Experimental Schmidt Decomposition and State Independent Entanglement Detection

We introduce an experimental procedure for the detection of quantum entanglement of an unknown quantum state with a small number of measurements. The method requires neither a priori knowledge of the state nor a shared reference frame between the observers and can thus be regarded as a perfectly state-independent entanglement witness. The scheme starts with local measurements, possibly supplemented with suitable filtering, which essentially establishes the Schmidt decomposition for pure states. Alternatively we develop a decision tree that reveals entanglement within few steps. These methods are illustrated and verified experimentally for various entangled states of two and three qubits.

Figure 1

The decision tree for efficient two-qubit entanglement detection. No shared reference frame is required between Alice and Bob; i.e., they choose their local $x$, $y$, $z$ directions randomly and independently, which effectively gives rise to a basis $\{x_A,y_A,z_A$} for Alice and $\{x_B,y_B,z_B\}$ for Bob (not detailed in the figure or the main text). The scheme starts with measuring $T_{zz}$ and follows at each step along the dashed arrow if the modulus of correlation is less than $\frac{1}{2}$ and otherwise along the continuous arrow. The algorithm succeeds as soon as $\sum _{}^{}{T}_{ij}^2>1$. The measurements in the blue shaded area suffice to detect all maximally entangled pure states with Schmidt-base vectors along $x$,$y$, or $z$.

Figure 2

Scheme of the experimental setup. The state $|\Psi ⟩=\frac{1}{\sqrt{2}}(|H⟩|H⟩+e^{i\delta }|V⟩|V⟩)$ is created by type I SPDC process. An yttrium vanadate crystal (${\mathrm{YVO}}_4$) is used to manipulate the phase $\delta$ of the prepared state. For preparation and analysis of the state, half- (HWP) and quarter-wave plates (QWP) are employed. Brewster plates (BP) can be introduced to make the state asymmetric and to perform the filter operation, respectively.

Figure 3

Demonstration of Schmidt decomposition of a maximally entangled state prepared in unknown bases. The correlation tensor and corresponding density matrix are depicted for (a) the unknown state, (b) the state after applying local filtering, and (c) the state analyzed in the Schmidt bases. It is important to note that only the blue shaded elements of the correlation tensors will be measured, as this suffices to prove entanglement. The full correlation tensors and the corresponding states are only shown for completeness and didactic reasons.

Figure 4

Correlation tensors and density matrices of the experimental realization of two different states. The imaginary parts of the density matrices are negligible and therefore skipped. Using the decision tree, only the blue shaded correlations have to be measured for detecting entanglement. The errors of the correlations are $<0.015$ for (a) and $<0.023$ for (b).

Figure 5

Density matrices of the experimental realization of the $G$ and $W$ state. The corresponding fidelities are equal to 92.23% and 89.84%.