Exact resource costs of remote state preparation: Lo's conjecture beyond the oblivious and qubit cases, the mixed-state entanglement–communication tradeoff, multipartite preparation, and RSP capacities of noisy channels

Remote state preparation (RSP) is the variant of quantum teleportation in which the sender classically knows the state to be delivered to the receiver. The founding 2001 paper of Bennett, DiVincenzo, Shor, Smolin, Terhal, and Wootters left open the exact resource accounting for RSP: whether an exactly faithful, finite-blocklength protocol can beat teleportation's classical cost (Lo's conjecture), the full achievable region in the entanglement-versus-classical-communication plane, how closely the Holevo bound can be approached for ensembles of entangled states, multiparty preparation with separately shared entanglement, and the unassisted RSP capacity of a general noisy channel. The asymptotic pure-state tradeoff curve and the Holevo-bound approachability question were resolved by 2005 (Berry–Sanders; Bennett–Hayden–Leung–Shor–Winter), and Lo's conjecture was confirmed for oblivious protocols and for single qubits (Leung–Shor; Hayashi–Hashimoto–Horibe). The surviving open core is precise and nonempty: Lo's conjecture for dimension $d\geq 3$ without obliviousness, the full entanglement–communication region for mixed-state ensembles (one-shot and asymptotic), RSP entanglement cost beyond Schmidt rank, the multiparty bit-allocation question as posed, and a tight evaluation of the RSP capacity of noisy channels.

Background

Quantum teleportation transmits one instance of an unknown quantum state from a sender (Alice) to a receiver (Bob) using prior entanglement and forward classical communication: for a qubit, two classical bits plus one ebit (a shared maximally entangled pair of qubits), and both resources are necessary, with no tradeoff between them. In remote state preparation (RSP) the goal is the same — Bob ends up holding a single specimen of the state — but Alice is given the full classical description of the state rather than a quantum system. Pati and Lo showed that special ensembles (e.g. equatorial qubit states $\frac{1}{\sqrt{2}}(|0\rangle+e^{i\varphi}|1\rangle)$) can be remotely prepared with one classical bit and one ebit, and Lo conjectured that, for an arbitrary $N$-dimensional pure state, RSP requires $2\log_2 N$ classical bits, exactly as in teleportation (Lo's conjecture). The subject of this task is the set of resource-cost questions posed by Bennett, DiVincenzo, Shor, Smolin, Terhal, and Wootters in the founding RSP paper (Phys. Rev. Lett. 87, 077902 (2001)).

A few conventions are needed. A protocol is exactly faithful if it always produces the target state exactly; it is asymptotically faithful if the output fidelity (the squared overlap, generalized to mixed states) approaches one as the number $n$ of prepared states grows; a protocol is oblivious to the receiver if Bob learns nothing about the prepared state beyond the single specimen he obtains (as in teleportation, where his correction depends only on the classical message). Entanglement is measured in ebits and classical communication in bits; the tradeoff region is the set of achievable pairs (ebits per state, forward bits per state) — denoted $(e,b)$. For an ensemble of states $\{p_i,\rho_i\}$, the Holevo information is $\chi=S(\bar{\rho})-\sum_i p_iS(\rho_i)$, where $S$ is the von Neumann entropy and $\bar{\rho}=\sum_i p_i\rho_i$; causality forces the forward classical cost of RSP to be at least $\chi$, since the remotely prepared ensemble could otherwise be used to signal more information than was sent. For a noisy quantum channel $\mathcal{N}$, the classical capacity $C$ is its asymptotic rate for reliable classical transmission, the quantum capacity $Q$ its rate for reliable quantum-state transmission, and the entanglement-assisted classical capacity $C_E$ its classical rate with unlimited shared entanglement.

The 2001 paper established that, with a large amount of shared entanglement, the asymptotic classical cost of RSP for general states is one bit per qubit — half that of teleportation — with most of the entanglement recoverable afterward using backward classical communication; it gave one point $(e_0\approx 3.79,b=1)$ on the tradeoff curve, schematic low-entanglement curves, and (with Devetak and Berger) a low-entanglement upper bound conjectured to be tight. It proved Lo's conjecture only for a restricted 'teleportation-type' class (Bob restricted to a unitary correction, message probabilities independent of the state, giving a $2\log d$ lower bound), and explicitly recorded: no exactly faithful RSP protocol for finite $n$ was known that always uses less classical communication than teleportation; the full $(e,b)$ achievable region was undetermined; how closely the Holevo-type bound $S(\bar{\rho})-\frac{1}{n}\sum_iS(\rho_i)$ could be approached for ensembles of states entangled with Alice's laboratory was 'under investigation'; whether Alice, sharing entanglement separately with Bob and Charlie, can remotely prepare an arbitrary tripartite state by sending at most $\log d_B$ bits to Bob and $\log d_C$ bits to Charlie was posed as a question; and the unassisted RSP capacity $R^{(d)}(\mathcal{N})$ (the asymptotic rate of remotely preparing known $d$-dimensional states over $\mathcal{N}$ without prior entanglement, defined by an achievability expression with block encoders and decoders) was left with only the bounds $Q\leq R^{(d)}\leq C$, the equality $R_E=C_E$ in the entanglement-assisted case, and the observation $R^{(2)}>Q$ for a strongly dephasing qubit channel.

The later literature resolved a substantial part of this agenda and thereby fixes what remains. Berry and Sanders proved that, with entanglement treated as an unlimited free resource, an ensemble of non-commuting mixed states — including states entangled between Alice and Bob — can be remotely prepared with average fidelity arbitrarily close to one at an asymptotic classical cost per state equal to the Holevo information, settling the Holevo-approachability question in the high-entanglement regime. Bennett, Hayden, Leung, Shor, and Winter (IEEE Trans. Inf. Theory 51, 56 (2005)) then determined the exact ebit/cbit tradeoff curve for memoryless sources of pure states, $E^*(R)=\min\{S(A:B|C):S(A:BC)\leq R\}$ per copy, including arbitrarily varying sources and ensembles of pure entangled states, via a universal protocol costing one bit of classical communication and one ebit per qubit whose optimality is shown by a dichotomy argument; for qubits the curve coincides with the one Devetak and Berger had constructed, confirming the tightness they conjectured. In the opposite, exact regime, Leung and Shor proved that any deterministic, exact, receiver-oblivious, forward-communication RSP protocol for a set of states whose projectors form an operator basis needs at least as much classical communication as teleportation; Hayashi, Hashimoto, and Horibe removed obliviousness for a single qubit, showing two classical bits are necessary, but stated that for higher dimensions it 'is still open' whether abandoning obliviousness helps; Bennett, Hayden, Leung, Shor, and Winter concurred that for dimension $\geq 3$ the analogous statement 'is unknown'. The mixed-state side of the tradeoff — equivalently, one-shot quantum state compression with entanglement — progressed through Jain, Bab Hadiashar, Nayak, and Renner, and Bab Hadiashar and Nayak, but as the 2026 work of Kundu and Lalonde records, entanglement cost for RSP of mixed states, and entanglement measures other than Schmidt rank (the number of terms in the state's Schmidt decomposition, its expression as a sum of orthogonal product states) for pure resource states, 'has not been studied at all'; they obtain the first nearly matching upper and lower bounds, and only for the flat states $P/k$ (mixed states with flat spectrum supported on a rank-$k$ projector). The multiparty bit-allocation question and a tight evaluation of $R^{(d)}(\mathcal{N})$ remain open.

Problem Statement

The overarching objective is to complete the resource accounting for remote state preparation — the full, tight characterization of the entanglement and classical communication required to deliver known quantum states — in the regimes that the 2001 founding paper left open and that two decades of subsequent work have not closed, with the parts since resolved by that later work explicitly set aside. The surviving open core is: (1) Lo's conjecture beyond the settled cases: does there exist an exactly faithful, deterministic RSP protocol — not assumed oblivious to the receiver, with no restriction on Bob's operations beyond the shared entanglement and forward classical communication — that remotely prepares arbitrary pure states of dimension $d\geq 3$ using strictly fewer than $2\log_2d$ classical bits per state? Equivalently, prove that no such protocol exists, or exhibit one. For a single qubit the worst-case version is settled — Hayashi, Hashimoto, and Horibe proved that two classical bits are necessary even without obliviousness, while the expected-cost, deterministic-exact version is achievable with one ebit and one classical bit (Bennett, Hayden, Leung, Shor, and Winter 2005) — so the open instance is $d\geq 3$; the same question covers the original finite-blocklength formulation, an exactly faithful protocol for finite $n$ that always uses less classical communication than teleportation. (2) The full achievable tradeoff region between entanglement consumption $e$ and forward classical communication $b$ for RSP of ensembles of mixed states (the pure-state and pure-entangled-state asymptotic region is settled): determine achievability and matching converses for all $(e,b)$, in both the asymptotic and one-shot settings, including the entanglement cost of RSP measured beyond Schmidt rank. (3) The multiparty question as posed: if Alice shares prior entanglement separately with Bob and with Charlie, can she remotely prepare an arbitrary tripartite state by sending at most $\log d_B$ classical bits to Bob and $\log d_C$ bits to Charlie, and what are the optimal classical-communication allocations and entanglement usage for multipartite RSP in general? (4) The operational structure and tight evaluation of the unassisted RSP capacity $R^{(d)}(\mathcal{N})$ of a general noisy quantum channel, together with the behavior of $R^{(d)}$ as the local dimension $d$ grows. These four deliverables jointly serve one scientific goal — the exact boundary between the communication cost of transmitting known and unknown quantum states, and the capacity of noisy channels for the known-state task — and a solution to any one part is expected to require the same machinery (information-theoretic converses over LOCC (local operations and classical communication), causality/teleportation-equivalence arguments, and rate-region constructions) as the others. A solution to any one of (1)-(4) constitutes a solution to the task.

References

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