entanglement witness. (4) Imprecision of the measurement $\varepsilon$ then translates into additional uncertainty in the final figure of merit.

Figure 2

Intended and actual measurement directions for the tomography of $|{\psi }_s\rangle$. The actual measurement directions $\left\{{\widehat{n}}_k\right\}$ given in Eq. (10) correspond to the triad of $\left\{{\widehat{m}}_k\right\}$ “opened” uniformly towards $-\widehat{s}=-\frac{1}{\sqrt{3}}(1,1,1)$ while satisfying $\arccos ({\widehat{m}}_k·{\widehat{n}}_k)=\varepsilon$.

Figure 3

Fidelity of the reconstructed state with respect to the initial state ${|{\psi }_s^{+}\rangle }^{\otimes n}$ when the actual measurement settings are those described in Fig. 2 and the intended measurement bases are those given in Eq. (3). The vertical dotted line corresponds to $\varepsilon =2^{\circ }$ as discussed in the text.

Figure 4

Susceptibility of two-qubit pure state $|{\psi }_{\alpha }\rangle$ to misalignment error $\varepsilon$ as a function of the entanglement present in $|{\psi }_{\alpha }\rangle$ (parametrized by the concurrence [24] of $|{\psi }_{\alpha }\rangle$). For given $\alpha$, $\mathcal{S}\left(\alpha \right)$ gives the rate of decrease of the fidelity with respect to $\varepsilon$ as $\varepsilon \to 0$; $\mathcal{S}\left(0\right)$ is the corresponding initial slope for the pure product state. The 257 numerical data points obtained from $4\times {10}^4$ optimizations are plotted in a solid line. The dashed lines represent segments of the plot that can be very well approximated using the explicit parametrizations given in Appendix 6c.

Figure 5

A schematic diagram showing the effect of misalignment error on the evaluation of an entanglement witness. An optimized entanglement witness $\mathcal{W}$ is tangent to the boundary of the set of biseparable states. (a) When evaluated using misaligned measurements, the witness can cross the boundary (as ${\mathcal{W}}^{\varepsilon }$) and thus some biseparable states may appear to be genuinely $n$-partite entangled. (b) To correct the problem, one can evaluate the correction factor $w\left(\varepsilon \right)$ such that no biseparable state gives a false positive result.

Figure 6

The worst expectation value of ${\mathcal{W}}^{\varepsilon }$ found using biseparable states and systematic error $\varepsilon \le {10}^{\circ }$. When $\varepsilon =0$, ${\mathcal{W}}^{\varepsilon }$ reduces to ${\mathcal{W}}_{|\text{GHZ}\rangle }$ given in Eq. (32). The curves for $n\ge 4$ are computed using Eqs. (33) and (37), whereas the curve for $n=3$ has been obtained numerically.

Figure 7

Evaluating the entanglement witness ${\mathcal{W}}_{|\text{GHZ}\rangle }$ with the intended measurement settings $M_k^{\left(j\right)}$ on the $XY$ plane (drawn here for $n=4$ parties). These settings correspond to equally spaced measurement directions ${\widehat{m}}_k^{\left(j\right)}$ on the $XY$ plane, with neighboring ones separated by an angle $\frac{k\pi }{n}$. The systematic errors considered here correspond to having these rays moving in pairs towards each other as $\varepsilon$ increases. Measurement directions are deviated according to the black arrows for qubits numbered $j=1,2$ and to the gray arrows for $j=3,4$. Measurement in the ${\sigma }_z$ basis is assumed to stay unperturbed.

Figure 8

A schematic view of the set $\mathcal{C}$ of vector $\vec{c}$ compatible with an actual state $\tau \left(\vec{t}\right)$ measured with maximal misalignment error $\varepsilon$; the set $\mathcal{C}$ is shaded in this two-dimensional projection. In our proof, we work with its enclosing sphere $\mathcal{B}(\vec{t},t\lambda )$ whose boundary is marked with the dashed line.

Figure 9

Reconstructed state $\vec{r}{}^{*}$ using MLE, compatible with outcome data $\vec{c}$ outside the Bloch sphere. We show in the text that $\vec{c}$ is contained inside a ball of radius $t\lambda$ centered at $\vec{t}$ (dashed line). The reconstructed Bloch vector $\vec{r}{}^{*}$ lies on the surface of the Bloch sphere. Moreover, since the function $\mathcal{L}\left(\vec{r}\right)$ is strictly concave—schematic representation of the contour lines of $\mathcal{L}\left(\vec{r}\right)$ are plotted using dots—$\vec{r}{}^{*}$ is constrained on a spherical cap such that the line segment (drawn with a double edge) between $\vec{r}{}^{*}$ and $\vec{c}$ does not cross the boundary of the Bloch sphere. The maximal angle $\theta$ between vectors $\vec{t}$ and $\vec{r}{}^{*}$ is then obtained (see text) when the mentioned line segment is tangent to both balls.

Figure 10

Loss of fidelity in the (numerically determined) worst-case scenario for low-entanglement states, for small $\varepsilon =\frac{\pi }{180}$. The normalized loss of fidelity is plotted as a solid line, contribution from the marginal terms $\mathrm{tr}\left[{\rho }_s({\rho }_s-{\tau }_s)\right]$, and the rest are plotted as dashed lines.

Figure 11

Comparison of the loss of fidelity, as defined in Eq. (A24) and for $\varepsilon =\frac{\pi }{180}$, in the (numerically determined) worst-case scenario with correlated and local systematic errors. This graph is comparable at first order to the one in Fig. 4.