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.

Helpful background. Multipartite separability distinguishes pairs from all-party entanglement; LOCC specifies which local measurements and classical messages are allowed. A network node may hold several qubits, so always distinguish laboratories from individual tensor factors.

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:

Pure three-qubit state	Entangled pair marginals	Genuinely tripartite entangled?
$\lvert\Phi^+\rangle_{AB}\otimes\lvert0\rangle_C$	$AB$ only	No
$\lvert\mathrm{GHZ}_3\rangle$	None	Yes
$\lvert W_3\rangle$	$AB$, $AC$, and $BC$	Yes

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

Fix the target pair $i,j$, an entanglement measure $E$, and the allowed local measurements on all assisting parties. The localizable entanglement is the supremum of the average output entanglement,

$$E_{\rm loc}(i,j)=\sup_{\text{allowed local measurements}}\sum_m p_m E(\rho_{ij|m}).$$

The outcome record is available for conditional corrections. This is not the entanglement of the unconditional marginal, and allowing joint quantum measurements on separated helpers would be a different operational constraint.

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:

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

Both outcomes have probability $1/2$. Each branch has one ebit; a $Z$ correction in the minus branch produces the same Bell state every time. One ebit is also the maximum for two qubits, so this protocol attains the localizable entanglement for the entropy measure. Thus the pair $AB$ has a separable unconditional marginal while a helper can supply a useful outcome record. 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 initial pairs and outcome record have distinct roles:

• 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.

Without the middle record, the endpoint state is the equal mixture of the four Bell states, $I_A/2\otimes I_C/2$. It is not merely an unknown but usable Bell pair. The swapping derivation gives the conditional states and corrections, and explains the effects of imperfect links and probabilistic analyzers.

Entanglement across a network cut

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 representation with crossing bond dimensions $D_e$, the Schmidt rank obeys $R\le\prod_e D_e$, hence pure-state entropy obeys $S\le\sum_e\log_2D_e$. These are upper bounds, not guaranteed saturation; tensor networks explains the representation. 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

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

A local success operator $K=\operatorname{diag}(\sqrt{\lambda_1/\lambda_0},1)$ produces $\sqrt{\lambda_1}(|00\rangle+|11\rangle)$, whose norm squared is $2\lambda_1$; the complementary branch is product. This constructs the probability, while the pure-state conversion theorem establishes its optimality. Here $\lambda_1$ is the smaller Schmidt probability, not the amplitude $\sqrt{\lambda_1}$.

For $(\lambda_0,\lambda_1)=(0.8,0.2)$, a single edge succeeds with probability $0.4$. Two independent edges both succeed with probability $0.16$; subsequent ideal deterministic swapping then joins their endpoints. A swapping analyzer that succeeds only with probability $s$ changes this simple route’s success probability to $0.16s$.

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. Under the assumed independent-edge model on an appropriate infinite lattice, probabilities above the bond-percolation threshold allow an infinite connected cluster. A finite network instead has a size- and geometry-dependent connection probability, with no sharp infinite-system transition.

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.

Independent edge conversion is one protocol. Whether another strategy improves the connection probability depends on the particular states, geometry, and allowed measurements.

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.

References

• A. Acín, J. I. Cirac, and M. Lewenstein, “Entanglement percolation in quantum networks”, Nature Physics 3, 256–259 (2007).
• C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, “Concentrating partial entanglement by local operations”, Physical Review A 53, 2046–2052 (1996).
• S. Perseguers, J. Wehr, A. Acín, M. Lewenstein, and J. I. Cirac, “Entanglement distribution in pure-state quantum networks”, Physical Review A 77, 022308 (2008).
• F. Verstraete, M. Popp, and J. I. Cirac, “Entanglement versus correlations in spin systems”, Physical Review Letters 92, 027901 (2004).
• G. Vidal, “Entanglement of pure states for a single copy”, Physical Review Letters 83, 1046–1049 (1999).
• M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “‘Event-ready-detectors’ Bell experiment via entanglement swapping”, Physical Review Letters 71, 4287–4290 (1993).

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.

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. Without the record, their state is $\tfrac14\sum_k|B_k\rangle\langle B_k|=I_4/4$, a product state with no usable endpoint entanglement. Conditioned on the record, each branch is a Bell state. Communicating the two-bit result identifies that frame, allowing a local correction or consistent reinterpretation of later measurements. The entanglement-swapping calculation derives these conditional states.

4. Network cuts. 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 probability:

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

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.