Quantum State Tomography via Reduced Density Matrices

Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However, it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work, we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally, we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results demonstrate the advantages and possible pitfalls of quantum state tomography with local measurements.

Figure 1

Three-dimensional caricatures of the possible shapes of state space and the space of reduced density matrices as projections. Pure states are given by the extreme points. (a) A sphere, for which all boundary points are extreme points. Only the points on the boundary of the projected circle have a unique preimage in the state space, and so are UDA. All the interior points have multiple extreme points in their preimage, so they are not UDP. Thus, UDP implies UDA. (b) A polytope, for which the five vertices are extreme points. The four corner points have a unique preimage in the state space, and so are UDA. However, one interior point located at the center has multiple preimages where only one is an extreme point, so it is UDP but not UDA.

Figure 2

(a) GHZ-type states (class A) such as ${\rho }_{+}^G$ in Eq. (5) are neither UDP nor UDA. The four-qubit fidelities between ${\rho }_{+}^G$ and ${\rho }_{-}^G$ (blue) and ${\rho }_{+}^G$ and ${\rho }_{\mathrm{mix}}^G$ (yellow) are completely different, but they do have the same 2-RDMs (red and green, where the worst-case fidelity out of six possible 2-RDM fidelities is shown) up to minor experimental errors. The error bars are calculated from the imperfection of the GRAPE pulses and fitting procedure. (b) States in class C are not UDA, so there can exist mixed states between which they have very low four-qubit fidelity (red) but the same 2-RDMs (blue). However, these types of states are UDP, so there do not exist any other four-qubit pure states with the same 2-RDMs.

Figure 3

Stability test against experimental noise for $|W⟩$ and $|{\psi }_S⟩$. The noise is artificially added in Gaussian distribution to the measured 2-RDMs under experimental conditions, by randomly sampling 90 distinct sets of 2-RDMs. The arrows indicate the mean for each sampled results. (a) Fidelities of the $|W⟩$ (class B) in a noisy environment. The $x$ axis is the coefficient $a$ defined in Eq. (2). (b) Fidelities of the $|{\psi }_S⟩$ (class C) in a noisy environment, as a function of $t$ defined in Eq. (3).