Graph-associated entanglement cost of a multipartite state in exact and finite-block-length approximate constructions

We introduce and analyze graph-associated entanglement cost, a generalization of the entanglement cost of quantum states to multipartite settings. We identify a necessary and sufficient condition for any multipartite entangled state to be constructible when quantum communication between the multiple parties is restricted to a quantum network represented by a tree. The condition for exact state construction is expressed in terms of the Schmidt ranks of the state defined with respect to edges of the tree. We also study approximate state construction and provide a second-order asymptotic analysis.

Figure 1

Schematic representation of our LOCC construction of a multipartite entangled state under graph $G$. Each solid circle represents a quantum system, possibly of a different dimension, while each rounded square corresponds to a party, which may possess several quantum systems within. The parties are connected by noiseless quantum communication channels specified by the edges of $G$. Under LOCC, each use of a channel is equivalent to consuming a maximally entangled state, given by a pair of blue solid circles with a line in-between (initial). The target state $\rho$ is represented by red solid circles encircled in a bubble (final). The maximally entangled states all together comprise a resource state $\left|{\mathrm{\Phi }}_{\text{res}}\left(G\right)\right.〉$ for LOCC construction of $\rho$.

Figure 2

A graphical representation of a tree. The vertices and edges are labeled where $v_1$ is the root of the tree and the other vertices $v_2,v_3,...,v_N$ and the edges $e_1,e_2,...,e_{N-1}$ are in order of a breadth-first search starting from $v_1$. $C_v, D_v$, and $D_v^{\prime }$ denote the set of $v$'s children, the set of $v$'s descendants, and the set of $v$ itself and $v$'s descendants, respectively. $D_v^{\prime }$ is decomposed into $v$ itself and $D_c^{\prime }$'s for all $c\in C_v$.

Figure 3

Schematic representation of a reduced density operator ${\rho }_e$ of the target state $\left|\psi \right.〉$ with respect to an edge $e$ of the rooted tree. For each edge $e=\left\{p\right(v),v\}$, the tree can be divided into two connected subgraphs corresponding to $D_v^{\prime }$ and its complement. We write the reduced density operator of $\left|\psi \right.〉$ corresponding to $D_v$ as ${\rho }_e$. The Schmidt basis of ${\mathcal{H}}_t^{D_v^{\prime }}$ for $\left|\psi \right.〉$ is written as ${\left\{\left|w_l\right.〉\right\}}_l$. We can represent ${\rho }_e$ using a subset of ${\left\{\left|w_l\right.〉\right\}}_l$ corresponding to nonzero Schmidt coefficients.

Figure 4

Schematic representation for the notations of the Hilbert spaces of the initial resource states and the target states. For each nonroot $v$ ($v\ne v_1$), an initial resource state shared between $v$ and its parent $p\left(v\right)$ is denoted by ${\left|{\mathrm{\Phi }}_{m_e}^{+}\right.〉}^e=\left(1/\sqrt{m_e}\right){\sum }_{l_v}{\left|l_v\right.〉}^{p\left(v\right)}\otimes {\left|l_v\right.〉}^v$, where $e=\left\{p\right(v),v\}, {\left|l_v\right.〉}^{p\left(v\right)}\in {\mathcal{H}}_{r_e}^{p\left(v\right)}$, and ${\left|l_v\right.〉}^v\in {\mathcal{H}}_{r_e}^v$. For each $v$, the computational basis of ${\mathcal{H}}_t^v$ for the $v$'s part of the target state is denoted by ${\left\{\left|l\right.〉\right\}}_l$.

Figure 5

Schematic description of examples for exact state construction of the four-qubit $W$ state $\left|W_4\right.〉$ over four parties connected by a line-topology graph. Theorem t5 implies that one Bell state $|{\mathrm{\Phi }}_2^{+}\rangle$ (a maximally entangled two-qubit state) is required at each edge for the initial resource state in these examples. The measurement for a party $v$ on an initial resource qubit (represented as ${\left\{M_{x,z}^v\right\}}_{x,z}$ in a blue rectangle) can be implemented by a unitary transformation $U_v$ (represented as a red rectangle) on the incoming initial resource qubit, an auxiliary qubit, and a target qubit followed by the Bell measurement (represented as a purple rectangle) on the auxiliary qubit and the outgoing initial resource qubit for quantum teleportation. Example t7 is a case in which each party has at most one child, but the child may have descendants. Example t8 describes a case in which a party $v_1$ has multiple children $v_2$ and $v_3$.

Figure 6

The second-order coefficient $b$ of the resource consumption rate at edge $e_{N/4}$ of our approximate state construction algorithm for the $N$-qubit $W$ states over $N$ parties under line-topology graphs for $N\in \{4,8,...,80\}$. The error threshold is set to be ${\epsilon }^{\prime }\left(e\right)={(N-1)}^{-1/2}\epsilon$ for all $e\in E$ with $\epsilon =\frac{1}{25}$. The second-order coefficient increases as $N$ increases, whereas the first-order asymptotic rate is independent of $N$.

Figure 7

Comparison of resource consumption rates at edge $e_1=\{v_1,v_2\}$ up to the second order for approximate construction of the $W$ state under the line-topology graph for $N=4$. The blue dotted-dashed curve represents the resource consumption rate $R_{\text{const}}$ when error thresholds at edges are set constantly, and the red solid curve the rate $R_{\text{opt}}$ when error thresholds are optimized in order to save the total resource consumption. The purple dotted curve represents their lower bound $\underline{R}$ up to the second order. Optimization of error thresholds at edges provides an efficient algorithm for approximate state construction.