In qubit teleportation, entanglement is necessary but not sufficient for beating the classical fidelity limit of 2/32/3: the classical channel between Alice and Bob must also carry enough information. Banik and Gazi (2013) quantified the classical communication of teleportation as the Shannon mutual information I(X;Y)I(X;Y) between Alice's classical input and Bob's output, and showed that roughly 0.2080.208 bits are sufficient to reach non-classical fidelity F>2/3F>2/3 with a Werner-state resource, proving necessity of 0.2080.208 bits only for channels that transform the teleported state isotropically. The exact minimal amount of classical communication that is necessary for non-classical fidelity in the general case — arbitrary protocols beyond the isotropic/standard Bell-measurement setting, and arbitrary two-qubit entangled resources beyond the Werner family — remains undetermined. Weinar, Laskowski, and Pawlowski (2013) extended the threshold to qudits within standard Bell-basis protocols and explicitly conjectured, without proof, that it holds in general; Bej, Halder, and Sengupta (2021 preprint, published 2023) formulated the resource-dependent cost as a constrained optimization but obtained only stationarity conditions. This record poses the open problem: determine the exact necessary (and jointly sufficient) mutual-information cost of the classical channel for achieving non-classical teleportation fidelity in full generality.

Background

Quantum teleportation (Bennett et al., 1993) transfers an unknown qubit from Alice to Bob using a shared entangled pair and classical communication: Alice performs a joint measurement on the unknown qubit and her half of the pair (in the standard protocol, a Bell measurement — a projection onto the four maximally entangled two-qubit Bell states), sends the outcome to Bob, who applies a corrective unitary. The quality of teleportation is measured by the average fidelity FF over all pure input states. Without entanglement, any measure-and-prepare strategy (measure the incoming state, then prepare a new state conditioned on the outcome) achieves at most Fcl=2/3F_{\mathrm{cl}}=2/3 for qubits (more generally 2/(d+1)2/(d+1) for dd-dimensional systems), so F>2/3F>2/3 — 'non-classical fidelity' — certifies genuinely quantum transmission. Horodecki, Horodecki, and Horodecki (1999) showed that in the standard protocol with resource state ρ\rho, F=(2f+1)/3F=(2f+1)/3 where ff is the fully entangled fraction (the maximal overlap of ρ\rho with a maximally entangled state), so a resource is useful exactly when f>1/2f>1/2; the Werner states Wα=(1−α)∣ψ−⟩⟨ψ−∣+α 1/4W_\alpha=(1-\alpha)|\psi^-\rangle\langle\psi^-|+\alpha\,\mathbb{1}/4, mixtures of a singlet with white noise, form the canonical noisy-resource family.

While the entanglement requirements of teleportation were thoroughly characterized (see the Horodecki et al. review, 2009), Banik and Gazi (2013) asked the complementary question: how much classical communication is needed? They kept a shared (Werner) entangled resource but replaced the perfect two-bit classical channel of the standard protocol by noisy ones, quantifying the classical communication actually transmitted as the Shannon mutual information I(X;Y)I(X;Y) between Alice's channel input XX and Bob's channel output YY. They studied two models: (i) two independent noisy 1-bit channels (each a binary symmetric channel that flips its bit with some probability), with total cost 2−H(η)−H(δ)2-H(\eta)-H(\delta) where HH is the binary Shannon entropy and η,δ\eta,\delta the channel reliabilities; and (ii) a single noisy 2-bit channel with transition probabilities p1,…,p4p_1,\dots,p_4, with cost I=2+∑ipilog⁡2piI=2+\sum_i p_i\log_2 p_i. Their results: on average (0.208+ϵ)(0.208+\epsilon) bits suffice for F>2/3F>2/3 with a singlet resource in model (ii), attained at p1=1/2p_1=1/2, p2=p3=p4=1/6p_2=p_3=p_4=1/6 where the channel acts isotropically; about 0.2550.255 bits suffice in model (i); and 0.2080.208 bits are necessary if the channel transforms the teleported state isotropically — proved by a dense-coding argument (dense coding is a protocol in which shared entanglement boosts the classical information carried by a transmitted quantum system) in which Alice encodes unitaries on the shared singlet and the Holevo bound (the upper bound on extractable classical information from a quantum ensemble) limits what a cheaper classical channel could convey. They also showed the sufficient cost grows as the Werner resource weakens. Their Discussion states explicitly that the optimal necessary amount 'still remains an open problem' and that the necessary amount for general entangled resources is unstudied.

Later work sharpened but did not close this. Weinar, Laskowski, and Pawlowski (2013) extended the analysis to qudits (dd-level quantum systems): within standard teleportation protocols (Bell-basis measurement), the minimal classical-channel capacity below which no shared entangled state can beat the classical limit is Cmin⁡(d)=log⁡2d−(1−1d)log⁡2(d+1)C_{\min}(d)=\log_2 d-(1-\tfrac{1}{d})\log_2(d+1), reproducing 0.2080.208 bits at d=2d=2 and reaching a non-monotonic maximum of 0.2590.259 bits at d=4d=4; they explicitly state that they restricted to standard protocols with maximally entangled states and Bell-basis measurements and that proving the general case 'is an open avenue of research.' Bej, Halder, and Sengupta (2021) studied combined noisy resources (general two-qubit states in canonical form ρ=14(1⊗1+∑itiiσi⊗σi)\rho=\tfrac{1}{4}(\mathbb{1}\otimes\mathbb{1}+\sum_i t_{ii}\sigma_i\otimes\sigma_i)) and noisy classical channels, derived the optimal fidelity, and formulated the minimum classical communication cost as a constrained minimization of I(X;Y)I(X;Y) — but the Lagrange conditions are transcendental, no closed-form solution is known even in this Bell-measurement setting, and no necessity result beyond the protocol class was obtained. The exact classical-communication threshold for non-classical teleportation fidelity therefore remains open in the general case.

Problem Statement

Determine the exact minimal amount of classical communication, quantified as the Shannon mutual information I(X;Y)I(X;Y) between Alice's classical-channel input and Bob's output, that is necessary for achieving non-classical average teleportation fidelity (F>2/3F>2/3 for a qubit, with the corresponding 2/(d+1)2/(d+1) limit in dimension dd). Concretely: (1) Prove or refute that the threshold 0.2080.208 bits for qubits — equivalently Cmin⁡(d)=log⁡2d−(1−1d)log⁡2(d+1)C_{\min}(d)=\log_2 d-(1-\tfrac{1}{d})\log_2(d+1) for qudits (dd-level quantum systems), as conjectured by Weinar, Laskowski, and Pawlowski — is the exact boundary below which no entangled resource, measurement strategy, or protocol whatsoever can beat the classical limit, extending the known necessity results (which hold only for isotropic transformations and for standard Bell-basis protocols) to arbitrary protocols, including general POVMs (positive-operator-valued measures, the most general form of quantum measurement) on Alice's side, arbitrary recovery operations on Bob's side, and any use of the classical channel. (2) For a general two-qubit entangled resource beyond the Werner/isotropic families — e.g. a canonical-form state with correlation-matrix entries tiit_{ii} — determine the minimal mutual-information cost required to attain F>2/3F>2/3, without restriction to any particular protocol class; the constrained minimization formulated (but not solved) by Bej, Halder, and Sengupta within Bell-measurement protocols is the closest known partial formulation of this question, not a restriction of its scope. A complete answer consists of a matching pair: a converse (necessity) bound valid in the stated generality, and an explicit protocol achieving non-classical fidelity at that cost. Sub-questions (1) and (2) are separate mandatory deliverables, and a refutation resolves the sub-question it refutes on its own: an explicit protocol attaining F>2/3F>2/3 at a mutual-information cost strictly below the claimed threshold — below 0.2080.208 bits in the qubit case of (1) (respectively below Cmin⁡(d)C_{\min}(d) in dimension dd), or below the claimed minimal cost for the resource in (2) — is a complete answer to that sub-question, with no matching converse bound required; the matching-pair form above is the completion standard for an affirmative answer.

References

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