A resource-efficient and noise-robust entanglement witness based on the swap test

TL;DR

This work presents a resource-efficient entanglement witness based on the SWAP test that applies to arbitrary two-qubit states, providing a lower bound on the concurrence and enabling entanglement certification without full state tomography. By measuring the ancilla outcome probability $\mathds{P}(1)$, entanglement is certified when $\mathds{P}(1) > \tfrac{1}{2}$, and a convex bound on the pure-state concurrence $C(\Phi)$ is obtained via a function $f$, extendable to mixed states by convexity. The authors implement a linear photonic integrated circuit with path-encoded qubits to realize the swap test on a room-temperature chip, including a reconfigurable local unitary pre-processing stage to boost detection rates, and they analyze robustness to MMIs imperfections and phase noise. Experimental results across Bell, random, and Werner-like states demonstrate high accuracy (up to about 99%) in entanglement witnessing and partial quantification, validating the method as a practical, platform-agnostic tool for two-qubit entanglement detection. The approach also links to QRNG certification through bounds on guessing probability and min-entropy, highlighting potential applications in quantum technologies beyond fundamental tests of entanglement.

Abstract

Quantum entanglement is an essential resource for quantum technologies, and the controlled swap test provides a versatile tool for its detection and quantification. Here, we propose a SWAP-based entanglement witness that applies to arbitrary two-qubit states - both pure and mixed - and provides a lower bound on the concurrence. The method is resource-efficient, robust to noise, and platform-independent. As an example, we validate the approach on a room-temperature photonic chip, where the swap test is carried out using only linear and well-established integrated optical components. The robustness of the method against photonic-hardware noise is also analysed. Our results establish a simple and reliable tool for entanglement witnessing.

Figures (6)

• Figure 1: (a) Gate representation of the swap test algorithm performed on the states $\ket{\psi}$ and $\ket{\xi}$. The input qubits are initialised so that the first two are in the states $\ket{\psi}$ and $\ket{\xi}$ and the third one, which is an auxilliary qubit, is in the state $\ket{0}$. At the earth of the swap test there is a controlled-SWAP (CSWAP) gate, or Friedkin gate fredkin1982conservative, that swaps the two target qubits only if the control qubit is in state $\ket{1}$. The Hadamard (H) gate nielsen2010quantum is applied to the ancilla before the CSWAP gate and before the ancilla measurement. (b) The swap test circuit operating as an entanglement witness. For an arbitrary pure two-qubit input state, the probability of measuring the ancilla in $\ket{1}$ is extracted at the output. A value above $1/2$ guarantees that the state is entangled. The measured probability additionally yields a lower bound for the concurrence of the input state.
• Figure 2: Modified swap test circuit with a reconfigurable local unitary $U = U_A \otimes U_B$ applied to the input state. The circuit is run multiple times with different settings of $U$ to enhance entanglement detection.
• Figure 3: (a) Schematic gate representation of the linear photonic implementation of the swap test circuit. The H gates are implemented through MMIs, i.e. integrated beam-splitters, and the gate $\text{PS}_3$ represents the action of PSs needed to counterbalance spurious phases between the optical paths. (b) Design of the PIC that implements the swap test circuit baldazzi2024linear. Waveguides are depicted in black while the red stripes identify the sections where thermal phase shifters are present. The red arrow indicates the waveguide where light is injected. The red box contains the preparation stage, where integrated MZIs and PSs are used to encode the input state $\ket{\Phi} \otimes \ket{0}$, Eq. \ref{['eq:generic_two_qubit_state']}. The green box contains the swap stage where MMIs, CRs and PSs are exploited to implement the swap test algorithm. Referring to (a), we see the sequence of MMIs (orange), the CSWAP gate (violet), the $\text{PS}_3$ gate (white), and the second set of MMIs (orange). Finally, single photons are detected at the output.
• Figure 4: (a) Experimental results for the family of states described by Eq. \ref{['eq:psi_omega']}. Theoretical values of $\mathds{P}(1)$ are represented by blue bars connected by a light blue curve, while experimental data with error bars are shown in orange. (b) Subset of $50$ two-qubit states randomly selected from a total of $2050$. Theoretical values of $\mathds{P}(1)$ are shown in blue, and experimental results with error bars in orange. The dashed red line at $\mathds{P}(1) = 0.5$ indicates the threshold for entanglement witnessing of pure states, as established in Theorem \ref{['theo:ent_witness_pure']}. The shaded grey region between $0.5$ and $0.5085$ represents the range of $\mathds{P}(1)$ values where entanglement cannot be confidently detected due to non-idealities in the PIC (see Sec. \ref{['sec:section_3.2']}). (c) Histogram of the distance in modulus between the theoretical and experimental results for $\mathds{P}(1)$. The vertical lines indicate the mean and median. The width of the bins, computed using Scott's rule, is $0.005$, and the bins are centered around multiple values of $0.0025$. (d) Confusion matrix summarizing the performance of the circuit as a classifier. States with a theoretical $\mathds{P}(1)\in [0.5,0.5085]$ are excluded from the analysis since, even if the measured $\mathds{P}(1)$ perfectly matches the theory: entanglement cannot be confidently identified for these cases. The overall accuracy is $99.0(1)\%$.
• Figure 5: Results of the probability $\mathds{P}(1)$ for the family of Werner-like mixed states described by Eq. \ref{['eq:Werner-state']} with $\ket{\Phi}=\ket{\Psi^-}$. Real-time experimental data with error bars are shown in orange, while data obtained through the post-postprocessing procedure described in the text are displayed in blue. The black solid line represent the theoretical prediction given by Eq. \ref{['eq:P1-werner']}. A linear model $y = mx + q$ is fitted to both datasets, giving $m = 0.73(5)$ and $q = 0.24(3)$ for the real-time data, and $m = 0.742(7), \ q = 0.240(3)$ for the post-processed data.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2