Entanglement Sharing

Entanglement sharing asks how quantum correlations are distributed across many named subsystems. The answer is not determined by a single number. A multipartite state can contain direct pairwise entanglement, global entanglement visible only across large cuts, entanglement that can be localized by measurements, or a network resource that can be routed through intermediate nodes.

This page is a bridge between the state-classification language of multipartite separability and the operational language of LOCC. Its theme is simple: entanglement is not a fluid that can be poured freely from one edge to another. It is constrained by tensor-product structure, cuts, measurements, and the allowed operations.

Pairwise Versus Global Entanglement

For a three-party state $\rho_{ABC}$, the pairwise marginals are

$$\rho_{AB}=\operatorname{Tr}_C\rho_{ABC}, \qquad \rho_{AC}=\operatorname{Tr}_B\rho_{ABC}, \qquad \rho_{BC}=\operatorname{Tr}_A\rho_{ABC}.$$

One can draw an edge between two parties when the corresponding marginal is entangled. This pairwise entanglement graph is often useful, but it is incomplete.

The standard three-qubit examples separate the possibilities:

$$\begin{array}{cccc} \text{state} & \text{pairwise entanglement} & \text{genuine tripartite entanglement} & \text{typical lesson} \\ \hline \lvert\Phi^+\rangle_{AB}\otimes\lvert0\rangle_C & AB\ \text{only} & \text{no} & \text{one entangled link} \\ \lvert\mathrm{GHZ}_3\rangle & \text{none in two-party marginals} & \text{yes} & \text{global coherence} \\ \lvert W_3\rangle & AB,\ AC,\ BC & \text{yes} & \text{distributed pairwise entanglement} \end{array}$$

The GHZ row is the warning. Its two-qubit marginals are separable, but the state is still genuinely tripartite entangled. The W row gives the opposite warning. Pairwise entanglement can be present throughout the graph, but that does not reduce the state to a collection of independent Bell pairs.

Localizable Entanglement

Pairwise marginals are not the only way to ask whether two parties can become entangled. Sometimes entanglement is present globally and can be localized onto a chosen pair by measuring the other parties.

For example,

$$\lvert\mathrm{GHZ}_3^+\rangle = \frac{1}{\sqrt2} \bigl( \lvert000\rangle+\lvert111\rangle \bigr)$$

has a separable $AB$ marginal:

$$\rho_{AB} = \frac12 \lvert00\rangle\langle00\rvert + \frac12 \lvert11\rangle\langle11\rvert.$$

But if $C$ is measured in the $X$ basis and the outcome is kept, the conditional $AB$ state is a Bell state up to a known local phase:

$$\lvert\phi_\pm\rangle_{AB} = \frac{1}{\sqrt2} \bigl( \lvert00\rangle \pm \lvert11\rangle \bigr).$$

So the pair $AB$ has no unconditional entangled marginal, yet one party can help localize entanglement onto $AB$ by performing a measurement and communicating the outcome. This is not a contradiction. The measurement consumes or redirects global multipartite entanglement.

Graph states and cluster states make this idea systematic. Measurements on intermediate qubits can move, concentrate, or reshape entanglement among the remaining qubits.

Network Language

A quantum network description has nodes and links. Depending on context, a node may be a laboratory, a memory, a qubit register, a lattice site, or a mode. A link may mean one of several things:

• a physical quantum channel;
• a shared Bell pair or partially entangled pair;
• an interaction edge in a Hamiltonian;
• a useful correlation inferred from a reduced state;
• a graph-state stabilizer edge.

These are not interchangeable. A graph of entangled marginals, a hardware connectivity graph, and a graph-state preparation graph can have different meanings even when they have the same drawing.

For resource networks, the cleanest model is this: neighboring nodes initially share bipartite states, and allowed operations are local quantum operations at nodes plus classical communication. The task is to create entanglement between selected distant nodes.

Entanglement Swapping Preview

Entanglement Swapping is the elementary routing move. Suppose $A$ shares a Bell pair with one qubit at $B$, and another qubit at $B$ shares a Bell pair with $C$:

$$\lvert\Phi^+\rangle_{AB_1} \otimes \lvert\Phi^+\rangle_{B_2C}.$$

A Bell measurement at $B$ projects $A$ and $C$ into a Bell state labeled by the two-bit measurement record. The two end nodes become conditionally entangled even though they never interacted directly.

The accounting is important:

• the original links are consumed;
• the measurement outcome must be communicated classically;
• no entanglement is created by local operations alone;
• entanglement is rerouted through the initial resource crossing the relevant cuts.

This is why entanglement swapping belongs naturally with LOCC. It is an allowed redistribution of an already nonlocal resource. The canonical protocol page derives the Bell-basis identity and treats Pauli frames, unconditioned endpoint states, imperfect links, probabilistic analyzers, and repeater-rate accounting.

Cut Accounting

For any subset $S$ of network nodes, the bipartition $S\vert\bar S$ is a cut. If no quantum channel or preexisting entanglement crosses that cut, LOCC cannot create entanglement across it.

This gives a useful discipline:

1. Identify the cut relevant to the desired final entanglement.
2. Ask what resources cross the cut initially.
3. Track which operations are local on each side of the cut.
4. Use an entanglement monotone appropriate to the setting.

For example, in the swapping setup above, the cut $A\vert B_1B_2C$ initially has one Bell pair crossing it, and after a successful swapping outcome the cut $A\vert C$ has one Bell pair when the middle node is discarded. Entanglement has moved, not appeared from nowhere.

Cut language is also the bridge to many-body entanglement. In a spin chain, one studies the entropy across spatial cuts. In a tensor network, the bond dimension across a cut limits how much entanglement the representation can carry. In a communication network, the available entangled links across a cut constrain what distant nodes can share.

Entanglement Distribution

Entanglement distribution is the task of producing useful entanglement between selected parties. There are several regimes:

• Direct distribution: a quantum system is sent through a channel, or a source emits entangled systems to two nodes.
• Swapping: neighboring entangled links are consumed to entangle farther nodes.
• Concentration: weak pure-state entanglement is probabilistically converted into stronger entanglement.
• Distillation: many noisy entangled pairs are processed into fewer higher-quality pairs.
• Network routing: multiple paths and intermediate measurements are chosen to optimize success probability, fidelity, latency, or resource cost.

For a pure two-qubit bond with Schmidt form

$$\lvert\psi\rangle = \sqrt{\lambda_0}\lvert00\rangle + \sqrt{\lambda_1}\lvert11\rangle, \qquad \lambda_0\ge\lambda_1, \qquad \lambda_0+\lambda_1=1,$$

the optimal probability of converting one copy to a Bell pair by LOCC is

$$p_{\mathrm{singlet}} = 2\lambda_1.$$

This simple number is one ingredient in entanglement-percolation models.

Entanglement Percolation Preview

Entanglement percolation studies large networks whose edges initially contain imperfect entangled states. A basic strategy is:

1. Try to convert each edge independently into a Bell pair.
2. Treat successful conversions as open bonds.
3. Ask whether the open-bond graph contains a path between distant nodes.

This is classical bond percolation applied after local filtering. If the success probability lies above the relevant percolation threshold, long connected paths appear with nonzero probability in large networks.

Quantum strategies can do better than this naive route. Measurements before or during conversion can change the effective network geometry, correlate outcomes, or optimize the final entanglement more directly than independent edge conversion. The details depend strongly on the lattice, the available measurements, and the resource states.

The safe lesson is not “quantum networks always beat classical percolation.” The safe lesson is that entanglement distribution is an operational optimization problem, and independent edge conversion is only one possible protocol.

Many-Body Viewpoint

In many-body systems, entanglement sharing is usually organized by regions rather than named laboratories. The same finite-party ideas reappear with different emphasis:

• pairwise entanglement between sites is only a small part of the correlation structure;
• block entropies describe entanglement across spatial cuts;
• mutual information includes both classical and quantum correlations between regions;
• localizable entanglement asks what measurements on the rest of the system can concentrate onto two chosen sites;
• tensor networks encode entanglement sharing through virtual bonds and cut dimensions.

Area-law behavior is a statement about how entanglement across regions scales with boundary size, not a statement that only neighboring pairs are entangled. Critical systems, topological phases, thermal states, and highly excited states require additional tools beyond the few-qubit examples on this page.

Common Mistakes

• Treating an entanglement graph as if it uniquely specified the quantum state.
• Inferring absence of global entanglement from separable two-party marginals.
• Inferring a product of Bell pairs from the presence of many entangled marginals.
• Confusing hardware connectivity with entanglement connectivity.
• Saying entanglement swapping creates entanglement by local operations alone.
• Forgetting that measurement outcomes and classical communication are part of the protocol.
• Treating entanglement percolation as a universal network-capacity theorem.
• Importing many-body area-law intuition into arbitrary few-party states without checking the cut.

Cross-Links

• Quantum Teleportation
• Multipartite Systems
• Multipartite Separability
• Monogamy of Entanglement
• GHZ States
• W States
• Graph States
• Stabilizer States Preview
• LOCC Preview
• Concurrence for Two Qubits
• Entanglement Entropy
• Mutual Information
• Marginals and Correlations

References

• C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels,” Physical Review Letters 70, 1895-1899, 1993.
• M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-Ready-Detectors Bell Experiment via Entanglement Swapping,” Physical Review Letters 71, 4287-4290, 1993.
• C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating Partial Entanglement by Local Operations,” Physical Review A 53, 2046-2052, 1996.
• A. Acin, J. I. Cirac, and M. Lewenstein, “Entanglement Percolation in Quantum Networks,” Nature Physics 3, 256-259, 2007.
• S. Perseguers, J. Wehr, A. Acin, M. Lewenstein, and J. I. Cirac, “Entanglement Distribution in Pure-State Quantum Networks,” Physical Review A 77, 022308, 2008.
• H. J. Kimble, “The Quantum Internet,” Nature 453, 1023-1030, 2008.
• S. Wehner, D. Elkouss, and R. Hanson, “Quantum Internet: A Vision for the Road Ahead,” Science 362, eaam9288, 2018.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Pairwise graph versus global entanglement. For each state below, draw the pairwise entanglement graph and state whether the pure state is genuinely tripartite entangled:

$$\lvert\Phi^+\rangle_{AB}\otimes\lvert0\rangle_C, \qquad \lvert\mathrm{GHZ}_3\rangle, \qquad \lvert W_3\rangle.$$

Solution

The Bell-pair state has one entangled edge, $AB$, and is not genuinely tripartite entangled because it is product across $AB\vert C$.

The GHZ state has no entangled two-party marginals, but it is genuinely tripartite entangled.

The W state has entangled two-party marginals for all three pairs and is also genuinely tripartite entangled.

Cross-Links

• Tensor Networks Preview — virtual-edge capacities, graph cuts, and the distinction between a network representation and an operational entanglement network.

2. Localizing GHZ entanglement. Measure qubit $C$ of $\lvert\mathrm{GHZ}_3^+\rangle$ in the $X$ basis. What conditional states appear on $AB$?

Solution

Use

$$\lvert0\rangle = \frac{1}{\sqrt2} \bigl( \lvert+\rangle+\lvert-\rangle \bigr), \qquad \lvert1\rangle = \frac{1}{\sqrt2} \bigl( \lvert+\rangle-\lvert-\rangle \bigr).$$

Then

$$\lvert\mathrm{GHZ}_3^+\rangle = \frac12 \Bigl[ \lvert+\rangle_C \bigl( \lvert00\rangle+\lvert11\rangle \bigr)_{AB} + \lvert-\rangle_C \bigl( \lvert00\rangle-\lvert11\rangle \bigr)_{AB} \Bigr].$$

After normalization, the $+$ outcome gives $\lvert\Phi^+\rangle_{AB}$ and the $-$ outcome gives $\lvert\Phi^-\rangle_{AB}$.

3. Swapping record. A middle node performs a Bell measurement during entanglement swapping but does not communicate its outcome. Can $A$ and $C$ use the output as a known Bell pair? Explain what changes when the outcome is communicated.

Solution

No. Conditioned on the middle outcome, $A$ and $C$ share one of four Bell states, but without the record they do not know which Pauli frame describes their pair. Communicating the two-bit result identifies that frame, allowing a local correction or consistent reinterpretation of later measurements. The conditional-state derivation belongs to Entanglement Swapping.

4. Cut accounting. In the swapping setup, why does the final $AC$ Bell pair not violate the rule that LOCC cannot create entanglement?

Solution

The initial state already contains entanglement across the relevant cuts. Across $A\vert B_1B_2C$, the pair $AB_1$ contributes one ebit. Across $AB_1B_2\vert C$, the pair $B_2C$ contributes one ebit. The Bell measurement at $B$ and classical communication reroute this entanglement; they do not create entanglement from a product resource.

5. Independent conversion. A network edge contains

$$\lvert\psi\rangle = \sqrt{\lambda_0}\lvert00\rangle + \sqrt{\lambda_1}\lvert11\rangle, \qquad \lambda_0\ge\lambda_1.$$

What is the optimal single-copy probability of converting it to a Bell pair by LOCC?

Solution

For a two-qubit pure state, the optimal singlet-conversion probability is twice the smaller Schmidt coefficient:

$$p_{\mathrm{singlet}} = 2\lambda_1.$$

This is the probability used by the simplest independent-edge percolation strategy.

6. Why the graph is not the state. Give one reason two quantum states can have the same pairwise entanglement graph but different multipartite structure.

Solution

The graph records only whether chosen two-party marginals are entangled. It does not record phases, higher-order coherences, entanglement across larger bipartitions, localizable entanglement, or operational convertibility. For example, many inequivalent pure states can have all three pairwise marginals entangled, yet differ in three-tangle, stabilizer structure, or behavior under measurements.