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Positive maps from the walled Brauer algebra

Maria Balanzó-Juandó, Michał Studziński, Felix Huber

Abstract

We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.

Positive maps from the walled Brauer algebra

Abstract

We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements from the walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.
Paper Structure (17 sections, 10 theorems, 88 equations, 9 figures, 1 table)

This paper contains 17 sections, 10 theorems, 88 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Matrix operations
  4. Entanglement
  5. Walled Brauer algebra
  6. Tripartite scenarios
  7. The Bardet-Collins-Sapra map
  8. Eggeling-Werner maps
  9. Multilinear maps and blockpositivity
  10. From $k$ to $1$ registers
  11. From $1$ to $k-1$ registers
  12. Maps and matrix functions from irreps of $U^{\otimes (n-k)}\otimes \bar{U}^{\otimes k}$
  13. Conclusions
  14. Primer on representation theory of the symmetric group and the walled Brauer algebra
  15. Eggeling-Werner maps
  16. ...and 2 more sections

Key Result

Lemma 1

Let $\varrho$ be a $U^{\otimes 3}$-invariant state with $r_k=\mathop{\mathrm{tr}}\nolimits(\varrho R_k)$, $k=+,-,1,2,3.$ Then $\varrho^{T_1}\geq 0$ if and only if where $F_1=(1-r_1-5r_--r_+)(-1-r_1+r_-+5r_+)/3$ and $F_2=(1-r_1-r_--r_+)(1+r_1-r_--r_+)/3$.

Figures (9)

  • Figure 1: Separability classes of all tripartite states. The set of fully separable states $\text{SEP}$ (blue) is strictly included in the union of the biseparable sets $A|BC\,,B|AC\,,C|AB$. Thus there are states that are biseparable for every possible bipartition, but which are not fully separable Bennett99Acin2001 (shaded region). States belonging to the convex hull of $A|BC\,,B|AC\,,C|AB$ are called biseparable, $\text{BISEP}$.
  • Figure 2: Graphical representation of the permutation operator $(1234)$ --left-- and the representation of the partially transposed permutation operator $(1234)^{T_4}$ --right--. The vertical dotted line depicts the 'wall' in the walled Brauer algebra. This wall separates transposed elements from the untouched ones. For more details on this graphical representations see Appendix \ref{['app:graphicaltrick']}.
  • Figure 3: The striped region represents the pairs of $(\alpha,\beta)$ for which $P(\alpha,\beta)$ of Eq. \ref{['P(a,b)']} is a witness of entanglement in the bipartition $1|23$. That is, these are the points for which the map is positive according to condition \ref{['itm:secondcondition']} of Bardet2020, and for which the operator $P(\alpha,\beta)$ is not positive semidefinite.
  • Figure 4: Graphical representation of the reshuffling of sites $k,l$. The action of the reshuffling operation interchanges the $k$'th "ket" leg with the $l$'th "bra" leg.
  • Figure 5: Graphical representation of Example \ref{['example11']}. The dashed line represents the partial trace.
  • ...and 4 more figures

Theorems & Definitions (23)

  • Lemma 1: Lemma 10 of Ref. Eggeling2001
  • Remark 2
  • Example 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Example 6: $3\to 2$ tensor factors map
  • Proposition 7
  • Definition 8
  • ...and 13 more