Many-Body Entanglement Swapping Protocol

• The protocol is a quantum communication method extending standard pairwise swapping to enable non-signaling parties to share multipartite states with preserved Schmidt structures.
• It employs local state preparation, intermediary unitary operations, and postselection to achieve high fidelity by managing Schmidt coefficient variance.
• This method supports flexible, resource-efficient quantum networking and fault-tolerant sharing of complex states, with successful demonstrations on IBM quantum hardware.

A many-body entanglement swapping protocol is a quantum communication technique that generalizes standard pairwise entanglement swapping to enable two non-signaling parties to share arbitrary many-body quantum states along arbitrary partitions. The protocol preserves the complex entanglement structure of a given target state—including its Schmidt vectors—and typically achieves high fidelity, especially as the variance of the Schmidt coefficients decreases. Beyond traditional pairwise scenarios, this approach provides a mechanism for flexible, efficient, and potentially fault-tolerant sharing of complex quantum states necessary for advanced quantum networking and distributed quantum information applications (Huhtanen et al., 27 Jun 2025).

1. Principles and Conceptual Motivation

Traditional entanglement swapping protocols allow distant parties to share Bell pairs by projecting qubits of intermediate pairs onto a Bell basis via joint measurement. The many-body protocol extends this to arbitrary multipartite states: instead of merely swapping entanglement between two pairs of qubits, it enables non-signaling parties—Alice and Bob, with arbitrarily partitioned Hilbert spaces—to share the same entanglement structure as a chosen “target” many-body state.

Both Alice and Bob independently generate local copies of the target state |$\psi_T\rangle$, constructed as

$|\psi_T\rangle = U |\sigma_0\rangle$

where $U$ is a known unitary, and $|\sigma_0\rangle$ a fixed product state. The state definition incorporates arbitrary bipartitions, and its entanglement is revealed in its Schmidt decomposition: $$|\psi_T\rangle = \sum_i \sqrt{\lambda_i} \, |\lambda_i^A\rangle \otimes |\lambda_i^B\rangle$$ where $\{\lambda_i\}$ are Schmidt coefficients and $|\lambda_i^A\rangle, |\lambda_i^B\rangle$ are corresponding Schmidt vectors for Alice and Bob.

2. Protocol Construction and Theoretical Details

The protocol operates in the following steps:

1. State Preparation: Both Alice and Bob prepare local copies of the target state, then partition their qubits—each party keeps their own partition and sends the remainder to a third, intermediary party ("Eve").
2. Central Unitary and Measurement: Eve receives the complementary sets from both Alice and Bob, applies a unitary operation $U_E$ (chosen as the inverse $U^\dagger$ in a suitably reordered basis), and then measures her register in the computational basis.
3. Postselection: The protocol postselects on a particular measurement outcome, typically $|\sigma_0\rangle$; Eve communicates this outcome to Alice and Bob.
4. Resultant Shared State: When Eve observes $|\sigma_0\rangle$, the final shared state between Alice and Bob is

$$|\psi_{AB}\rangle = \frac{1}{\sqrt{p_E}} \sum_{i,j} \sqrt{\lambda_i \lambda_j}\, r_{ij}^E\, |\lambda_i^A\rangle \otimes |\lambda_j^B\rangle$$

where $$r_{ij}^E = \langle \sigma_E | U_E (|\lambda_i^B\rangle \otimes |\lambda_j^A\rangle)$$.

If $U_E$ and the measurement outcome are chosen ideally, $r_{ij}^E = \sqrt{\lambda_i}\, \delta_{ij}$, yielding

$$|\psi_{AB}\rangle = \sum_i \frac{\lambda_i^{3/2}}{\sqrt{p_0}} |\lambda_i^A\rangle \otimes |\lambda_i^B\rangle$$

with normalization $p_0$ (see below).

The resulting state reproduces the full entanglement structure—i.e., the Schmidt vectors—of the target state, differing only by a transformation of the Schmidt coefficients.

3. Fidelity, Postselection Probability, and Resource Cost

The protocol’s practical efficiency and accuracy are determined by:

• Success (Postselection) Probability:

$$p_0 = \sum_i \lambda_i^3 = \exp(-2 S_3(\{\lambda_i\}))$$

where $S_3$ is the third Rényi entropy of the Schmidt distribution. Cost is thus directly linked to the entanglement entropy of the target state—states with low entropy (i.e., less entangled or with uniform Schmidt spectrum) result in higher success probability and resource efficiency.

• Fidelity:

$$F = \langle \psi_T | \psi_{AB} \rangle = \frac{\sum_i \lambda_i^2}{\sqrt{\sum_j \lambda_j^3}} = \exp[S_3(\{\lambda_i\}) - S_2(\{\lambda_i\})]$$

Fidelity achieves unity for maximally entangled states (uniform $\lambda_i$), and remains high when variance in $\lambda_i$ is small. For large systems with uniform Schmidt spectrum, resource cost grows only slowly.

• Scaling: The protocol’s resource overhead grows with the entropy, not directly with the number of qubits. In low-entanglement (area-law) states, this enables high scalability.

Quantity	Formula	Significance
Postselection p₀	$\sum_i \lambda_i^3$	Sets implementation cost
Fidelity F	$(\sum_i \lambda_i^2) / \sqrt{\sum_j \lambda_j^3}$	Measures closeness to target state
Dependence on entropy	$p_0 = \exp(-2 S_3)$, $F = \exp[S_3 - S_2]$	Shows link to Rényi entropies

4. Preservation of Schmidt Structure and Theoretical Guarantees

A key feature is the exact preservation of the target state’s Schmidt vectors across the protocol. While the protocol modifies the Schmidt coefficients (cubing and normalizing them), the entanglement "structure"—i.e., the nonlocal basis—is exactly those of the target state. As a result, applications that depend critically on the presence of particular Schmidt vectors (e.g., distributed quantum algorithms or simulations of many-body physics) can operate on the swapped state as if it were the original.

In the limit where all $\lambda_i = 1/d_S$ (maximally entangled state of rank $d_S$), the protocol is lossless: success probability $p_0 = d_S^{-2}$ and fidelity $F$ approaches 1.

5. Implementation and Demonstration on Quantum Hardware

The protocol has been benchmarked on IBM superconducting quantum hardware, with demonstrations for GHZ states up to 12 qubits (with 6 qubits in the entangled partition). The standard sequence involves local GHZ state creation (Hadamard + CNOT chain), transmission of half the qubits to Eve, and Eve’s action (inverse unitary plus measurement). Outcomes differing from the ideal measurement induce only a local bit-flip on the shared state, correctable by local operations.

Multiple readout approaches—including raw measurement ("Sampler"), expectation values ("Estimator"), and error mitigation (dynamic decoupling)—were tested. Even in the presence of considerable gate and readout noise, practical fidelities remained high, and the protocol was compatible with error mitigation and detection schemes.

6. Extensibility: Fault Tolerance, Error-Detection, and Complex State Sharing

Beyond the basic many-body state sharing, the protocol offers several functionalities:

• Flexible sharing: It accommodates arbitrary, possibly non-Clifford many-body states, including those with complex or non-uniform entanglement structure beyond simple Bell/GHZ/W states.
• Fault-tolerant swapping: By encoding information in logical qubits (e.g., via stabilizer codes), the protocol can be combined with quantum error correction for error-detected or error-corrected swapping. Redundancy and error patterns in Eve’s measurement outcomes reveal likely error syndromes, assisting in fault tolerance.
• Hardware efficiency: The cost in required repetitions depends primarily on entanglement entropy, not raw qubit count, making the protocol well suited for systems where traditional pair swapping would be intractable.

7. Implications and Outlook

The many-body entanglement swapping protocol establishes a method for non-signaling parties to establish entanglement resources suitable for advanced quantum communication, networked computation, and the remote simulation of complex quantum systems. Its entropic cost metric and inherent compatibility with fault-tolerant techniques position it as a promising element in the development of scalable, error-tolerant quantum networks and distributed quantum processors.

The protocol’s demonstration on actual quantum hardware, along with its design for hardware efficiency and compatibility with error correction, underscores its utility for both near-term and future quantum technologies where networking of large, complex quantum states is a necessity (Huhtanen et al., 27 Jun 2025).