Teleportation-based quantum state tomography (QST), introduced by Rigolin (APS Open Science 1, 000046, 2026), inverts the roles of the standard teleportation protocol: known probe states are teleported through an unknown shared state, and Bell measurements plus classical communication alone let one party fully reconstruct that shared state, for arbitrary n-qubit and (after Rigolin's 2026 follow-up) n-qutrit density matrices. Two linked questions that this framework explicitly poses remain open. First, can the continuous-variable (CV) teleportation protocol be modified into a tomography scheme that fully reconstructs CV states, such as a two-mode or n-mode Gaussian state, using only local operations and classical communication in the reconstruction? Second, what is the minimum number of teleportation experiments and known input states needed to reconstruct structured state classes, such as matrix-product states or nearly pure states, which standard tomography already handles with far fewer resources than generic states require?

Background

Quantum state tomography (QST) is the task of reconstructing an unknown quantum state, described by a density matrix, from measurement statistics on many identically prepared copies. The quantum teleportation protocol of Bennett et al. (1993) transfers an unknown qubit from a sender (Alice) to a receiver (Bob): Alice jointly measures her unknown qubit and her half of a shared maximally entangled pair using a Bell measurement — a projection onto the four maximally entangled Bell states — and sends the classical outcome to Bob, who applies a corresponding correction to his half. Rigolin (2025, published 2026) observed that this protocol can be repurposed for tomography by exchanging what is known and what is unknown: if the state shared between Alice and Bob is the unknown object to be reconstructed, while Alice teleports a sequence of known pure input states through it, then Bob's single qubit at the end of each teleportation run depends linearly on the density-matrix elements of the shared state. A minimum of four known input states, lying on the equator and poles of the Bloch sphere, suffices to make the resulting linear system invertible, so Bob can fully reconstruct the shared two-qubit state; the construction extends to three-qubit and arbitrary n-qubit states and works in a distributed setting where every party except Bob performs only local Bell measurements and communicates classically. Rigolin's follow-up (Physics Letters A 590, 131811, 2026) extended the same idea to qutrits (three-level systems, the spin-1 analog of qubits) using nine known single-qutrit probe states and the spin-1 analog of the Bell basis, reconstructing arbitrary single and multipartite qutrit states; without prior information the resources scale exponentially with the number of parties, as in standard QST. For continuous variables — quantum systems with infinite-dimensional Hilbert spaces, such as optical field modes described by position-like and momentum-like quadrature operators — teleportation is also well developed: Vaidman (1994) and Braunstein and Kimble (1998) formulated it with a two-mode squeezed (Einstein–Podolsky–Rosen-type) resource and a joint quadrature measurement playing the role of the Bell measurement, and Furusawa et al. (1998) demonstrated it experimentally. CV states commonly targeted by tomography are Gaussian states, whose Wigner phase-space distribution is Gaussian and which are therefore completely characterized by finitely many parameters: the mean values of the quadratures and their covariance matrix. Notably, LOCC-only reconstruction (each party acts only on its own subsystem and communicates classically) of two-mode Gaussian states is already known outside the teleportation framework: Haruna, de Oliveira, and Rigolin (2007) identified a minimal set of local measurements for quantifying two-mode Gaussian entanglement, and Rigolin and de Oliveira (2008, 2009) achieved complete reconstruction of a two-mode Gaussian state via local operations and classical communication. Efficient tomography of Gaussian processes probed with a few coherent states is also known (Wang et al., 2013). What does not yet exist is the teleportation-based CV analog of Rigolin's qubit and qutrit protocols, and correspondingly the source works pose as open problems whether the CV teleportation protocol itself can be modified so that the shared CV state — for example a two-mode or n-mode Gaussian state — is the object reconstructed, with only local operations and classical communication allowed in the reconstruction. The same works also leave open the resource question for structured states: matrix-product states (MPS), which are one-dimensional tensor-network states with limited entanglement, specified by local tensors and a bond dimension, and nearly pure states, can both be reconstructed by standard tomography with far fewer measurement settings than generic states; whether the teleportation-based framework admits such resource reductions, and what the true minimum number of teleportation experiments is for these classes, is undetermined.

Problem Statement

The teleportation-based quantum state tomography framework is established for arbitrary finite-dimensional states: for qubits, four known single-qubit input states and Bell measurements plus classical communication suffice to fully reconstruct an arbitrary n-qubit density matrix; for qutrits, nine known single-qutrit states and spin-1 Bell measurements do the same. Two linked extensions of this framework, posed as open problems by the source works, remain unresolved. Both extend teleportation-based QST from generic finite-dimensional states to regimes where it currently does not exist, and each corresponds to a distinct open question posed by the sources.

(1) Continuous-variable extension. Modify the continuous-variable teleportation protocol (Vaidman; Braunstein–Kimble) so that it works as a tomography protocol in the sense above: known continuous-variable probe states are teleported through an unknown shared continuous-variable state — an arbitrary n-mode Gaussian state, for every mode number n, with the two-mode case (n = 2) as the basic instance — and the shared state is fully reconstructed from the measurement outcomes using only local operations and classical communication, in analogy with the qubit and qutrit protocols. Full reconstruction here means that the protocol determines all parameters of an arbitrary n-mode Gaussian state — all quadrature mean values and all entries of the covariance matrix — for every mode number n, rather than of a restricted subfamily such as a single known state or a class with only one free parameter. A complete answer should specify the probe states and the continuous-variable analog of the Bell measurement, give the reconstruction procedure, demonstrate that the map from the shared state's parameters to the observed outcome statistics is invertible so that the reconstruction is unique, and characterize the resources used (number and type of probe states and measurements).

(2) Minimal resources for structured states. Within the teleportation-based framework (qubit or qutrit version), determine the minimum number of teleportation experiments and known input states needed to reconstruct structured classes of states — such as matrix-product states or nearly pure states — for which standard tomography techniques are known to require far fewer resources than generic-state tomography. A complete answer should give, for at least one precisely specified structured class, matching upper and lower bounds: an explicit protocol achieving the claimed number of teleportations and a proof that fewer cannot suffice. The structured class must be a genuinely unknown family: membership in the class alone must not determine the state, and the protocol must determine the unknown parameters of an arbitrary member of the class (as is the case, for example, for matrix-product states with a specified bond dimension, or for nearly pure states). Singleton classes, or classes consisting only of a priori known states, do not qualify.

References

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