Security of multiparty quantum protocols under general superselection rules beyond compact symmetry groups

Superselection rules (SSRs) restrict the quantum operations available to a local agent: certain conserved charges cannot be created, destroyed, or coherently superposed by local means. Kitaev, Mayers, and Preskill showed that SSRs cannot enhance the information-theoretic security of quantum cryptographic protocols, but their two methods have complementary coverage: a reference-system simulation theorem (Theorem 1) handles any number of parties yet only SSRs whose superselection sectors are unitary irreducible representations of compact groups, while a message-format verification method handles general SSRs — including those of non-Abelian anyons with nontrivial braiding — yet only two-party protocols. The problem left open is whether SSRs that do not arise from compact symmetry groups, in particular anyonic SSRs with nontrivial fusion and braiding, can enhance the security of protocols with more than two parties. Solving it requires formulating security definitions for multiparty protocols with path-dependent (braided) charge transport and proving or refuting an extension of the simulation theorem to fusion-braiding superselection structure. The same source separately identifies a quantitative theory of the size of finite truncated reference systems needed as a function of protocol parameters. The question is explicitly stated to remain open for multi-party protocols in the 2007 review of the field.

Background

A superselection rule (SSR) is a fundamental limitation on the states and operations a local agent can realize: the Hilbert space of a system decomposes into orthogonal superselection sectors labeled by conserved charges, and no local operation can create a coherent superposition of states in different sectors or change the total charge. Familiar examples are electric charge, fermion parity, and, in two spatial dimensions, the topological charge of anyons. For non-Abelian anyons the charge structure is especially rich: charges obey fusion rules (fusing charges $q$ and $q'$ generally decomposes into a direct sum of possible outcomes, with an associated fusion space) and braiding rules (exchanging two charges acts by a nontrivial unitary transformation on the fusion space), so the effect of transporting a charge depends on the path taken around other charges.

Aharonov and Susskind observed already in 1967 that a charge SSR can be effectively circumvented by parties who share a suitable reference system, and Verstraete and Cirac showed in 2003 that SSRs enable data hiding, i.e. correlations inaccessible to local operations. This motivated the question, posed by Sandu Popescu (unpublished, as recounted in the source paper and the review literature), of whether SSRs might place inviolable limits on the cheating strategies available to dishonest parties and thereby enhance information-theoretic security — security against an adversary with unlimited quantum-computational power — of quantum cryptographic protocols such as quantum bit commitment (a scheme in which a committer fixes a bit value that is neither changeable nor revealed early) and quantum coin flipping (a protocol in which distant parties generate a shared random bit that neither party can bias). Mayers began a cryptographic analysis of SSRs in 2002.

Kitaev, Mayers, and Preskill answered this question negatively within the scope their methods could reach. They distinguish the invariant world (I-world), in which all operations conserve charge, from the unrestricted world (U-world), in which arbitrary operations are allowed. First, for SSRs whose superselection sectors are the unitary irreducible representations (representation spaces that cannot be decomposed into smaller invariant subspaces) of a compact symmetry group $G$, a shared reference system transforming as the (left) regular representation of $G$ (the representation space spanned by the group elements themselves, on which $G$ acts by left multiplication) allows parties to simulate any U-world operation by G-invariant ones; their Theorem 1 shows that for an $n$-party protocol with up to $k<n$ cheating parties, any U-world cheating strategy can be simulated in the I-world, so security in the I-world is maintained when the SSR is relaxed. Because a Lie group's regular representation is infinite dimensional, they also observed that a finite truncated reference suffices when the charge change is bounded: for $G=U(1)$ with charge changes bounded by $r$ units, a $(2r+1)$-dimensional truncation gives perfect fidelity, and similar remarks apply to arbitrary compact Lie groups — but they develop no general quantitative theory of the required reference size. Second, for two-party protocols under completely general SSRs (including anyonic ones), they use the concept of the format of a message: since, with trivial total charge, the parties hold conjugate charges, the recipient can verify that a received message is consistent with charge-conserving operations and abort otherwise. This proves (their Theorem 4) that two-party protocols under general SSRs are no more secure than in the unrestricted world, so unconditionally secure bit commitment and strong coin flipping with arbitrarily small bias remain impossible. The two methods thus have complementary coverage: reference systems cover any number of parties but only compact-group SSRs, where braiding is trivial; format verification covers braided SSRs but only two parties. For three or more parties with braided charges, genuinely new phenomena appear — e.g. a party can split its charge and send part of it on a path encircling another laboratory, changing charges in a path-dependent way that is neither strictly local nor detectable by two-party format checks — and the paper explicitly declines to address this case.

Problem Statement

Determine whether superselection rules that do not arise from compact symmetry groups — in particular, superselection rules with nontrivial fusion and braiding structure, such as those governing non-Abelian anyons in two spatial dimensions — can enhance the information-theoretic security of quantum cryptographic protocols with more than two parties, where the source paper's two methods (reference-system simulation for compact groups, and two-party message-format verification) both fail. The determination is jointly posed by two sub-questions: (1) formulate appropriate definitions of security for multiparty protocols in which the effect of sending a subsystem from one party to another is path-dependent because of braiding, and characterize cheating strategies that exploit braiding and path dependence, such as a party splitting its charge and winding part of it around other parties' laboratories; and (2) under those definitions, either prove that an $n$-party protocol secure in the charge-conserving invariant world remains secure when the superselection rule is relaxed — an extension of the reference-system simulation theorem (Theorem 1 of the source) from superselection sectors labeled by unitary irreducible representations of compact groups to general superselection rules with fusion-braiding (tensor-categorical) structure — or exhibit a protocol whose information-theoretic security genuinely relies on such a superselection rule. The source identifies a further, independent open direction: (3) develop a quantitative theory of the size of finite truncated reference systems required to simulate unrestricted-world operations that change charge by a bounded amount, as a function of the protocol's parameters, beyond the single $U(1)$ example (a $(2r+1)$-dimensional truncation for charge changes bounded by $r$ units) given in the source.

References

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