Max Teleportation Fidelities

• Average of maximum teleportation fidelities is a performance measure that defines the optimal mean fidelity after protocol-specific optimizations in quantum teleportation.
• It employs a two-step process of optimizing parameters like correction unitaries and gains, then averaging over input states or network node pairs.
• This metric provides insights into fidelity distributions, benchmarks, and network topologies, guiding both theoretical and experimental designs.

Average of maximum teleportation fidelities denotes a family of optimization-and-averaging performance measures used to characterize teleportation protocols and quantum networks. In network theory, the phrase is used literally for the average, over source-target pairs, of the largest teleportation fidelity achievable for each pair under an allowed entanglement-distribution scheme. In much of the protocol literature, the nearest object is instead the maximum average teleportation fidelity: the largest input-averaged fidelity attainable for a fixed resource after optimizing over correction unitaries, measurements, gains, or controller settings. The common structure is an optimization followed by an averaging step, but the optimized variable and the averaging ensemble differ across settings (Mylavarapu et al., 2024, Roy et al., 17 Jun 2026, Cho et al., 26 Nov 2025).

1. Definitions and terminological scope

For standard teleportation of a pure input state $|\phi\rangle$, the single-shot fidelity is the input-output overlap $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$, and the average teleportation fidelity is the Haar average

$F=\int d\phi\, f(\phi),$

with normalized Haar measure $\int d\phi=1$. Several papers augment this first moment with the fidelity deviation

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$

which quantifies the spread of fidelities over the input ensemble rather than the mean alone (Cho et al., 26 Nov 2025, Bang et al., 2018, Song et al., 2021).

Within this protocol-level usage, the relevant “maximum” is usually not $\int d\phi\,\max f(\phi)$. Rather, it is the largest value of the average $F$ obtainable after optimizing over admissible protocol parameters. For isotropic shared resources, this optimization is typically over Bob’s correction structure $\{\hat V_\alpha\}$, equivalently over the composed unitaries $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$ (Cho et al., 26 Nov 2025). In two-qubit analyses it is often an optimization over local-unitary strategies within the standard protocol (Ghosal et al., 2019). In continuous-variable teleportation it is the gain optimization $\max_g \mathcal F$ (Patra et al., 2022). In controlled teleportation it is an average over branchwise maximal fidelities, followed by an optimization over Charlie’s measurement basis (Wei et al., 9 Jun 2026).

A further notational ambiguity comes from the Horodecki-type relation

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$0

used in qudit analyses where $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$1 denotes the maximal overlap with a maximally entangled state, rather than the averaged teleportation fidelity itself (Weinar et al., 2013). By contrast, network papers use the exact phrase “average of maximum teleportation fidelity” for a graph-level quantity,

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$2

namely the average over node pairs of the best end-to-end teleportation fidelity permitted by the network and protocol class (Mylavarapu et al., 2024, Roy et al., 17 Jun 2026).

2. Maximum average fidelity for fixed resources and protocol classes

For $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$3-dimensional teleportation with isotropic shared resource

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$4

the average teleportation fidelity takes the closed form

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$5

Its maximum is achieved when all composed corrections are identity,

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$6

equivalently $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$7, and then

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$8

The corresponding classical benchmark is

$$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$9

and for isotropic states one has

$F=\int d\phi\, f(\phi),$0

Thus the optimum average fidelity is fixed exactly by the isotropic visibility parameter and the dimension (Cho et al., 26 Nov 2025).

In the qubit Werner specialization, the same structure yields

$F=\int d\phi\, f(\phi),$1

with the optimum attained by matching Bob’s correction unitaries to Alice’s Bell-basis labels, $F=\int d\phi\, f(\phi),$2. The classical threshold is $F=\int d\phi\, f(\phi),$3, so useful qubit teleportation requires $F=\int d\phi\, f(\phi),$4 (Bang et al., 2018, Song et al., 2021).

For an arbitrary two-qubit resource state $F=\int d\phi\, f(\phi),$5, the maximal average fidelity achievable within the standard protocol and local-unitary strategies is

$F=\int d\phi\, f(\phi),$6

where $F=\int d\phi\, f(\phi),$7 is the fully entangled fraction. In the canonical form with diagonal correlation matrix $F=\int d\phi\, f(\phi),$8, the closed expression is

$F=\int d\phi\, f(\phi),$9

The state is useful for teleportation iff $\int d\phi=1$0 (Ghosal et al., 2019).

Qudit teleportation with noisy classical communication provides another realization of the same optimization logic. In the Bell-measurement protocol considered there, the average fidelity is

$\int d\phi=1$1

where $\int d\phi=1$2 is the average probability that the Bell outcome is transmitted correctly. The same paper connects this to the Horodecki form

$\int d\phi=1$3

with $\int d\phi=1$4 for the induced noisy resource. The classical threshold remains $\int d\phi=1$5, and beating it requires $\int d\phi=1$6 (Weinar et al., 2013).

3. Fidelity deviation, full fidelity distributions, and universality

The protocol literature increasingly treats the average fidelity as insufficient by itself. For isotropic $\int d\phi=1$7-dimensional teleportation, the deviation obeys the tight bound

$\int d\phi=1$8

Hence $\int d\phi=1$9 forces $D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$0, and the $D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$1 plane acquires a witness geometry. Writing

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$2

the teleportation-advantage and Bell-nonlocality witnesses become

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$3

The identical slope and different intercepts separate the regions of teleportation advantage and CGLMP-inequality violation (Cho et al., 26 Nov 2025).

For optimal teleportation with an arbitrary two-qubit state, the fidelity deviation can itself be computed exactly. In the useful sector $D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$4, the universality condition is

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$5

Moreover, for every prescribed maximal average fidelity $D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$6, there exist states with $D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$7. This identifies a large class of states that are simultaneously useful and dispersion-free (Ghosal et al., 2019).

A stronger statistical extension replaces the first two moments by the full probability density function of actual teleportation fidelities. For single-qubit teleportation, the actual fidelity is treated as a random variable over Haar-random inputs and protocol outcomes, and the general PDF is

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$8

For the optimal classical measure-and-prepare protocol this reduces to

$D=\sqrt{\int d\phi\, f(\phi)^2-F^2},$9

with $\int d\phi\,\max f(\phi)$0 and support on the full interval $\int d\phi\,\max f(\phi)$1. The paper shows that protocols with identical average fidelities can have markedly different distributions, and that local amplitude damping can introduce asymmetries invisible at the level of $\int d\phi\,\max f(\phi)$2 alone (Bussandri et al., 15 Oct 2025).

This suggests that the “maximum” relevant to teleportation quality can refer not only to an optimized mean, but also to support endpoints, upper-tail probabilities, or prior-weighted certification scores. In that sense, the average of maxima and the maximum of averages are only two members of a broader family of fidelity-sensitive diagnostics (Bussandri et al., 15 Oct 2025).

4. Benchmarks beyond Haar-uniform pure inputs

When the input ensemble is not Haar-uniform over pure qubits, the benchmark itself changes. For a von Mises–Fisher prior on the Bloch sphere,

$\int d\phi\,\max f(\phi)$3

the highest mean fidelity achievable without entanglement depends on the concentration parameter $\int d\phi\,\max f(\phi)$4. For the single-qubit projective strategy optimized over measurement axis and Bayesian reconstruction, the exact benchmark is

$\int d\phi\,\max f(\phi)$5

attained at a measurement axis perpendicular to the prior peak, $\int d\phi\,\max f(\phi)$6. As $\int d\phi\,\max f(\phi)$7, this reduces to $\int d\phi\,\max f(\phi)$8; as $\int d\phi\,\max f(\phi)$9, it approaches $F$0. Hence nonuniform prior information can raise the classical no-entanglement threshold far above the uniform-input value (Opatrný et al., 24 May 2026).

A different generalization concerns isotropic distributions of mixed input states. For single-qubit mixed inputs drawn isotropically from the Bloch ball, the standard teleportation protocol with arbitrary resource state admits a maximal average fidelity

$F$1

For Werner resources this becomes

$F$2

and for fixed-purity inputs of Bloch radius $F$3,

$F$4

In this mixed-input setting, the paper shows that separable Werner states and even Bell-diagonal classical-quantum states can exceed the corresponding classical benchmark, whereas the computational basis on Alice’s side never does so for any resource state (Bussandri et al., 2021).

Noisy X-state channels exhibit a different threshold structure. For the standard qubit protocol,

$F$5

Thus entanglement is necessary but not sufficient for quantum advantage; the threshold concurrence is

$F$6

After unambiguous state extraction, the total average fidelity decreases, $F$7, but the successful postselected branch satisfies $F$8, so fidelity can be redistributed into a higher-quality subensemble (Roa et al., 2015).

5. Network-level averages of pairwise maxima

In quantum-network studies, average of maximum teleportation fidelities is a literal graph-level figure of merit. For a pair $F$9, one first identifies the path or entanglement-distribution strategy that yields the largest end-to-end teleportation fidelity, and then averages that optimal pairwise value over all source-target pairs: $\{\hat V_\alpha\}$0 For loopless networks this simplifies because the path is unique, so the “maximum” is formal rather than nontrivial (Mylavarapu et al., 2024, Roy et al., 14 Aug 2025).

With Werner-state links, entanglement swapping along a path $\{\hat V_\alpha\}$1 yields

$\{\hat V_\alpha\}$2

In a homogeneous network with link parameter $\{\hat V_\alpha\}$3 and path length $\{\hat V_\alpha\}$4, this becomes

$\{\hat V_\alpha\}$5

The classical benchmark remains $\{\hat V_\alpha\}$6, so for a path of length $\{\hat V_\alpha\}$7 quantum advantage requires $\{\hat V_\alpha\}$8 (Mylavarapu et al., 2024, Roy et al., 17 Jun 2026).

For loopless repeater networks, this metric sharply distinguishes topologies. In stars and chains with identical links, the star maximizes the network average while the chain minimizes it. In the large-$\{\hat V_\alpha\}$9 limit,

$\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$0

Hence a large star retains quantum advantage iff $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$1, whereas large chains lose it on average (Mylavarapu et al., 2024).

The metric becomes more operationally flexible when multiple paths are allowed and sequential purification is permitted. In that setting, the average network fidelity is evaluated after producing several end-to-end states and purifying them with the Deutsch protocol,

$\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$2

Because purification is commutative but not associative, the ordering matters. For homogeneous networks, the best ordering is Shortest Path Last (SPL), and purification can substantially enhance average teleportation fidelity in highly connected topologies such as complete graphs, triangular lattices, square lattices, and Erdős–Rényi networks, while offering little benefit in large rings (Roy et al., 17 Jun 2026).

Binary-tree repeater networks provide a finer topology comparison. For directed asymmetric, directed symmetric, undirected asymmetric, and undirected symmetric binary trees, closed analytic expressions can be written as

$\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$3

where $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$4 counts allowed length-$\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$5 paths. The directed symmetric binary tree (DSBT) is the best of the four classes. For $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$6, the threshold link qualities for $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$7 are

$\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$8

and for $\hat X_\alpha=\hat V_\alpha\hat U_\alpha^\dagger$9 they are

$\max_g \mathcal F$0

respectively. All four topologies satisfy

$\max_g \mathcal F$1

but DSBT approaches that limit slowest and is therefore the most favorable finite-size binary-tree architecture under this metric (Roy et al., 14 Aug 2025).

6. Generalizations, experiments, and persistent ambiguities

Controlled teleportation introduces a nested optimization structure that is especially close to the literal phrase “average of maximum teleportation fidelities.” After Charlie’s two measurement outcomes, the paper defines the branchwise maximal fidelities

$\max_g \mathcal F$2

and then averages them: $\max_g \mathcal F$3 A further optimization over Charlie’s measurement angles $\max_g \mathcal F$4 gives the optimal average fidelity $\max_g \mathcal F$5. Under generalized noisy channels interpolating between amplitude damping and dephasing, the dependence on the evolution parameter need not be monotonic: when only Charlie’s qubit is noisy and $\max_g \mathcal F$6, the optimal average fidelity first decreases and then increases, whereas for $\max_g \mathcal F$7 it decreases monotonically (Wei et al., 9 Jun 2026).

In continuous-variable teleportation, the analogous quantity is the gain-optimized average fidelity $\max_g \mathcal F$8, evaluated for constrained-uniform or Gaussian-suppressed input ensembles and for Gaussian and non-Gaussian resources. The paper emphasizes that average fidelity and fidelity deviation must be considered jointly, and finds that photon-subtracted resources often give the highest average fidelity in noiseless high-energy regimes, while two-mode squeezed vacuum states are generally more robust under noise (Patra et al., 2022).

Experimental work with imperfect quantum dots shows the same optimization principle in a directly operational form. For a fixed imperfect source, the average teleportation fidelity depends strongly on Bell-state measurement design. A minimal 25% Bell-state measurement gave an average fidelity of $\max_g \mathcal F$9, below the classical threshold, a polarization-selective 50% Bell-state measurement raised it to $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$00, and adding a $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$01 GHz etalon increased it further to $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$02 (Basset et al., 2020). This is not a new abstract metric, but it is an explicit experimental realization of protocol-dependent maximization of an average fidelity.

A persistent ambiguity across the literature is therefore structural rather than notational. Most single-protocol papers do not study the average of per-input maxima. They study the maximum of an average over inputs, or the average over branchwise maxima, or the average over pairwise path maxima. The distinction is substantive: $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$03, $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$04, and $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$05 are different objects, attached to different operational questions (Cho et al., 26 Nov 2025, Bang et al., 2018, Song et al., 2021).

Taken together, these results place average of maximum teleportation fidelities within a broader hierarchy of teleportation metrics. At one end are optimized input averages such as $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$06 or $$f(\phi)=\langle\phi|\rho_{\mathrm{out}}|\phi\rangle$$07; at another are network averages over pairwise optima; and beyond both lie fidelity-deviation, full-distribution, and prior-weighted criteria. The unifying theme is that teleportation quality is governed not only by entanglement strength, but also by what is optimized, over which ensemble the average is taken, and which aspect of the fidelity distribution is operationally relevant.