Imbalance entanglement: Symmetry decomposition of negativity

In the presence of symmetry, entanglement measures of quantum many-body states can be decomposed into contributions from distinct symmetry sectors. Here we investigate the decomposability of negativity, a measure of entanglement between two parts of a generally open system in a mixed state. While the entanglement entropy of a subsystem within a closed system can be resolved according to its total preserved charge, we find that negativity of two subsystems may be decomposed into contributions associated with their charge imbalance. We show that this charge-imbalance decomposition of the negativity may be measured by employing existing techniques based on creation and manipulation of many-body twin or triple states in cold atomic setups. Next, using a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in a Riemann geometry, we compute this decomposed negativity in critical one-dimensional systems described by conformal field theory. We show that it shares the same distribution as the charge-imbalance between the two subsystems. We numerically confirm our field theory results via exact calculations for noninteracting particles based on a double-Gaussian representation of the partially transposed density matrix.

Figure 1

Schematic block structure of density matrices in the basis of $|N_1{N}_2\rangle$ with blocks labeled by $Q=N_1-N_2^{{\mathrm{T}}_2}$. (a) A density matrix ${\rho }_A$ has a block structure with respect to $N_A=N_1+N_2$ as shown by the thick lines. (b) Following a partial transposition, ${\rho }_A^{{\mathrm{T}}_2}$ has a block structure determined by $Q=N_1-N_2^{{\mathrm{T}}_2}$ as shown by the thick lines. The partial transposition is indicated by the arrows in (a) and relates the blocks according to Eq. (12); for a specific example of these matrices see Eqs. (3) and (5).

Figure 2

Depiction of the experimental protocol to measure the imbalance-resolved Rényi negativity. (a) The system is tripartitioned into regions $A_1$, $A_2$, and $B$. (b) $n$ copies of the system are prepared. (c) A copy-space Fourier transform is performed on a region containing $A_1$ and $A_2$. (d) The $N_{A_1}^k$ and $N_{A_2}^k$ are measured in regions $A_1$ and $A_2$ ($k=1,\cdots ,n)$, and $q=\frac{1}{n}{\sum }_k(N_{A_1}^k-N_{A_2}^k)$ and $r_n=exp\left\{{\sum }_k\frac{2\pi i}{n}k(N_{A_1}^k-N_{A_2}^k)\right\}$ of Eq. (34) are calculated. The contribution of imbalance sector $Q$ to the Rényi negativity is then given by the average over ${\delta }_{Q,q}\times r_n$.

Figure 3

Schematic representation of the Rényi negativity $\mathrm{Tr}\left\{{\left({\rho }_A^{{\mathrm{T}}_2}\right)}^n\right\}$ as an $n$-sheet Riemann surface with $A_1=[-{\ell }_1,0]$ and $A_2=[0,{\ell }_2]$. The phase factor $exp\left\{i\alpha (N_1-N_2^{{\mathrm{T}}_2})\right\}$ is implemented by flux insertions ${\mathcal{V}}_{\alpha }(-{\ell }_1){\mathcal{V}}_{-2\alpha }\left(0\right){\mathcal{V}}_{\alpha }\left({\ell }_2\right)$ and represented by the vertical arrows. The winding line serves only as a visual aid.

Figure 4

Exact numerical calculations of ${\left[R_3\right]}_Q$ for a half-filled tight-binding chain of free fermions are displayed on a logarithmic scale and fitted to the CFT predictions (solid lines) of Eq. (52) with $K=1$. The inset shows the same data displayed on a linear scale. All graphs are for ${\ell }_1+{\ell }_2=4000a$ and infinite-length environment $B$.