Comparing the three W-like states with the W state

Technical Summary: Comparing Three W-like States with the W State

Problem Statement
While W-like states (of the form $\gamma|001\rangle + \beta|010\rangle + \alpha|100\rangle$) have been extensively utilized in quantum communication schemes, including distillation, teleportation, and superdense coding, a direct comparative analysis between specific W-like states and the canonical W state has been absent. The W state is known for its robustness against particle loss (remaining entangled when one qubit is traced out), unlike the GHZ state. However, it is not a maximally entangled state under local unitary (LU) operations, nor does it possess maximal von Neumann entanglement entropy (vNEE) for single-qubit reductions. This paper investigates whether specific W-like states can exhibit superior entanglement properties compared to the W state, specifically regarding maximal vNEE, tangles, and suitability for perfect quantum protocols.

Methodology
The author, Dafa Li, focuses on three specific W-like states denoted as $|\vartheta'\rangle$, $|\eta'\rangle$, and $|\xi'\rangle$:
$$|\vartheta'\rangle = \frac{1}{2}|001\rangle + \frac{1}{2}|010\rangle + \frac{1}{\sqrt{2}}|100\rangle$$
$$|\eta'\rangle = \frac{1}{2}|001\rangle + \frac{1}{\sqrt{2}}|010\rangle + \frac{1}{2}|100\rangle$$
$$|\xi'\rangle = \frac{1}{\sqrt{2}}|001\rangle + \frac{1}{2}|010\rangle + \frac{1}{2}|100\rangle$$

The analysis proceeds through the following steps:

1. Schmidt Decomposition (SD): The states are transformed into their LU-equivalent Schmidt decomposition forms ($|\vartheta\rangle$, $|\eta\rangle$, and $|\xi\rangle$) to facilitate calculation. For instance, $|\vartheta\rangle = \frac{1}{\sqrt{2}}|000\rangle + \frac{1}{2}|101\rangle + \frac{1}{2}|110\rangle$.
2. Entanglement Measures: Four specific measures are calculated and compared across the W state, the GHZ state, and the three new W-like states:
   • 2-tangle ($\tau_{xy}$): Measuring bipartite entanglement between two qubits.
   • Negativity ($N_{xy}$): Derived from the partial transpose of the reduced density matrix.
   • von Neumann Entanglement Entropy (vNEE, $S(\rho_x)$): Measuring the entanglement of a single qubit with the rest of the system.
   • Tangle ($\tau_{x(yz)}$): Measuring the entanglement between one qubit and the remaining pair.
3. Optimization Analysis: The paper derives conditions under which a W-class state achieves maximal vNEE ($S(\rho_x) = \ln 2$) for a specific qubit and proves that the constructed states are unique maximizers of average entanglement measures within those constrained sets.
4. Protocol Verification: The suitability of these states for perfect quantum teleportation and superdense coding is demonstrated via explicit protocol derivations, contrasting them with the known limitations of the W state.

Key Results

• Maximal vNEE and Tangle: The paper proves that $|\vartheta\rangle$ (and its LU equivalent $|\vartheta'\rangle$) is the unique state within the set of W-class states having $S(\rho_A) = \ln 2$ that maximizes the average 2-tangle ($A\tau_{xy}$), average vNEE ($Avnee$), average tangle ($A\tau_{x(yz)}$), and average negativity ($Aneg$). Analogous results hold for $|\eta\rangle$ (maximizing properties relative to qubit B) and $|\xi\rangle$ (relative to qubit C).
• Comparison with the W State:
  • Single-Qubit Mixedness: For the W state, the reduced density matrix of any single qubit is not maximally mixed ($S(\rho) \approx 0.637 < \ln 2$). In contrast, for $|\vartheta'\rangle$, $|\eta'\rangle$, and $|\xi'\rangle$, the reduced density matrix of one specific qubit is maximally mixed ($S(\rho) = \ln 2$).
  • Robustness against Particle Loss: When one qubit is traced out, the remaining two-qubit system of the W-like states exhibits higher entanglement than that of the W state. Specifically, for $|\vartheta'\rangle$, tracing out qubit B or C leaves the remaining pair with a 2-tangle of $1/2$ and negativity of $1/4$, whereas the W state yields a 2-tangle of $4/9$ and negativity of $(\sqrt{5}-1)/6$. Since $1/2 > 4/9$ and $1/4 > (\sqrt{5}-1)/6$, the W-like states are more robust against specific particle losses.
  • Average Measures: The average entanglement measures ($A\tau_{xy}$, $Avnee$, $A\tau_{x(yz)}$, $Aneg$) for the W state and the three W-like states are found to be nearly identical, though the W-like states achieve specific maximal values for individual qubit reductions.
• Global Entanglement: The global entanglement measures (Meyer-Wallach $E_{MW}$ and average determinant $E_{AD}$) are calculated as $5/8$ for the W-like states, compared to $8/9$ for the W state.

Significance and Claims
The paper claims that the three W-like states ($|\vartheta'\rangle$, $|\eta'\rangle$, $|\xi'\rangle$) offer distinct advantages over the standard W state in specific quantum information processing tasks:

1. Perfect Teleportation and Superdense Coding: The author demonstrates that while the W state cannot be used for perfect teleportation or superdense coding, the states $|\vartheta'\rangle$, $|\eta'\rangle$, and $|\xi'\rangle$ (and their SD forms) can serve as perfect resources for these protocols. Explicit derivations show how Alice and Bob can achieve perfect state transfer or transmit 2 classical bits via 1 qubit using these states.
2. Enhanced Robustness: Due to the higher entanglement remaining in the two-qubit subsystem after the loss of a specific particle, these states are claimed to be more suitable for transmitting information over noisy quantum channels and for reliable quantum communication networks.
3. Maximal Mixedness: The ability of these states to render a specific single-qubit reduced state maximally mixed (unlike the W state) is highlighted as a key feature distinguishing them within the W SLOCC class.

The paper concludes that these W-like states possess higher robustness against specific particle loss and maximal mixedness properties that make them superior to the W state for perfect teleportation and superdense coding, despite having similar average global entanglement measures.