Deterministic Teleportation in GPT and Quantum

• Deterministic teleportation is a protocol that transfers an unknown quantum state with unit probability through joint measurements, classical communication, and correction maps.
• It relies on regular composites which preserve product states and positivity under partial evaluations, and employs group symmetries to ensure consistent error correction.
• The framework distinguishes deterministic from probabilistic teleportation by highlighting key structural differences across classical, quantum, and postclassical theories.

Deterministic teleportation is a class of protocols in which an unknown quantum state or operation is transferred with unit probability (i.e., every attempt succeeds, without postselection) from a sender to a receiver, with a success metric quantified by retrieval fidelity. These protocols have been investigated across generalized probabilistic theories, as well as within quantum information science in various physical platforms and abstraction levels. Deterministic teleportation is contrasted with conclusive (probabilistic) teleportation, which relies on measurement outcomes or postselection filtering, and whose success is intrinsically less than one. This article presents an overview of deterministic teleportation, including its operational definitions, structural requirements, mathematical formalism, distinction from classical and generic nonclassical models, and its broader implications for the theory and implementation of quantum (and beyond-quantum) information processing (0805.3553).

1. Definitions and Core Concepts

Deterministic teleportation refers to a protocol in which, for every allowed state $\alpha$ of an input system $A$, the outcome of a suitable measurement on a composite system—combined with classical communication and local correction—allows the receiver to recover $\alpha$ with certainty and without the need for postselection.

Operationally, such a protocol involves:

• A shared bipartite state (channel) $\omega$ between sender and receiver.
• A joint measurement (often encoded as an “effect” $f$) on the sender’s side.
• A (possibly conditional) correction map $\tau$ on the receiver’s side, which is typically a symmetry or reversible transformation.

In formal language, the protocol defines an “auxiliary” map: $\mu = \omega \circ f : A \to B$ The process is deterministic if, for all normalized states $\alpha$ in the state space $\Omega$,

$\tau(\mu(\alpha)) = \mu(\alpha) \eta(\alpha)$

where $t=1$ for perfect deterministic teleportation and $\eta$ is an isomorphism ($A \simeq B$). In “strong” deterministic teleportation, even the application of $\tau$ is not required—the protocol outputs $\alpha$ unconditionally at the receiver.

A necessary resource for such a protocol is a composite system (of three or more parties) with sufficient regularity and symmetry to support the required state, measurement, and correction structure. The paper introduces the notion of “regular” composites to capture these requirements.

2. Regular Composites and Structural Requirements

A regular composite is a class of system tensor products in generalized probabilistic theories (GPTs) that is closed under coarse-graining and partial evaluation (i.e., tracing out subsystems or forming marginals does not violate the positivity or dynamical closure properties required by the protocol).

Mathematically, for a composite $A$ of factors $A_1, A_2, ..., A_n$ (state spaces in the GPT sense), regularity requires:

• The composite contains all product states/effects.
• The positivity cone is stable under all “partial evaluations,” meaning that tracing out in any subsystem yields a positive state in the remaining factor(s).
• Dynamical admissibility: positive maps on factors extend naturally to the composite.

Regularity plays an essential role because teleportation protocols, and their necessary correction operations, involve partitioning and recombining subsystems in different ways (e.g., Alice’s vs Bob’s side). Without regularity, constructions of the necessary joint measurement and correction maps may fail.

3. Conditions for Deterministic Teleportation

Within the convex framework of GPTs, the conditions for deterministic (and conclusive) teleportation are rigorously formulated. For a regular tripartite composite—systems $A_1, A_2, B$—the necessary and sufficient condition for conclusive teleportation is the existence of an effect $f$ and bipartite state $\omega$ such that $\mu = \omega \circ f : A \to B$ is invertible (more precisely, an order-isomorphism) onto its image, and a contractive correction mapping $\tau$ exists satisfying, for all normalized states $\alpha$: $$\tau(\mu(\alpha)) = t_\alpha \mu(\alpha) \eta(\alpha)$$ with $t_\alpha > 0$. For deterministic teleportation, $t_\alpha$ is independent of $\alpha$ (in the strong protocol, $t_\alpha = 1$ for all $\alpha$).

A sufficient condition for deterministic teleportation is the presence of a group symmetry such that:

• The state spaces are weakly self-dual.
• There exists an order-isomorphism (e.g., $\phi: A \to A$ or $\eta: A \to B$) compatible with a finite group $G$ acting transitively on the set of pure states $\Omega$.
• An observable is constructed as the $G$-orbit of a given effect. The protocol then selects a measurement outcome $f_i$ from the orbit; for each outcome, the correction is a known symmetry, guaranteeing recovery of $\alpha$.

Conclusive teleportation is more general (possibly postselected and/or probabilistic), while deterministic is the limiting case where all outcomes yield success.

4. Mathematical Framework

The rigorous structure of deterministic teleportation in generalized probabilistic theories relies on the following mathematical elements:

• The normalized state space,

$\Omega = \{ \alpha \in A_+ : u(\alpha) = 1 \}$

where $u$ is the order unit.

• The composite is equipped with a bipartite state $\omega$ and measurement effect $f$, defining the conditional state via: $\mu = \omega \circ f : A \to B$
• For conclusive/deterministic teleportation, a correction map $\tau$ (typically an automorphism/symmetry of $B$) must exist such that: $$\tau(\mu(\alpha)) = t_\alpha \mu(\alpha) \eta(\alpha)$$ Strong deterministic protocols set $t_\alpha = 1$ and $\eta=$ identity.

In the symmetric case, observables ($\{f_g\}$) are constructed by group averaging: $f_g = \frac{1}{|G|} \omega^{-1} \circ g$ so that

$\omega \circ f \circ \mu = \text{identity}$

up to the relevant isomorphism. This exhibits the central role of order-isomorphisms and group actions.

These constructions do not require the underlying theory to be quantum or classical; similar structures can arise in nonclassical, nonquantum theories provided the necessary regularity and symmetry are present.

5. Theoretical Implications and Distinctions

The existence of deterministic teleportation protocols articulates a crucial structural difference between classical, quantum, and general nonclassical probabilistic theories:

• Classical probabilistic models (simplicial state spaces) render teleportation trivial or degenerate, since all “copying” is allowed without restriction.
• Quantum theory supports deterministic teleportation by virtue of entanglement and the structure of quantum measurements.
• Most nonclassical GPTs lack the specific entanglement structure and group symmetry to enable teleportation: the no-cloning and no-broadcasting theorems are generic in nonclassical GPTs, but teleportation is exceptional rather than generic.
• The work establishes that families of theories neither classical nor quantum can support deterministic teleportation if their composite state spaces are regular and possess the requisite group symmetry (self-duality, equivariant isomorphisms, transitive group actions on pure states).

This result implies that teleportation is a more delicate and demanding phenomenon than other quantum-information-theoretic features, and its generic appearance in quantum theory is underpinned by special structural properties.

6. Broader Impact and Further Directions

The axiomatic characterization of deterministic teleportation in the GPT framework enables several consequences and research directions:

• Identification of the minimal convex-geometric and symmetry conditions for reliable state transfer, guiding the search for “postclassical” information theories.
• Systematic exploration of operational protocols (error correction, cloning, entanglement swapping) in theories not restricted to quantum or classical mechanics.
• Implications for physical theories beyond standard quantum mechanics, including possible models arising in quantum gravity, categorical quantum mechanics, or exotic probabilistic settings.
• Insights into the compositional structure (minimal vs maximal tensor products), which bears directly on the physical realizability of multipartite protocols in alternative theories.
• Potential applications to the general design of communication protocols where deterministic state transfer is desired in non-standard information processing models.

These generalized insights set the stage for extending teleportation and related protocols to diverse physical and mathematical frameworks, enriching both theoretical foundations and the catalog of admissible information-processing architectures (0805.3553).