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Entanglement and distillation from symmetric positive maps

Albert Rico

TL;DR

This work develops a symmetric framework for entanglement detection by building positive maps and witnesses from immanant inequalities, unifying reduction-type criteria with a broader class of multilinear constructions. Through the Choi–Jamiołkowski isomorphism and Young projectors, it yields both linear and multicopy entanglement witnesses, extendable to multipartite settings and adaptable via local filtering with operators E. A key contribution is the generalized maps Ψ_{ar{a}}^{E}, their extension to random-state detection, and the state–witness contraction method that produces highly symmetric witnesses capable of detecting entanglement even in states with local PPT, including multicopy scenarios. The results show that projecting onto low-dimensional subspaces and combining maps can substantially enhance detection power, potentially enabling distillation and broader PPT-compliant entanglement detection, with practical implications for randomized measurement schemes and symmetry-inspired entanglement testing.

Abstract

Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.

Entanglement and distillation from symmetric positive maps

TL;DR

This work develops a symmetric framework for entanglement detection by building positive maps and witnesses from immanant inequalities, unifying reduction-type criteria with a broader class of multilinear constructions. Through the Choi–Jamiołkowski isomorphism and Young projectors, it yields both linear and multicopy entanglement witnesses, extendable to multipartite settings and adaptable via local filtering with operators E. A key contribution is the generalized maps Ψ_{ar{a}}^{E}, their extension to random-state detection, and the state–witness contraction method that produces highly symmetric witnesses capable of detecting entanglement even in states with local PPT, including multicopy scenarios. The results show that projecting onto low-dimensional subspaces and combining maps can substantially enhance detection power, potentially enabling distillation and broader PPT-compliant entanglement detection, with practical implications for randomized measurement schemes and symmetry-inspired entanglement testing.

Abstract

Recently, a toolkit of highly symmetric techniques employing matrix inequalities has been developed to detect entanglement in various ways. Here we unifiedly explain in detail these methods, and expand them to a new family of positive maps with further detection capabilities. In the simplest case, we generalize the reduction map to detect more generic states using both multiple copies and local filters. Through the Choi-Jamiołkowski isomorphism, this family of maps leads to a construction of multipartite entanglement witnesses. Discussions and examples are provided regarding the detection of states with local positive partial transposition and the use of multiple copies.

Paper Structure

This paper contains 21 sections, 3 theorems, 62 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entanglement detection: positive maps and witnesses
  4. Immanants
  5. Immanant inequalities
  6. Entanglement witnesses from immanant inequalities
  7. Positive maps from immanant inequalities
  8. Extension and combination of multilinear positive maps
  9. Extension with local degrees of freedom
  10. Detection of random states
  11. Combination of multilinear maps
  12. New witnesses from state-witness contractions
  13. New multipartite witnesses
  14. Detection of states with local positive partial transpositions
  15. Detection through multicopy witnesses
  16. ...and 6 more sections

Key Result

Proposition 1

Let the immanant inequality hold for any $k\times k$ positive matrix $G\geq 0$, and let $E,\varrho_1,...,\varrho_{k-1}$ be positive matrices of size $d\times d$. The following multilinear map from $\mathds{C}^{d^{k-1}}\times \mathds{C}^{d^{k-1}}$ to $\mathds{C}^d\times\mathds{C}^d$, is positive, namely $\Psi_{\Vec{a}}^{E}(\varrho_1\otimes...\otimes\varrho_{k-1})\geq 0$ when $\varrho_i\geq 0\,\,\

Figures (4)

  • Figure 1: Detection of two-qutrit random states by $\Psi_{(0,1)}^{E}\otimes\mathop{\mathrm{id}}\nolimits(\varrho_{AB})\not\geq 0$ with $E=\mathop{\mathrm{diag}}\nolimits(a,b,c)$. The axes parametrize $a$, $b$ and $c$ as $u:=(a-b+1)/2$ and $v:=c$ in order to plot the results in the probability simplex of size $3$, and the color bar indicates the percentage of Ginibre random bipartite states ginibre1965 detected with in a sample of size 2000. At the baricenter of the triangle lays the case $E=\mathop{\mathrm{diag}}\nolimits(1,1,1)=\mathds{1}$, which evaluates the reduction map \ref{['eq:RedMap']} and detects $27\%$ of the random states approximately. The edges contain the case $E=\mathop{\mathrm{diag}}\nolimits(1,1,0)$, which evaluates the reduction map on a two-dimensional subspace \ref{['eq:RedMapProj2dim']} and increases the detection rate to $74\%$.
  • Figure 2: Entanglement detection scheme proposed in this work. The method employed here works in the following steps for $n=2$ matrix inequalities, whose extensions to an arbitrary number $n$ are straightforward. (1) Chose two immanant inequalities with coefficients $\{a_\lambda\}$ and $\{b_\lambda\}$ given by $\vec{a}$ and $\vec{b}$, acting on matrices of size $k$ and $k'$. (2) Construct the corresponding $k$- and $k'$-partite witnesses or positive operators $W_{\vec{a}}=E^{\otimes k}\sum_\lambda a_\lambda P_\lambda$ and $W_{\vec{b}}=F^{\otimes k'}\sum_\lambda b_\lambda P_\lambda$, in the spirit of MaassenSlides. Following Eq. \ref{['eq:CombineMaps']}, these will be used as Jamiołkowski matrices jamiolkowski1972linear of positive maps $\Psi_{\vec{a}}:\mathcal{L}(\mathds{C}^d)^{\otimes k-1}\rightarrow \mathcal{L}(\mathds{C}^d)$ and $\Psi_{\vec{b}}:\mathcal{L}(\mathds{C}^d)^{\otimes k'-1}\rightarrow \mathcal{L}(\mathds{C}^d)$ constructed in Proposition \ref{['prop:GeneralMap']}, where we set $k\geq k'$. In their joint input space we place a $(k+k'-2)$-partite quantum state $\varrho$ to be detected. (3) If such a state is separable (a convex combination of product states $\varrho_{A_1}\otimes...\otimes\varrho_{A_{k-1}}\otimes\varrho_{B_1}\otimes...\otimes\varrho_{B_{k'-1}}$), the joint action of the two maps outputs a product of positive matrices $\Psi_{\vec{a}}\otimes\Psi_{\vec{b}}$. (4) Positivity of the output is then evaluated with a $(n=2)$-partite state or witness $\tau$, that detects entanglement in $\varrho$ if $\mathop{\mathrm{tr}}\nolimits(\Psi_{\vec{a}}\otimes\Psi_{\vec{b}}\cdot\tau)<0$. In the dual formulation of this procedure, $\mathcal{W}_\tau=\mathop{\mathrm{tr}}\nolimits_{A_kB_{k'}}(W_{\vec{a}}\otimes W_{\vec{b}}(\mathds{1}\otimes\tau))$ is a witness for $\varrho$ ( state-witness contraction technique of short).
  • Figure 3: Nonlinear entanglement detection with the action of $\Psi^{\mathds{1}}_{(0,...,0,1)}$ on the party $A$ of $k$ copies of a bipartite state $\varrho_{AB}$, namely Eq. \ref{['eq:REdmapkcopies']}. A state is detected if the resulting operator acting on $B_1...B_{k-1}A_k$ is not positive semidefinite, see Eq. \ref{['eq:REdCritkcopies']}. The case $k=2$ recovers the (single-copy) reduction criterion \ref{['eq:RedCrit']}, whose detection capabilities on generic states are significantly improved by the (two-copy) criterion obtained for $k=3$ (Obs. \ref{['obs:MultCopyGenRMapGinibre']}).
  • Figure 4: Construction of a four-partite witness from two matrix inequalities. (a) Consider two immanant inequalities on $3\times 3$ positive semidefinite matrices $G$. (b) Each inequality can be used to construct a three-partite witness $W$ or positive operator $M$MaassenSlides. (c) Take the tensor product $W_A\otimes M_B$, which can be used for example as a witness on three copies $\varrho_{A_1B_1}\otimes\varrho_{A_2B_2}\otimes\varrho_{A_3B_3}$TP_Rico24. (d) Here we contract the last subsystem with a Bell state $\phi^+=| \phi^+\rangle \langle \phi^+|$ shared between parties $A_3$ and $B_3$, as displayed in Eqs. \ref{['eq:WitGenkn']} and \ref{['eq:Wit4partNonDeco']}. Proposition \ref{['prop:WitPartContr']} shows that we are left with a four-partite entanglement witness $\mathcal{W}_\tau$ in the subsystems $A_1A_2B_1B_2$, and Observation \ref{['obs:4PartWitDetBoundEnt']} shows that it can detect states with local positive partial transpositions.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 4
  • proof
  • Corollary 1
  • proof
  • proof