Statistics of entanglement transformation with hierarchies among catalysts

The distribution of typical bipartite pure states is studied within the framework of state transformation via local operations and classical communication (LOCC). We report the statistics of comparable and incomparable states in different dimensions for single- and multicopy regimes, and we establish a connection between state transformation and the difference between the entanglement contents of the initial and the target states. From the analysis of catalyst resources, required to further otherwise impossible LOCC transformations between pairs, we demonstrate a universal pattern in the average and minimum entanglement of the randomly generated catalysts. Furthermore, we introduce a concept of hierarchy between different kinds of catalysts, and we show how they can not only aid in the conversion of incomparable states, but they can also act as a less costly resource toward this goal. We confirm the existence of catalysts, referred to as strong catalysts, which can activate LOCC transformation between pairs at the single-copy level, when it is initially impossible even with multiple copies.

Figure 1

Entanglement patterns of random pure bipartite states in $d\otimes d$. The normalized frequency distribution of entanglement (quantified by entanglement entropy), $f_E^N$ (ordinate), is shown against $E\left(\psi \right)$ (abscissa) for (a) one copy, (b) two copies, and (c) three copies. (d) Mean entanglement $\langle E\rangle$ (ordinate) with respect to the dimension $d$ (abscissa) for one copy (bottom line with squares), two copies (middle line with circles), and three copies (top line with triangles). We Haar uniformly simulate ${10}^6$ states in each dimension. All the axes are dimensionless.

Figure 2

Incomparable vs comparable states. Normalized frequency distributions of incomparable, $f_{\text{incomp}}^N$ (solids), and comparable states, $f_{\text{comp}}^N$ (stripes) (vertical axis), vs ${\mathrm{\Delta }}_E^{\psi \to \varphi }$ (horizontal axis) when ${10}^6$ states are generated Haar uniformly with $E\left(\psi \right)\ge E\left(\varphi \right)$ in (a) $3\otimes 3$, (b) $4\otimes 4$, and (c) $5\otimes 5$. (d) The percentage of incomparable bipartite states (ordinate) with respect to the number of copies $k$ (abscissa) in $4d$ (red stars), $5d$ (blue squares), $6d$ (black circles), $7d$ (brown triangles), and $8d$ (yellow diamonds). (In short, $d\otimes d$ is referred to as $dd$ systems.) All the axes are dimensionless.

Figure 3

(a) Mean entanglement, $\langle E\left(\chi \right)\rangle$ (square), and minimum entanglement, $E_{\text{min}}\left(\chi \right)$ (circle), of catalyst states (ordinate) vs ${\mathrm{\Delta }}_E^{\psi \to \varphi }$ (abscissa) when random states with $E\left(\psi \right)\ge E\left(\varphi \right)$ are used to check incomparability as well as for catalysts in $4\otimes 4$. (c) Triangles and diamonds represent $\langle E\left(\chi \right)\rangle$ and $E_{\text{min}}\left(\chi \right)$, respectively, in $5\otimes 5$. (b) and (d) The average number of catalyst states, $\langle n_{\chi }\rangle$ (ordinate), against ${\mathrm{\Delta }}_E^{\psi \to \varphi }$ (abscissa), for random states in $4d$ (striped columns) [in (b)] and $5d$ (solid columns) [in (d)]. In all the plots, states are incomparable when a single copy of the input-target pair is available. The data are computed by generating $5\times {10}^4$ pairs to find incomparable states, while for a given incomparable pair, ${10}^5$ catalysts are simulated. All the axes are dimensionless.

Figure 4

(a) $E_{min}\left(\chi \right)$ of catalysts for two-copy incomparable states (ordinate) in $4d$ (circles) and $5d$ (diamonds) with respect to ${\mathrm{\Delta }}_E^{2(\psi \to \varphi )}$ (abscissa). (c) The same as in (a) except the ordinate is for $\langle E\left(\chi \right)\rangle$ (squares and triangles are used for states in $4d$ and $5d$, respectively). (b) The average number of catalysts, $\langle n_{\chi }^3\rangle$, for three-copy incomparable states, and (d) $\langle n_{\chi }^2\rangle$ for two-copy incomparable states with $4d$ (stripes) and $5d$ (solid) (ordinate) against ${\mathrm{\Delta }}_E^{k(\psi \to \varphi )}$. All the axes are dimensionless.

Figure 5

Entanglement patterns of one-step-assisted catalysts. (a) $E_{min}{\left(\chi \right)}_{2\left(1\right)}$ (solid circles) and $E_{min}{\left(\chi \right)}_{3\left(2\right)}$ (hollow circles) (ordinate) with respect to ${\mathrm{\Delta }}_E^{k(\psi \to \varphi )}$ (abscissa) having $k=1,2$, respectively, for randomly generated states in $4\otimes 4$. (c) A similar plot to (a) for $\langle E{\left(\chi \right)}_{2\left(1\right)}\rangle$ (solid squares) and $\langle E{\left(\chi \right)}_{3\left(2\right)}\rangle$ (hollow squares). Parts (b) and (d) are for random states in $5\otimes 5$. Part (b) is similar to (a) with circles being replaced by diamonds, while (d) is similar to (c) except squares are changed to triangles. All the axes are dimensionless.

Figure 6

$\langle n_{\chi }\rangle$ (ordinate) against ${\mathrm{\Delta }}_E^{k(\psi \to \varphi )}$ (abscissa) with $k=1,2$ for catalysts ${|\chi \rangle }_{2\left(1\right)}$ (stripes) and ${|\chi \rangle }_{3\left(2\right)}$ (solid), respectively. (a) Distribution of $\langle n_{\chi }\rangle$ in $4\otimes 4$ while (b) is for $5\otimes 5$. All the axes are dimensionless.

Figure 7

Frequency distribution of average number of strong catalysts, $\langle n_{{\chi }_{n\left(1\right)}}\rangle$ (ordinate) with ${\mathrm{\Delta }}_E^{\psi \to \varphi }$ (abscissa) found in $4\otimes 4$ (stripes) and $5\otimes 5$ (solid). (a) Strong catalysts $\langle n_{{\chi }_{3\left(1\right)}}\rangle$ for three-copy incomparable states. (b) Strong catalysts $\langle n_{{\chi }_{4\left(1\right)}}\rangle$ for four-copy incomparable states. Unlike one-step-assisted catalysts, they can be found in good numbers for low values of ${\mathrm{\Delta }}_E^{\psi \to \varphi }$. All the axes are dimensionless.

Figure 8

Quantifying cost-efficient catalysts. $E_{min}\left(\chi \right)-E\left(\psi \right)$ [in (a)] and $E_{min}\left(\chi \right)-E\left(\varphi \right)$ [in (b)] (ordinate) against the single-copy entanglement difference, ${\mathrm{\Delta }}_E^{\psi \to \varphi }$ (abscissa). Circles and diamonds represent the Haar uniformly generated states in $4\otimes 4$ and $5\otimes 5$, respectively. (c) and (d) Similar plots to (a) and (b) where the vertical axis is for $\langle E\left(\chi \right)\rangle -E\left(\psi \right)$ and $\langle E\left(\chi \right)\rangle -E\left(\varphi \right)$ in (c) and (d), respectively. Squares and triangles correspond to $4d$ and $5d$ catalyst states. All the axes are dimensionless.