Inequivalent multipartite coherence classes and two operational coherence monotones

Quantum coherence has received significant attention in recent years, but the structure of multipartite coherent states is unclear. In this paper, we generalize important results in multipartite entanglement theory to their counterparts in quantum coherence theory. First, we give a necessary and sufficient condition for when two pure multipartite states are equivalent under local incoherent operations assisted by classical communications (LICC), i.e., two states can be deterministically transformed to each other under LICC operations. Next, we investigate and give the conditions in which such a transformation succeeds only stochastically. Different from the entanglement case for two-qubit states, we find that the stochastic LICC (sLICC) equivalence classes are infinite. Moreover, it is possible that there are some classes of states in multipartite entanglement that can convert into each other, whereas they cannot convert into each other in multipartite coherence. In order to show the difference among sLICC classes, we introduce two coherence monotones: accessible coherence and source coherence, following the logistics given in [Phys. Rev. Lett. 115, 150502 (2015)]. These coherence monotones have a straightforward operational interpretation, namely, the accessible coherence characterizes the proficiency of a state to generate other states via quantum incoherent operations, whereas the source coherence characterizes the set of states that can be reached via quantum incoherent operations acting on the given state of interest.

Figure 1

The hierarchy of LOCC, LICC, and LSICC.

Figure 2

In this schematic, the source set $M_s^{\mathcal{O}}\left(\rho \right)$ and the accessible set $M_a^{\mathcal{O}}\left(\rho \right)$ of state $\rho$ are depicted. Any state in $M_s^{\mathcal{O}}\left(\rho \right)$ can be transformed to $\rho$ via $\mathcal{O}$, and $\rho$ can be transformed into any state in $M_a^{\mathcal{O}}\left(\rho \right)$ via $\mathcal{O}$.

Figure 3

The source set (blue) $M_s\left(|\psi \rangle \right)$ and the accessible set (red) $M_a\left(|\psi \rangle \right)$ of state $|\psi \rangle$ are depicted. Meanwhile, the source set (cyan) $M_s\left(|\varphi \rangle \right)$ and the accessible set (magenta) $M_a\left(|\varphi \rangle \right)$ of state $|\varphi \rangle$ are depicted. Here, the $x$ axis represents the basis vectors $|00\rangle$, the $y$ axis represents the basis vectors $|01\rangle$, and the $z$ axis represents the basis vectors $|11\rangle$. In the figure, $|\psi \rangle =\sqrt{0.5}|00\rangle +\sqrt{0.3}|11\rangle +\sqrt{0.2}|22\rangle$ and $|\varphi \rangle =\sqrt{0.5}|00\rangle +\sqrt{0.3}|01\rangle +\sqrt{0.2}|02\rangle$. The source sets (accessible sets) are indeed different between these two states.

Figure 4

The source set (blue) $M_s\left(|\psi \rangle \right)$ and the accessible set (red) $M_a\left(|\psi \rangle \right)$ of state $|\psi \rangle$ are depicted. Any state in $M_s\left(|\psi \rangle \right)$ can be transformed to $|\psi \rangle$ via LSICC and LICC, and $|\psi \rangle$ can be transformed into any state in $M_a\left(|\psi \rangle \right)$ via LSICC and LICC. In the figure, $|\psi \rangle =\sqrt{0.6}|01\rangle +\sqrt{0.4}|10\rangle$.

Figure 5

As shown in the figure, the source set (blue color) $M_s\left(\rho \right)$ and the accessible set (red color) $M_a\left(\rho \right)$ of state $\rho$ are depicted. Any state in $M_s\left(\rho \right)$ can be transformed to $\rho$ via IC and SIO, and $\rho$ can be transformed into any state in $M_a\left(\rho \right)$ via IC and SIO. In the figure, the single-qubit $\rho$ has the Bloch vector $\left(\frac{1}{2},0,\frac{\sqrt{2}}{2}\right)$.

Figure 6

As shown in the figure, the source set (blue color) $M_s\left(\rho \right)$ and the accessible set (red color) $M_a\left(\rho \right)$ of state $\rho$ are depicted. Any state in $M_s\left(\rho \right)$ can be transformed to $\rho$ via PIO, and $\rho$ can be transformed into any state in $M_a\left(\rho \right)$ via PIO. In the figure, the single-qubit $\rho$ has the Bloch vector $\left(\frac{1}{2},0,\frac{3}{4}\right)$.

Figure 7

As shown in the figure, the source set (blue) $M_s\left(|\psi \rangle \right)$ and the accessible set (red) $M_a\left(|\psi \rangle \right)$ of state $\rho$ are depicted. Any state in $M_s\left(|\psi \rangle \right)$ can be transformed to $|\psi \rangle$ via SIO and IC, and $|\psi \rangle$ can be transformed into any state in $M_a\left(|\psi \rangle \right)$ via SIO and IC. In the figure, $|\psi \rangle =\sqrt{0.5}|00\rangle +\sqrt{0.3}|01\rangle +\sqrt{0.2}|10\rangle$.