[Paper Review] All bipartitions of arbitrary Dicke states

George Moreno, Fernando Parisio|arXiv (Cornell University)|Jan 2, 2018

Quantum Information and Cryptography3 citations

TL;DR

This paper derives closed-form analytical expressions for the Schmidt decompositions of all possible bipartitions of arbitrary Dicke states using their permutation symmetry. It computes the entanglement entropy for all such partitions and reveals that W states maintain constant entanglement under increasing qubit number, while the potential for multipartite entanglement decays as $ n^{-1/2} $. The study further introduces a family of entanglement witnesses and shows a trade-off between random and systematic errors in detecting entanglement.

ABSTRACT

By exploiting the permutation symmetry of Dick states, we derive closed analytical expressions of Schmidt decompositions for {\it all} possible bipartitions of a system described by this kind of state. This allows us to exhaustively compute the entropy of entanglement of the bipartitions and, thus, compare the their entanglement extent. We also address the multipartite character of Dicke states by calculating the purity of balanced bipartitions to determine the potential of multipartite entanglement (the average purity). In particular, we found that the entanglement of $W$ states remains constant as the number of qubits is increased. As a final application we define a family of multipartite entanglement witnesses and compute their resistance against random and systematic imperfections. It is shown that in some circumstances, for a fixed white noise fraction, the entanglement becomes detectable only if one {\it increases} the amount of systematic imperfection in the state.

Motivation & Objective

• To systematically analyze the entanglement structure of all possible bipartitions of arbitrary Dicke states using their permutation symmetry.
• To compute the entropy of entanglement for every bipartition and compare their entanglement extent.
• To quantify the potential for multipartite entanglement via the average purity of balanced bipartitions.
• To design and evaluate a family of entanglement witnesses for experimental robustness against random and systematic imperfections.

Proposed method

• Leveraging the permutation symmetry of Dicke states, the authors derive a recursive decomposition that expresses any Dicke state as a superposition of states on a single qubit and the remaining $(n-1)$ qubits.
• The recursive structure yields a Schmidt decomposition for any $(1|n-1)$ bipartition, enabling direct computation of the entanglement entropy via the von Neumann entropy of the reduced density matrix.
• The method is generalized to arbitrary bipartitions through a combinatorial identity proven by finite induction, allowing analytical treatment of all $\binom{n}{k}$-symmetric partitions.
• The potential for multipartite entanglement is quantified using the average purity of reduced density matrices from balanced bipartitions.
• A family of entanglement witnesses is constructed based on the state's symmetry and measured for resistance to white noise and systematic errors.
• Numerical and analytical analysis reveals a non-monotonic detection threshold: for fixed white noise, systematic errors must increase to restore detectability.

Experimental results

Research questions

• RQ1 What is the entanglement entropy for every possible bipartition of an arbitrary Dicke state, and how does it vary with system size and excitation number?
• RQ2 Does the entanglement of W states ($k=1$) remain constant as the number of qubits increases, and how does this compare to other Dicke states?
• RQ3 How does the potential for multipartite entanglement—measured by the average purity of balanced bipartitions—scale with the number of qubits?
• RQ4 What is the trade-off between random and systematic preparation errors in detecting entanglement using entanglement witnesses?
• RQ5 Can the proposed entanglement witnesses be experimentally realized and how robust are they under realistic noise models?

Key findings

• The entanglement entropy of a $(1|n-1)$ bipartition of a Dicke state $|D_n^{(k)}\rangle$ is given by $ S(n,k,1) = -\left(\frac{n-k}{n}\right)\log\left(\frac{n-k}{n}\right) - \left(\frac{k}{n}\right)\log\left(\frac{k}{n}\right) $, maximized at $k = n/2$ for even $n$.
• For W states ($k=1$), the entanglement entropy remains constant at $S(n,1,1) = -\frac{n-1}{n}\log\left(\frac{n-1}{n}\right)$, approaching 1 ebit as $n \to \infty$, indicating persistent bipartite entanglement.
• The potential for multipartite entanglement, measured by the average purity of balanced bipartitions, decays as $n^{-1/2}$ for large $n$, significantly above the $2^{-n/2}$ algebraic minimum.
• A family of entanglement witnesses is constructed that are particularly suitable for experimental implementation, especially those resembling the form in Bourennane et al. (2004).
• A counterintuitive trade-off is found: for a fixed white noise fraction, entanglement becomes detectable only when systematic errors are increased, indicating non-monotonic robustness.
• The analytical derivation of the Schmidt decomposition for all bipartitions is achieved via a finite induction proof of a generalized decomposition identity, confirming the recursive structure of Dicke states.