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Numerical approach for unstructured quantum key distribution

Patrick J. Coles, Eric M. Metodiev, Norbert Lütkenhaus

arXiv:1510.01294v2quant-ph

TL;DR

QKD key-rate calculations are difficult for arbitrary protocols because symmetry-based analytical methods do not generally apply, especially when imperfections break symmetry. The paper reformulates the calculation as a dual optimization with fewer parameters and reliable lower-bound outputs, then applies it to structured and unstructured protocols. The method exactly reproduces several known results, determines key rates for n-MUB protocols, and improves prior B92 bounds, while its tightness can be limited by the Golden–Thompson inequality.

  • Problem

    No general method computes tight key-rate bounds for arbitrary unstructured QKD protocols, although such protocols are relevant to experimental implementations and symmetry-breaking imperfections.

  • Method

    The paper transforms the key-rate optimization into a dual maximization problem whose parameter count is tied to independent experimental constraints.

  • Results

    The approach exactly reproduces known results for several structured protocols, determines previously unknown n-MUB key rates, and predicts positive B92 key rates for p ⩽0.053.

  • Takeaways & Limitations

    The method provides an efficient and reliable numerical route for studying QKD protocols lacking symmetry and for assessing protocol imperfections.

  • Takeaways & Limitations

    The bound can be loose when the Golden–Thompson inequality used in the derivation is not saturated, although it introduces no looseness for B92’s post-selection map.

Abstract

from arXiv · show

Quantum key distribution (QKD) allows for communication with security guaranteed by quantum theory. The main theoretical problem in QKD is to calculate the secret key rate for a given protocol. Analytical formulas are known for protocols with symmetries, since symmetry simplifies the analysis. However, experimental imperfections break symmetries, hence the effect of imperfections on key rates is difficult to estimate. Furthermore, it is an interesting question whether (intentionally) asymmetric protocols could outperform symmetric ones. Here, we develop a robust numerical approach for calculating the key rate for arbitrary discrete-variable QKD protocols. Ultimately this will allow researchers to study "unstructured" protocols, that is, those that lack symmetry. Our approach relies on transforming the key rate calculation to the dual optimization problem, which dramatically reduces the number of parameters and hence the calculation time. We illustrate our method by investigating some unstructured protocols for which the key rate was previously unknown.

RESULTS

The paper formulates QKD key-rate estimation as a reliable numerical optimization over experimentally constrained states, then demonstrates the approach across structured and unstructured protocols. The method reproduces known results, establishes tight bounds for several protocols, and improves bounds for protocols lacking symmetry.

  • Optimization framework: The key-rate problem constrains the shared state using Alice’s and Bob’s measurement data, represented by expectation-value constraints from their POVMs.These constraints define the admissible state set used in the optimization.
  • Optimization framework: Theorem 1 reformulates the primal minimization as a maximization whose outputs are reliable lower bounds on physically achievable key rates.The dual formulation is derived using convex optimization duality and entropic identities.
  • Optimization framework: The dual problem optimizes one parameter per independent experimental constraint, potentially reducing dimensionality substantially compared with the primal problem.The framework also incorporates source-replacement constraints for prepare-and-measure protocols, MDI constraints, and public announcements.
  • Structured protocols: The numerics exactly reproduce known key-rate dependences for BB84, six-state, MDI QKD with BB84 states, and higher-dimensional two-MUB protocols.For the protocols in Figures 1 and 3, the numerical optimization is perfectly tight.
  • Unstructured protocols: For n-MUB protocols in dimension d, the method determines key-rate dependence for intermediate n where the symmetry group and analytical key rate were previously unknown.The largest error-tolerance increase occurs when adding the third basis, from n = 2 to n = 3, followed by diminishing returns.
  • Unstructured protocols: For the rotated-basis qubit protocol, adding constraints produces bounds tighter than the entropic uncertainty-principle bound, and small angle variations around BB84 have little effect on the key rate.The analysis also indicates that retaining data normally discarded during sifting can be useful.

DISCUSSION

The paper presents an efficient and reliable numerical method for lower-bounding key rates of arbitrary QKD protocols. By reducing optimization parameters to experimental constraints, the method addresses unstructured protocols and supports future finite-key extensions.

  • DISCUSSION: The method calculates achievable lower bounds for key rates of arbitrary QKD protocols by reformulating the optimization as a maximization.The authors describe reliability as every computer-generated solution being an achievable key rate.
  • DISCUSSION: The optimization uses the number of experimental constraints rather than d^2 parameters, which can be independent of dimension.This reduction is the stated source of the method’s efficiency.
  • DISCUSSION: The approach targets unstructured protocols because experimental imperfections break symmetries and no general method previously handled their key-rate effects.It also permits questions about intentionally asymmetric protocols that cannot otherwise be evaluated.
  • DISCUSSION: The authors identify finite-key analysis as a future extension, while noting that finite-size effects generally reduce rates below asymptotic values.The present work focuses on asymptotic key rates.

METHODS

The methods transform the key-rate problem into a dual optimization and simplify it using entropic identities, coherence, and relative-entropy optimization. This yields a tractable formulation while retaining equality with the primal problem under strong duality.

  • METHODS: The proof combines optimization duality, entropic identities, and a relative-entropy optimization result to derive the main reformulation.These are the three technical tools identified in the proof outline.
  • METHODS: The primal problem minimizes the conditional entropy over states consistent with experimental constraints, with Eve modeled through a purifying system.The measured conditional-entropy term is separated from the part requiring optimization.
  • METHODS: Rewriting the problem in terms of coherence connects secret-key calculation to coherence optimization.The continuity of coherence is used to argue that strong duality holds.
  • METHODS: The dual formulation is an unconstrained optimization over Lagrange multipliers associated with the experimental constraints.Strong duality connects the dual solution to the primal solution.
  • METHODS: The dual problem is simplified through a coherence representation, a channel construction, minimization interchange, and a relative-entropy optimization solution.The resulting expression is summarized as the dual problem after these substitutions.
  • METHODS: A Golden–Thompson-based bound supplies a simpler lower bound on the dual objective for the projective-measurement case.The proof of the corresponding arbitrary-POVM result is deferred to supplementary material.

SUPPLEMENTARY NOTE 1: MDI QKD

The supplementary note elaborates the framework for measurement-device-independent QKD and gives further details on the example associated with Fig. 2.

  • SUPPLEMENTARY NOTE 1: MDI QKD: The note expands the framework for MDI QKD and provides additional details for the calculation illustrated in Fig. 2.

Framework for MDI QKD (continued)

The MDI framework models Alice, Bob, and the untrusted measurement outcome within a constrained tripartite state, allowing the numerical method to operate without modeling the untrusted channel. The example uses biased-basis BB84 states, and the numerics reproduce the known theoretical curve.

  • Framework for MDI QKD (continued): The framework uses a tripartite state ρABM, where M stores the outcome of the untrusted node’s measurement.A and B are the source-replacement systems held by Alice and Bob.
  • Framework for MDI QKD (continued): The untrusted node and Eve are combined into a channel mapping A′B′ to the classical register M, while A and B remain with Alice and Bob.The numerical method requires only constraints on ρABM, not details of that channel.
  • Framework for MDI QKD (continued): The method also fixes the marginal ρAB and need not explicitly enforce classicality of M, because the worst case already corresponds to M being classical.
  • Framework for MDI QKD (continued): Decohering M does not change Eve’s ignorance about Alice’s key when the constraint set remains compatible with that decoherence.The proof uses data processing to construct a decohered state with no greater Eve ignorance.
  • Framework for MDI QKD (continued): The example uses BB84 signal states with biased basis choices, distills key from both bases without sifting, and reproduces the known theoretical curve.The bias is implemented with pz = 1 − ε for 0 < ε ≪ 1.

SUPPLEMENTARY NOTE 2: ARBITRARY POST-SELECTION

For arbitrary post-selection, the method reformulates the key-rate optimization over the post-selected image of the feasible state set. This produces the standard optimization form, with an inequality unless the post-selection map has a completely positive inverse.

  • Constraint transformation: The approach transforms constraints on ρAB into constraints on the post-selected state G(ρAB).The transformation is constructed by analyzing the image of the original state space under G.
  • Constraint transformation: Gram-Schmidt decomposes the Hermitian operator space into constraint-fixed and free basis directions.The trace constraints fix coefficients along one orthonormal basis, while the remaining coefficients are unconstrained by those trace constraints.
  • Constraint transformation: After applying G, the transformed trace constraints determine the coefficients along the Ωn operators while leaving the Υm coefficients unrestricted.This converts the original constraints into trace constraints directly on G(ρAB).
  • Optimization reformulation: With post-selection, the primal optimization can be expressed over operators in the image of the feasible set, subject to transformed trace constraints.The post-selected state is normalized by ppass = Tr(G(ρAB)) in the key-rate formula.
  • Optimization reformulation: Because G(HAB)+ can strictly contain G(PAB), replacing the exact image with the positive semidefinite cone introduces an inequality and requires additional trace constraints when using a larger representation space.The extra constraints restrict the optimization back to the image of G.
  • Optimization reformulation: When G has a completely positive inverse, the reformulation is exact, and the B92 post-selection step introduces no looseness.In this case, G(PAB) = G(HAB)+.

SUPPLEMENTARY NOTE 3: TIGHTNESS FOR PROTOCOLS WITH MUBS

The numerical approach is proven perfectly tight for the entanglement-based protocols involving mutually unbiased bases. The proof identifies the Golden–Thompson inequality as the only possible source of looseness in the bound.

  • Tightness result: The numerical approach is perfectly tight for the entanglement-based protocols involving MUBs considered in the main text.This result is established analytically in the supplementary proof.
  • Tightness result: The only potential source of looseness in the bound is the use of the Golden–Thompson inequality in Eq. (60).The tightness question therefore reduces to determining when that inequality is saturated.

A general lemma

A sufficient lemma characterizes when the Golden–Thompson step is saturated, thereby guaranteeing that the numerical method is tight. The conditions concern a common eigenbasis for the measurement operators and the action of ZA in that basis.

  • Tightness criterion: The lemma provides sufficient conditions under which the Golden–Thompson inequality is saturated and the numerical method is tight.The criteria are sufficient but are not claimed to be necessary.
  • Tightness criterion: ZA must map each basis projector |ek⟩⟨ek| to a diagonal linear combination in that common eigenbasis.Equivalently, the off-diagonal matrix elements specified in condition (b) vanish for distinct basis indices.
  • Proof mechanism: These conditions make the relevant operators commute, which is the equality condition for the Golden–Thompson inequality.The proof uses commutation to establish saturation for an optimizing state.
  • Tightness criterion: The measurement operators {Γi} must be simultaneously diagonalizable in a common orthonormal eigenbasis {|ek⟩}.Under this condition, Q(λ) and exp(Q(λ)) are diagonal in the same basis.
  • Proof mechanism: Tightness only requires saturation for one optimizing σ*AB, even when the maximal eigenvalue defining it is degenerate.The maximizing state can be chosen from the common eigenbasis under the lemma’s conditions.
  • Proof mechanism: Under the two stated conditions, the Golden–Thompson inequality is saturated.This completes the lemma’s sufficient tightness argument.

Specific protocols

For the MUB protocols, tightness is established by showing that the relevant measurement operators are diagonal in the Bell basis and that this basis satisfies the lemma’s second condition. The construction uses generalized Pauli operators and the associated Bell states.

  • Specific protocols: The proof targets MUB protocols by verifying the two sufficient conditions from the general tightness lemma.The key task is to establish the required basis properties for these specific protocols.
  • Bell-basis construction: Generalized Pauli operators in dimension d are used to construct d^2 orthonormal Bell basis states.The Bell basis is written as {|φq,r⟩}.
  • Verification of tightness: The relevant Γi operators are shown to be diagonal in the Bell basis, completing the basis requirements for tightness.The proof also establishes that the resulting operator expression is diagonal in that basis.
  • Verification of tightness: The Bell basis satisfies condition (b) of the tightness lemma: ZA maps each Bell-basis projector without off-diagonal terms between distinct Bell-basis states.The corresponding matrix elements vanish for every distinct pair of Bell-basis labels.
  • Verification of tightness: For the protocols considered, ZA is analyzed using the standard basis representation and its action on an arbitrary operator.This provides the operator-level route for verifying the Bell-basis condition.

Two MUBs

The supplementary analysis verifies that the relevant measurement operators are diagonal in the Bell basis for two-MUB and related protocols, establishing tightness in the stated cases.

  • The error operators for the two-MUB protocol are shown to be diagonal in the Bell basis.
  • The six-state protocol is tight because its remaining error operator is also diagonal in the Bell basis.
  • For prime dimension d, the construction supplies d + 1 mutually unbiased bases, with Fig. 4 using a subset of n bases.
  • The n-MUB measurement operators combine the identity with error operators for the selected bases.
  • The additional error operators are likewise shown diagonal in the Bell basis, proving tightness for the protocol in Fig. 3.
  • The supplementary construction is restricted to odd prime dimensions because the even-prime case d = 2 was already covered.

SUPPLEMENTARY NOTE 4: ANALYSIS OF B92 PROTOCOL

The B92 analysis formulates the protocol through source replacement and dual optimization, finding a four-parameter dual problem instead of the primal problem’s twelve parameters.

  • B92 uses two non-orthogonal signal states, random measurements, and post-selection on selected outcomes.
  • The source-replacement scheme converts the prepare-and-measure protocol into constraints on the shared state ρAB.
  • The formulation includes depolarizing noise, error and success operators, Alice’s constraints, a key map, and Bob’s post-selection filter.
  • 4 parameters are required by the B92 dual problem, compared with 12 parameters for the primal problem.

SUPPLEMENTARY NOTE 5: ARBITRARY KEY-MAP POVMS

The paper extends its key-rate framework from projective key-map measurements to arbitrary POVMs by expressing the primal problem through a generalized coherence and then transforming it to the dual.

  • The main result is generalized from projective key-map POVMs to arbitrary measurements.
  • The generalized treatment begins by defining a coherence-like quantity for a POVM P = {Pj}.
  • The post-measurement state used in the generalized coherence need not be normalized.
  • The primal problem can be defined or lower-bounded using the generalized coherence and a purification of ρAB.
  • The resulting formulation is transformed to the dual problem, with the corresponding relation expressed as an inequality.

SUPPLEMENTARY NOTE 6: STRONG DUALITY

The strong-duality analysis perturbs the constraints to satisfy Slater’s condition, proves strong duality for the perturbed problem, and shows that the perturbation approaches the original solution as ε becomes small.

  • Weak duality gives a lower bound on the primal problem, while the paper proves equality for the perturbed problem through strong duality.
  • The unperturbed problem may lack a strictly positive feasible state, so the constraints are perturbed to guarantee Slater’s condition.
  • The perturbed problem replaces the original constraints with positive-definite states satisfying modified trace constraints.
  • Solving the perturbed dual problem yields a result arbitrarily close to the primal solution as ε becomes sufficiently small.
  • Strong duality holds for the perturbed problem because the perturbed feasible set contains a state satisfying Slater’s condition.
  • The coherence is continuous under small trace-distance changes, supporting convergence between perturbed and unperturbed objectives.
  • For sufficiently small ε, the perturbed solution is within O(ε) of an unperturbed feasible solution.
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