Detection of many-body entanglement partitions in a quantum computer

Abstract

We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, $m$-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.

Figures (3)

• Figure 1: Structures of three-partite separability. Three-partite states can be fully separable (SEP), with separability partition $\kappa=[1,1,1]$; biseparable ($\varrho_{AB|C}$, $\varrho_{A|BC}$, $\varrho_{AC|B}$ and their convex combination), with partition $\kappa=[2,1]$; or genuinely multipartite entangled (GME), with $\kappa=[3]$. We shift symmetric witnesses $W$ detecting biseparable states (green area) by a parameter $\alpha_\kappa$, to construct new witnesses detecting states outside of certain separability partitions (blue area).
• Figure 2: Detection of entanglement partitions $\kappa$ in $n$-partite systems.$(a)$ We perform weak Schur sampling on the $n$-partite state describing the system, for instance implementing the Schur transform gate ($U_{ST}$) and measuring the register $\lambda$ while discarding the registers $M_\lambda$ and $T_\lambda$. This step is tractable in a quantum computer, and efficiency can be gained with more direct methods of measuring $\lambda$ such as generalized phase estimation (see Table \ref{['tab:Scaling']}). $(b)$ We repeat the procedure multiple times, and store in a classical computer the probabilities $p_\lambda$ of obtaining each outcome $\lambda$. These probabilities are the projections of $\varrho$ onto each irreducible subspace labelled by $\lambda$. $(c)$ Based on the probabilities $p_\lambda$, we detect states that are not in certain separability partitions $\kappa$. In particular, the first component $k_1$ of $\kappa$ determines the entanglement depth of $\varrho$, and the number of components $m$ in $\kappa$ determines its separability length.
• Figure 3: We fully characterize all tripartite witnesses of the form $\sum_{\lambda \vdash_d n} c_\lambda p_\lambda < 0 \implies \varrho \notin \kappa\operatorname{-SEP}$. Up to normalization, they are all given by $q_1(3p_3-p_{2,1}+p_{1,1,1}) + q_2 (p_{2,1}-4p_{1,1,1}) + q_3 4p_{(1,1,1)}$ for $q_k \geq 0$, plotted here as a simplex. A negative value in the corresponding region implies $\kappa$-inseparability. Here, the dashed triangular region formed by the vertices $p_{2,1}-4p_{1,1,1}$, $\propto 4p_3 - p_{2,1}$, and $\propto p_{1,1,1}$ are known from immanant inequalities, but they do not detect bound entanglement, i.e., they are positive for all fully positive partial transpose (PPT) states $\varrho : \varrho^{T_1},\varrho^{T_2},\varrho^{T_3} \geq 0$.

Theorems & Definitions (9)

• proof
• Proposition 3
• Theorem 4
• Proposition 5
• proof
• Corollary 1
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• Corollary 2
• proof