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Simplified instantaneous non-local quantum computation with applications to position-based cryptography

Salman Beigi, Robert Koenig

arXiv:1101.1065v3quant-ph

TL;DR

The paper develops non-recursive teleportation-based protocols for instantaneous non-local measurement and quantum computation. It contrasts their entanglement use with Vaidman’s doubly exponential scheme, proves a linear entanglement lower bound for a non-local measurement, and derives position-based cryptography implications.

  • Problem

    Vaidman’s recursive scheme requires entanglement doubly exponential in the number of communicated qubits, motivating protocols that avoid this unfavorable behavior while addressing instantaneous non-local measurement and computation.

  • Method

    The paper uses port-based teleportation, whose correction operation is discarding a subsystem based on classical information, to construct non-recursive instantaneous protocols.

  • Results

    The proposed protocols realize instantaneous non-local measurements and computation with an entanglement-accuracy parameter N, while a certain 2n-qubit measurement requires at least n/2 ebits.

  • Takeaways & Limitations

    The results imply that position-based cryptographic schemes can be attacked with exponentially scaling entanglement, while some schemes remain secure below a linear entanglement bound under classical communication.

Abstract

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Instantaneous measurements of non-local observables between space-like separated regions can be performed without violating causality. This feat relies on the use of entanglement. Here we propose novel protocols for this task and the related problem of multipartite quantum computation with local operations and a single round of classical communication. Compared to previously known techniques, our protocols reduce the entanglement consumption by an exponential amount. We also prove a linear lower bound on the amount of entanglement required for the implementation of a certain non-local measurement. These results relate to position-based cryptography: an amount of entanglement scaling exponentially in the number of communicated qubits is sufficient to render any such scheme insecure. Furthermore, we show that certain schemes are secure under the assumption that the adversary has less entanglement than a given linear bound and is restricted to classical communication.

I. REVIEW OF VAIDMAN’S SCHEME: TELEPORTATION WITHOUT COMMUNICATION

Vaidman’s scheme uses shared EPR pairs to teleport quantum systems, but missing classical corrections make instantaneous non-local computation non-trivial. Recursive techniques address this restriction under local operations and limited communication.

  • Communication restriction: Vaidman’s techniques instead provide instantaneous implementations of non-local operations using prior shared entanglement and local operations with a single round of simultaneous classical communication.The procedure targets a joint unitary UAB while respecting the non-signaling constraint imposed by the restricted interaction.
  • Teleportation resource: Vaidman’s scheme uses shared EPR pairs as the entanglement resource for teleportation-based non-local operations.Alice and Bob share EPR pairs, and teleportation transfers quantum states up to Pauli corrections determined by measurement outcomes.
  • Teleportation resource: n shared EPR pairs teleport an arbitrary n-qubit state, with Bob’s state differing by an n-qubit Pauli correction.The correction is indexed by one of 4^n possible measurement outcomes and normally requires sending the outcome string.
  • Communication restriction: Instantaneous non-local computation cannot directly use teleportation because the teleportation outcomes cannot be communicated before the operation must occur.With free classical communication, Bob could teleport his system to Alice, who applies the unitary and teleports it back; the communication restriction prevents this direct procedure.

A. Reduction to a state held by one of the parties

A local, no-communication protocol can be converted into a protocol that leaves the desired non-local unitary state shared between Alice and Bob. Teleportation measurements and one simultaneous exchange of classical outcomes supply the needed corrections.

  • Reduction protocol: The reduction starts from a no-communication local protocol P′ that leaves Bob holding σsU|Ψ⟩ for a classical outcome string s.The outcome string is determined from Alice’s and Bob’s local measurement outcomes.
  • Reduction protocol: Bob teleports one register using n additional EPR pairs, producing a teleportation outcome v that is exchanged with Alice.Alice sends α, while Bob sends β and v in the single simultaneous communication round.
  • Reduction protocol: Alice and Bob compute the correction components from their exchanged outcomes and apply local Pauli corrections to their respective registers.Alice applies σsAσv, while Bob applies σsB.
  • Reduction protocol: The protocol ends with Alice and Bob sharing U|Ψ⟩ across their registers.Thus the reduction converts the local protocol P′ into the desired distributed implementation of the non-local unitary.

B. Vaidman’s recursive scheme

Vaidman’s recursive scheme repeatedly handles unknown teleportation corrections by branching over possible outcomes and preparing entangled resources for each branch. Success occurs when a round yields a trivial correction.

  • First round: Vaidman first teleports Bob’s system to Alice, who applies U and teleports the resulting state back without knowing Bob’s correction outcome.If Bob’s first outcome is 0^n, the desired state is obtained immediately, but this occurs with probability 4^-n.
  • Recursive branching: When the first correction is nontrivial, the protocol recursively implements the corresponding corrected unitary in a later round.Alice does not know Bob’s outcome, so separate entangled register sets are prepared for every possible outcome.
  • Recursive branching: Each recursive round uses teleportation measurements and outcome-indexed local operations, while Bob retains only the branch corresponding to his actual outcomes.Alice operates across all prepared branches, whereas Bob identifies the relevant registers from his measurement results.
  • Round success: A later round succeeds when Bob obtains the trivial outcome 0^{2n}, which has probability 4^-2n per round.Upon success Bob stops further measurements, while Alice continues applying operations to the remaining entangled states.
  • Round success: After success, the exchanged classical records determine the final correction, leaving Bob with σsRU|Ψ⟩ in the appropriate register.Alice’s message contains her measurement results, while Bob’s message contains the stopping round and his teleportation outcomes.

C. Entanglement consumption in Vaidman’s scheme

Vaidman’s scheme consumes doubly exponential entanglement because recursive branching grows exponentially with the number of rounds. The proposed simplified schemes avoid this recursive structure through teleportation with effectively trivial corrections.

  • Entanglement scaling: For constant ε, Vaidman’s scheme requires doubly exponential entanglement in n.The number of branches is exponential in R, while achieving success probability 1−ε requires R roughly proportional to log(1/ε) · 2^4n.
  • Entanglement scaling: The unfavorable scaling arises from recursively branching over prior teleportation outcomes to compensate for corrections that cannot be communicated interactively.Each additional round introduces resources for possible outcome sequences from earlier rounds.
  • Simplified approach: The simplified schemes avoid this recursive growth by using a teleportation scheme whose correction operations are effectively trivial.The resulting protocol is non-recursive.

II. PORT-BASED TELEPORTATION

Port-based teleportation replaces arbitrary correction operations with a measurement-selected output port that Bob keeps, and its channel quality is analyzed using worst-case diamond distance.

  • Protocol: Port-based teleportation lets Bob recover the teleported state by discarding every subsystem except the classically selected port.Alice performs a POVM and sends its index; Bob keeps subsystem B′i as the output register.
  • Protocol: N shared maximally entangled states support teleportation of a qudit through Alice’s POVM and Bob’s port-selection operation.The protocol maps input system A to Bob’s system B using auxiliary entanglement |Φ⟩⊗N.
  • Accuracy: Entanglement fidelity measures average preservation of quantum information over random pure inputs, whereas diamond norm gives a worst-case channel criterion.The diamond norm is introduced because average-case fidelity does not guarantee closeness for all inputs.
  • Accuracy: Diamond distance has an operational interpretation as channel distinguishability and cannot increase under composition of maps.This monotonicity supports analyzing composed teleportation-based protocols.

III. PROTOCOLS FOR INSTANTANEOUS MEASUREMENT AND COMPUTATION

The paper introduces two single-round classical-communication protocols: one for instantaneous non-local POVMs and one for non-local unitaries, with accuracy controlled by shared entanglement.

  • Protocol overview: Two protocols provide instantaneous implementations of non-local POVMs and bipartite unitaries using local operations and one round of classical communication.The protocols are parameterized by N, which captures entanglement consumption and achieved accuracy.
  • Instantaneous measurement: The POVM protocol uses port-based teleportation, measures every candidate port, and outputs the result associated with Alice’s selected index.Alice sends i while Bob sends {(j, γj)}j to Charlie, who outputs γi.
  • Instantaneous computation: The unitary protocol applies U to candidate systems, uses teleportation corrections, and retains the systems indexed by the exchanged classical outcomes.Alice and Bob’s local actions commute, so their prescribed ordering is only an analysis convenience.
  • Accuracy: N := 2^8n+4/ε^2 sets the entanglement parameter for protocols approximating the target operation to accuracy ε.The theorem states separate guarantees for the POVM and CPTP-map implementations.
  • Scope: The analysis extends straightforwardly to multipartite non-local POVMs and unitaries.The same protocol framework is not restricted to the bipartite setting.

IV. A LOWER BOUND

A mutually unbiased-basis construction yields a non-local measurement whose instantaneous implementation requires entanglement scaling linearly with the number of qubits.

  • Lower bound: A measurement on 2n qubits is not realizable instantaneously with fewer than n/2 ebits of shared entanglement.The construction is based on mutually unbiased bases.
  • Construction: For d = 2^n, mutually unbiased bases define an ensemble in which Alice and Bob must identify x from distributed inputs.The basis family exists when d is a prime-power dimension, with the paper specializing to d = 2^n.
  • Consequence: The lower bound is expressed through diamond-norm distinguishability between the constructed measurement and an instantaneous implementation.The theorem connects the measurement’s approximation error to the entanglement dimension.
  • Construction: The associated POVM exactly outputs x on the constructed states, linking successful instantaneous measurement to the lower-bound task.The success probability of an arbitrary instantaneous measurement is then bounded using the theorem’s ensemble analysis.

V. IMPLICATIONS FOR POSITION-BASED QUANTUM CRYPTOGRAPHY

The results translate instantaneous non-local protocols into position-based cryptography consequences: exponential entanglement enables attacks, while limited entanglement and classical communication can support exponential soundness.

  • Setting: Position-verification asks a prover at a claimed location to answer synchronized challenges, with soundness measuring adversarial acceptance probability.The one-dimensional example places the honest prover between two verifiers and defines ε-soundness against colluding adversaries.
  • Position verification: Protocol πn sends a basis choice and encoded quantum state simultaneously, then accepts a measurement outcome matching the encoded string and timing constraints.The prover measures in the basis selected by the classical challenge and broadcasts the result.
  • Insecurity: An amount of shared entanglement exponential in n is sufficient to make position-based schemes insecure, improving on earlier doubly exponential attacks.The attack reduces security to success in an instantaneous measurement task.
  • Security: Fewer than n/2 shared ebits and classical communication suffice for security of πn under the stated adversarial restriction.For at most m ebits, the soundness bound is 2 · 2^m−n/2 for n > 1.
  • Limitations: The one-round protocol achieves exponential security but becomes impractical for large n because honest parties must manipulate n qubits simultaneously.Security under unrestricted quantum communication remains unresolved in the cited discussion.
  • Alternative protocol: A separate composition-based protocol offers a comparable entanglement-soundness scaling and is described as more practical.Its analysis relates prior-entanglement security to the no-entanglement case.

Appendix A: Derivation of port-based teleportation

The appendix proves Theorem II.1 by reducing port-based teleportation to state discrimination and bounding the pretty good measurement's success probability.

  • The proof first establishes an equivalence between port-based teleportation and distinguishing a specified set of states.This equivalence is developed in Appendix A 1.
  • It then proves a general lower bound for the pretty good measurement's success probability.The bound is presented in Appendix A 2.
  • Combining the state-discrimination equivalence with the measurement bound yields Theorem II.1.The final combination appears in Appendix A 3.

1. From port-based teleportation to distinguishing quantum states

This section reformulates port-based teleportation as a quantum hypothesis-testing problem, relating teleportation fidelity to the success probability of distinguishing an associated state ensemble.

  • Port-based teleportation fidelity is directly related to distinguishing an associated ensemble of states with a POVM.The quantity is a simple function of the ensemble's average success probability under equal priors.
  • Lemma A.1 formalizes the equivalence between port-based teleportation and hypothesis testing.The construction uses states η_i and a POVM whose outcomes correspond to ports.
  • The entanglement fidelity of the induced port-based teleportation map is determined by the average state-discrimination success probability.The map is obtained from the POVM and the shared maximally entangled resource.
  • The derivation uses maximally entangled-state identities involving transposition and partial traces.These identities move operators between the two halves of the maximally entangled state.

2. A lower bound on the success probability of the pretty good measurement

This section develops a general lower bound on the pretty good measurement's success probability and prepares its application to port-based teleportation.

  • The section seeks a POVM that solves the state-distinguishing problem arising from port-based teleportation.The required POVM is constructed through the pretty good measurement.
  • The authors derive and generalize a lower bound on the pretty good measurement's success probability for uniform state families.This bound is stated as Lemma A.3 and is intended for application to port-based teleportation.
  • The proof relies on an inequality for non-negative operators and rank-based estimates.Lemma A.2 supplies the main technical inequality used in the argument.
  • The resulting bound is obtained by defining auxiliary operators and applying the preceding inequality.The proof uses rank(Y^-1/4η_iY^-1/4) = rankη_i.

3. Proof of Theorem II.1

The proof of Theorem II.1 combines the hypothesis-testing reformulation with the lower bound for the pretty good measurement to obtain a suitable port-based teleportation POVM.

  • The theorem proof combines Lemma A.1 with the lower bound from Lemma A.3.This transfers the measurement bound into the port-based teleportation setting.
  • The relevant state-distinguishing problem uses the states η_i from the earlier reformulation.The proof then applies the general success-probability bound to this ensemble.
  • The construction yields a POVM whose associated CPTP map achieves port-based teleportation with a stated entanglement fidelity.The fidelity follows from the correspondence established in Lemma A.1.
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