Multipartite entanglement outperforming bipartite entanglement under limited quantum system sizes

Multipartite quantum entanglement serves as a resource for spatially separated parties performing distributed quantum information processing. Any multipartite entangled state can be generated from appropriately distributed bipartite entangled states by local operations and classical communication (LOCC), and in this sense, any distributed process based on shared multipartite entanglement and LOCC is simulatable by using only bipartite entangled states and LOCC. We show here that this reduction scenario does not hold when there exists a limitation on the size of the local quantum system of each party. Under such a limitation, we prove that there exists a set of multipartite quantum states such that these states in the set cannot be prepared from any distribution of bipartite entanglement, while the states can be prepared from a common resource state exhibiting multipartite entanglement. We also show that temporal uses of bipartite quantum communication resources within a limitation of local system sizes are sufficient for preparing this common resource state exhibiting multipartite entanglement, yet there also exist other states exhibiting multipartite entanglement which cannot be prepared even in this setting. Hence, when the local quantum system sizes are limited, multipartite entanglement is an indispensable resource without which certain processes still cannot be accomplished.

Figure 1

The task of system-size-limited quantum state transformation, where the parties transform a common resource state represented by blue circles by LOCC into an arbitrary state in a given target set $\left\{\left|{\psi }_0\right.〉,\left|{\psi }_1\right.〉,...\right\}$ represented by red circles. To differentiate the capabilities of common resource states exhibiting multipartite entanglement at the top and those consisting only of bipartite entanglement at the bottom, where each connected pair of blue circles represents a bipartite entangled state, we consider the static setting where each party's local system size for storing the common resource state is limited. We also consider the dynamic setting where the parties have to prepare a common resource state within these limitations by performing quantum communication, in addition to storing the common resource state. The difference in the capabilities arises in terms of achievability of this task.

Figure 2

A simple example of a graph representing a graph state and a quantum circuit representing a class of states parameterized by $\alpha$ which can be deterministically prepared using this graph state. Given a graph state ${\left|\mathrm{\Phi }\right.〉}^{v_1,v_2,v_3}$ as illustrated on the left, by performing the unitary $exp\left(i\alpha X^{v_1}\right)$ parameterized by $\alpha$ and a measurement in the $Z$ basis $\left\{\left|0\right.〉,\left|1\right.〉\right\}$ on the qubit represented by the black vertex $v_1$, followed by local unitary corrections on the white vertices $v_2$ and $v_3$ conditioned by the measurement outcome, we can deterministically obtain a two-qubit state $\left|\psi \left(\alpha \right)\right.〉$ defined in Eq. (3) represented by $v_2$ and $v_3$. The state $\left|\psi \left(\alpha \right)\right.〉$ can also be represented as the output of the quantum circuit on the right, where a two-qubit gate $exp\left(i\alpha Z^{v_2}\otimes Z^{v_3}\right)$ parameterized by $\alpha$ is applied to ${\left|+\right.〉}^{v_2}\otimes {\left|+\right.〉}^{v_3}$.

Figure 3

A quantum circuit generating all the states in the target set $S_0:=\left\{\left|\psi \left(\mathbf{\alpha }\right)\right.〉\right\}$ for the system-size-limited quantum state preparation in Propositions t1 and t2, where $\mathbf{\alpha }=\left({\alpha }_1,...,{\alpha }_7\right)$ is a tuple of parameters. The wires of the circuit starting from the input ${\left|+\right.〉}^{v_1},...,{\left|+\right.〉}^{v_8}$ represent qubits held by the parties $v_1,...,v_8$, respectively. The circuit consists of seven two-qubit gates $exp\left(i{\alpha }_{i}Z\otimes Z\right)$ parameterized by ${\alpha }_i\in \left\{{\alpha }_1,...,{\alpha }_7\right\}$.

Figure 4

A graph representing a 15-qubit graph state $\left|{\mathrm{\Phi }}_{\mathrm{res}}\right.〉$ used as a common resource state exhibiting multipartite entanglement in Proposition t1. Each of the parties $v_k\in \left\{v_1,...,v_7\right\}$ holds two qubits ${\mathcal{H}}^{v_k}\otimes {\mathcal{H}}_{\mathrm{a}}^{v_k}$, while party $v_8$ holds one qubit ${\mathcal{H}}^{v_8}$. Eight of the 15 qubits ${\mathcal{H}}^{v_1},...,{\mathcal{H}}^{v_8}$ represented by white vertices are qubits which can be prepared in any state $\left|\psi \left(\mathbf{\alpha }\right)\right.〉$ in the target set $S_0$. The other seven ${\mathcal{H}}_{\mathrm{a}}^{v_1},...,{\mathcal{H}}_{\mathrm{a}}^{v_7}$ represented by black vertices are auxiliary qubits to be measured. To obtain $\left|\psi \left(\mathbf{\alpha }\right)\right.〉\in S_0$ parameterized by $\mathbf{\alpha }=\left({\alpha }_1,...{\alpha }_7\right)$, each party $v_i\in \left\{v_1,...,v_7\right\}$ performs the following protocol in order. First, a unitary $exp\left(i{\alpha }_{i}X\right)$ parameterized by ${\alpha }_i$ is performed on ${\mathcal{H}}_{\mathrm{a}}^{v_i}$. Then, the qubit ${\mathcal{H}}_{\mathrm{a}}^{v_i}$ is measured in the $Z$ basis $\left\{\left|0\right.〉,\left|1\right.〉\right\}$, and depending on the outcome, a unitary correction is applied to the remaining qubits. Using this protocol, the parties can deterministically transform $\left|{\mathrm{\Phi }}_{\mathrm{res}}\right.〉$ into $\left|\psi \left(\mathbf{\alpha }\right)\right.〉\in S_0$ for any $\mathbf{\alpha }$.

Figure 5

A line-topology tree representing a resource state consisting of bipartite entanglement to prepare a fully entangled state within the configuration ${\mathit{d}}_0$. Since the target set $S_0$ includes a fully entangled state $\left|\psi \left({\mathbf{\alpha }}_{\frac{\pi }{4}}\right)\right.〉$, the common resource states consisting of bipartite entanglement for $S_0$ have to be represented by the line-topology tree in the figure, which leads to a contradiction with the condition given in Inequality (1) as shown in the main text.

Figure 6

A quantum circuit representing a protocol for preparing the common resource state $\left|{\mathrm{\Phi }}_{\mathrm{res}}\right.〉$ for the target set $S_0$ by quantum communication in addition to LOCC within the configuration ${\mathit{d}}_0$. Each of the parties $v_k\in \left\{v_1,...v_7\right\}$ can perform local operations on at most two qubits ${\mathcal{H}}^{v_k}\otimes {\mathcal{H}}_{\mathrm{a}}^{v_k}$, while the party $v_8$ can perform local operations on one qubit ${\mathcal{H}}^{v_8}$. The dashed lines represent the separation of the parties. Each wire of the circuit corresponds to a qubit corresponding to the Hilbert space on the right, and the circuit consists of the controlled-$Z$ gates $CZ$ defined in Eq. (2) and quantum communication represented by crossings of the wires.