Entanglement Meter: Estimation of entanglement with single copy in Interferometer

Technical Summary: Entanglement Meter: Estimation of entanglement with single copy in Interferometer

Problem Statement
The efficient experimental certification and quantification of high-dimensional entanglement in composite systems remain significant challenges. Standard methods, such as quantum state tomography, require estimating a large number of independent parameters, necessitating numerous measurements and complex classical post-processing. Furthermore, evaluating entanglement monotones from a known density matrix is computationally NP-hard as dimensionality increases. Existing proposals for measuring entanglement often rely on multiple copies of the input state or averaging over random unitary operations, which can be resource-intensive. The authors aim to address these limitations by exploring whether a Mach-Zehnder interferometer can be used to design an "entanglement meter" capable of detecting and estimating entanglement directly from interference patterns using a single copy of the input state, without full state tomography.

Methodology
The paper utilizes a Mach-Zehnder interferometric setup where the control qubit (the interferometric arm) governs a controlled-unitary operation acting on a target quantum system described by a density operator $\rho$. The evolution involves a sequence of Hadamard gates, a phase shift, and a controlled-unitary $U$. The intensity pattern at the output yields a visibility $V = |\text{tr}(U\rho)|$ and a phase shift $\alpha = \arg[\text{tr}(U\rho)]$.

The authors propose specific choices for the unitary operator $U$ and the input state configuration to extract entanglement measures:

1. Linear Entanglement Entropy (Pure States):

   • Setup: A single copy of a reduced subsystem $\rho_A$ is fed into the interferometer alongside a maximally mixed state $I_B/d$ on the other arm.
   • Unitary: An oracle unitary $U = (I - 2P)$ is applied, where $P = |\Psi\rangle_{AB}\langle\Psi|$ is the projector of the unknown pure bipartite state.
   • Mechanism: The interference visibility is shown to be directly related to the purity of the reduced state, $\text{tr}(\rho_A^2)$. For two-qubit pure states, the visibility $V$ is equal to the linear entanglement entropy $E(|\Psi\rangle)$.
   • Extension to Mixed States: Using the convex roof construction, the authors derive an inequality showing that the visibility provides an upper bound for the convex roof extended linear entropy of arbitrary mixed states.

2. Negativity Measure:

   • Setup: A single copy of the joint state $\rho_{AB}$ is used.
   • Unitary: A specific unitary $U = e^{i\theta X_A} \otimes e^{i\theta X_B}$ is applied, where $X$ is a Hermitian operator constructed from off-diagonal basis elements.
   • Mechanism: For pure two-qubit states, the visibility is directly proportional to the negativity. For higher dimensions ($d > 2$), the authors derive a relationship between visibility and negativity using the properties of the operator $X$.
   • Mixed States: The method is shown to yield the exact negativity for a specific class of mixed states (mixtures of maximally entangled states and classical-classical states), whereas the linear entropy method only provides an upper bound for these cases.

3. Mutual Predictability and Entanglement Witnesses:

   • Mutual Predictability: Instead of performing local measurements in mutually unbiased bases (MUBs), the authors construct a unitary operator $U_{O_A, O_B}$ that acts as a projector onto the diagonal elements in a specific basis. The visibility of the interference pattern yields the mutual predictability value directly. Violation of a separability bound on the sum of visibilities indicates entanglement.
   • Entanglement Witness (SWAP Operator): The authors propose measuring the expectation value of the SWAP operator (an entanglement witness) using the interferometer. By setting the unitary $U$ to be the SWAP operator (or a function thereof), the phase shift of the interference pattern becomes sensitive to the sign of $\text{tr}(W\rho)$. A phase shift of $\pi$ indicates a negative expectation value, signaling an entangled state (specifically demonstrated for Werner states).

Key Results

• Single-Copy Efficiency: The paper demonstrates that linear entropy and negativity can be measured using only a single copy of the input state (either reduced or joint), eliminating the need for the two copies required by previous quantum network proposals.
• Direct Visibility Measures: For any two-qubit pure bipartite state, the interference visibility is a direct measure of entanglement (linear entropy or negativity).
• Phase Sensitivity: The phase shift in the interferometer serves as a signature for separability. Specifically, for the SWAP operator witness, entangled states induce a phase shift of $\pi$ relative to separable states.
• No Local MUB Measurements: The mutual predictability, a necessary and sufficient criterion for separability in certain dimensions, can be extracted from intensity patterns without performing explicit local measurements in mutually unbiased bases.
• Resource Economy: The proposed methods avoid full quantum state tomography and the computational complexity associated with evaluating entanglement monotones from a reconstructed density matrix.

Significance and Claims
The authors claim that their proposals demonstrate the power of interferometric setups for the efficient detection and quantification of entanglement in both pure and mixed states. The primary significance lies in the potential design of a portable "entanglement meter." Such a device would allow users to verify the presence and amount of entanglement in a pair of systems by simply observing the interference visibility or phase shift, without requiring complex state reconstruction or multiple copies of the state. The paper suggests this could have applications in quantum technology, such as verifying the quality of entangled pairs supplied by manufacturers. The authors maintain that these results can be tested with existing technology and pave the way for future generalizations to multipartite systems.