Bounding the entanglement of a state from its spectrum

Abstract

Recent efforts have focused on characterizing the set of separable states that cannot be made entangled by any global unitary transformation. Here we characterize the set of states whose entanglement content cannot be increased under any unitary. By employing linear maps (and their inverses), we derive constraints on the achievable degree of entanglement from the spectrum of the density matrix. In particular, we focus on the negativity and the Schmidt number. Our approach yields analytical and practical criteria for quantifying the entanglement content of full-rank states in arbitrary dimensions using only a subset of their eigenvalues. Moreover, some of the derived conditions can be used to bound the spectra of Schmidt number witnesses.

Figures (7)

• Figure 1: Pictorial representation of the set of quantum states bounded by a convex entanglement measure ($\text{EM}(\rho)\leq\gamma$) and of the sets with $\text{EFS}_{\gamma}$ , considering $0<\gamma_{1}<\gamma_{2}<\gamma_{\max}$. Note that when $\gamma=\gamma_{\max}$, the whole set of states is recovered.
• Figure 2: Geometry of the $\mathrm{EFS}_{\gamma}$ inner characterization using Lemma \ref{['Lemma:GeneralAlphaPM']} in the probability simplex described by the eigenvalues $\boldsymbol{\lambda}$ of the density matrix for $D=3$ in barycentric coordinates. The figure is illustrative as $D=3$ does not correspond to any bipartite splitting. $0\leq\gamma_{1}<\gamma_{2}<\gamma_{\max}$ is assumed, following Fig. \ref{['fig:Pictorial']}.
• Figure 3: Maximal negativity of PPS of Schmidt rank $\chi$ as a function of $p =NM/(NM+\alpha)$ for $N = M = 6$. The dashed lines indicate the maximal negativity for pure Schmidt rank $\chi$ states (or convex combinations of them). In black, we depict the negativity obtained for each spectrum, maximizing over a sample of $10^4$ Haar random unitaries.
• Figure 4: Maximal negativity obtained from the pectrum with $10^4$ random unitary matrices given different parametrized $2$-qubit spectra (dashed lines) along with our bound on the maximal negativity obtainable for each spectrum (solid line) following Theorem \ref{['thm:AlphaNegativity']}. The depicted spectra are of the form $(1-p/2,p/6,p/6,p/6)$ in blue, $((1-p)/2,(1-p)/2,p/2,p/2)$ in red , $(1-19p/30,p/3,p/5,p/10)$ in green and $(1-13p/15,p/3,p/3,p/5)$ in yellow.
• Figure 5: Maximal negativity compatible with $m_k$, for mixed bipartite states of local dimension $N=6\leq M$ with respect to the measure $m_k$ in Eq. \ref{['eq:PartialSums']}. Here $m_k$ is defined according to Eq. \ref{['eq:MixedMeasure']}. For each value of $k$, the horizontal dashed lines delimit the regions where the Schmidt number (SN) is above certain thresholds.

Theorems & Definitions (48)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1
• proof
• Lemma 1
• Definition 5
• Lemma 2
• proof