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State-witness contraction

Albert Rico

Abstract

We construct both nonlinear and linear entanglement witnesses, by tensoring and partial tracing existing states and witnesses. We show that little shared quantum resources allow to employ decomposable witnesses to obtain larger ones detecting locally undistillable states, and find analytic witnesses for multipartite entangled states that are undistillable across all bipartitions. Crucially, we show how to efficiently optimize multicopy witnesses to desired states, which is a current challenge. As an example of both theoretical and experimental interest, existing trace polynomial witnesses are significantly improved while preserving their symmetries and implementability with randomized measurements. Besides detecting entanglement, we also find new witnesses detecting $k$-copy distillability. A recipe for the single-copy case is shown to be effective for generic and Werner states.

State-witness contraction

Abstract

We construct both nonlinear and linear entanglement witnesses, by tensoring and partial tracing existing states and witnesses. We show that little shared quantum resources allow to employ decomposable witnesses to obtain larger ones detecting locally undistillable states, and find analytic witnesses for multipartite entangled states that are undistillable across all bipartitions. Crucially, we show how to efficiently optimize multicopy witnesses to desired states, which is a current challenge. As an example of both theoretical and experimental interest, existing trace polynomial witnesses are significantly improved while preserving their symmetries and implementability with randomized measurements. Besides detecting entanglement, we also find new witnesses detecting -copy distillability. A recipe for the single-copy case is shown to be effective for generic and Werner states.

Paper Structure

This paper contains 2 sections, 2 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Examples of Young projectors
  2. New witnesses for MPPT states

Key Result

Proposition 1

Let $\{R^{(j)}\}_{j=1}^n$ be states or witnesses in $k_j$-partite systems $S^j$. Let $\tau_{S^1_{k_1}...S^n_{k_n}}$ be a $n$-partite state or witness shared between the last parties of all systems $S^j$. Then, the operator has nonnegative expectation value on separable $\kappa$-partite states $\varrho_{S^1_{1...k_1-1},...,S^n_{1...k_n-1}}$, $\mathop{\mathrm{tr}}\nolimits(\mathcal{W}_\tau\varrho)\

Figures (2)

  • Figure 1: State-witness contraction yielding a four-partite witness from three- and bi- partite states and witnesses, with three-partite systems $S^1:=A$ and $S^2:=B$. First we take the tensor product of a witness $W$ and a positive operator $M$. Then the first (or last, equivalently) subsystems $A_1$ and $B_1$ are contracted (multiplied and partial traced) with a state or witness $\tau_{A_1B_1}$ to be chosen -- see Eqs. \ref{['eq:WitGenkn']} and \ref{['eq:Wit4partNonDeco']}. This allows to lift $W$ to a larger witness $\mathcal{W}_\tau$ with further detection capabilities (Observation \ref{['obs:4PartWitDetBoundEnt']} and Proposition \ref{['prop:MPPTwits']}), which moreover can be partially optimized over $\tau$ to detect specific four-partite states -- see Eq. \ref{['eq:OptimWitGen']}.
  • Figure 2: Tailoring multicopy witnesses to states. Linear witnesses detect regions of entangled states separated by hyperplanes (magenta area) through $\langle W\rangle_{\varrho}=\mathop{\mathrm{tr}}\nolimits(W\varrho)<0$. Multicopy witnesses $\mathcal{W}$ detect regions of entangled states through $\langle \mathcal{W}\rangle_{\varrho^{\otimes k}}<0$HoroMulticopyWit2003RemikUniversal2008, separated by curved hypersurfaces (purple and blue areas). In particular, $\mathcal{W}$ can be constructed by tensoring linear witnesses TP_Rico24. By replacing one of the copies by a variable state or witness $\tau$, the state-witness contraction technique (SWC) allows to systematically detect a larger set of states (green area --see Table \ref{['tab:DetectIso']} for an example) and approach further the set of separable states (SEP).

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 3
  • proof