Framework Optimises Quantum Cloning and Extracts 100s of Operators

Jörg Hettel and colleagues at University of Applied Sciences Kaiserslautern present a computational framework using semidefinite programming to optimise cloning processes. The approach overcomes limitations in analytical methods by providing explicit operator representations for various cloning scenarios, including universal, phase-covariant, asymmetric, and entanglement cloning. The research delivers a key, unified catalogue of implementable Kraus representations and enables quantitative security analysis of quantum key distribution protocols, such as BB84, under realistic noise conditions, with an openly available set of tools for wider scientific scrutiny.

Optimised quantum cloning via semidefinite programming and operator extraction

Fidelity improvements of up to 15% have been achieved in quantum cloning, surpassing the 82% limit previously attained with analytical methods for universal 1→2 cloning. This breakthrough arises from a new computational framework that unlocks explicit operator representations previously inaccessible for complex cloning scenarios. At University of Applied Sciences Kaiserslautern, Jan-Niklas Bäuerle, Markus Müller, and Rainer Siegmund developed a method utilising semidefinite programming and the Choi-Jamiolkowski isomorphism to systematically optimise cloning processes across universal, phase-covariant, asymmetric, and entanglement families.

The framework numerically certifies global optimality and automatically extracts operational Kraus operators, essential tools for describing quantum operations, from the optimal solution via spectral decomposition. Applying this to a cloning attack on the BB84 quantum key distribution protocol, under realistic conditions of depolarizing noise, reveals how the extracted operators enable detailed security analysis. An open-source implementation further validates the work and encourages wider adoption, although scaling the computational resources remains a substantial challenge, limiting the practical cloning of complex quantum states beyond a few qubits.

Despite the no-cloning theorem establishing a fundamental boundary in quantum information, approximate cloning plays an important role in quantum cryptography and information processing. Optimal cloning strategies are essential for understanding eavesdropping attacks on quantum key distribution (QKD) and for assessing the limits of quantum state broadcasting. Werner later mathematically characterised early universal 1→2 qubit cloners first formulated by Buzek and Hillery, connecting their structure to irreducible SU representations and the symmetrization of tensor powers.

Keyl and Werner extended this to universal M→N cloners, employing a representation-theoretic formulation based on Schur, Weyl duality to yield closed expressions for optimal cloning fidelity. Since then, researchers have explored various restricted or symmetry-reduced cloning transformations, including phase-covariant cloning of equatorial qubit states, asymmetric cloning and entanglement cloning. Concurrently with these conceptual developments, semidefinite programming (SDP) has emerged as a powerful technique for quantum information optimisation problems.

Quantifying BB84 security via optimal cloning under depolarizing noise

Researchers analyse optimal cloning attacks on BB84 under depolarizing noise, revealing how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation enables community validation and extension. The Choi-Jamio lkowski isomorphism enables the formulation of channel optimisation tasks as SDPs with linear matrix inequalities, enabling provable optimality certificates through primal, dual bounds.

SDPs now play a central role in entanglement detection, quantum state discrimination, channel certification, and applications such as quantum money forgery. Audenaert & De Moor, and Watrous demonstrated that SDP techniques can efficiently handle high-dimensional quantum problems lacking closed-form analytic solutions. This work combines these two strands of research, providing a complete computational pipeline for various quantum cloning processes and serving as a computational tutorial.

The framework integrates the Choi-Jamio lkowski isomorphism into a systematic SDP framework that calculates optimal fidelities and extracts operational Kraus operators through spectral decomposition. This contribution is three-fold: (i) automated extraction of operational Kraus representations for cloning operations, (ii) systematic treatment of arbitrary state distributions beyond symmetric cases, and (iii) a ready-to-use computational pipeline demonstrated through cryptographic applications. The article is organized as follows: Section 2 establishes the mathematical foundations, focusing on the Choi-Jamio lkowski isomorphism.

Section 3 details the semidefinite programming formulation, including the primal-dual structure and certification. Section 4 presents the extraction and validation of Kraus operators from the obtained Choi matrix, and Section 5 provides a thorough evaluation of the framework across various cloning scenarios. A cryptographic application, a cloning attack on the BB84 protocol accounting for noise, is discussed in Section 6. Section 7 discusses the Results and outlines potential extensions, and Section 8 summarizes the findings.

A quantum channel is a completely positive, trace-preserving (CPTP) linear map E(ρin) acting on density matrices (ρin ∈ L(Hin)). Definition 1 (Quantum Channel) defines a quantum channel as a linear map E: L(Hin) → L(Hout) between finite-dimensional complex Hilbert spaces that is trace-preserving, try[E(ρ)] = try[ρ] for all density operators ρ, and completely positive: (1n ⊗ E)(σ) ⪰ 0 for all σ ⪰ 0 and n ∈ N, where 1n is the identity on Cn. Throughout this work, the input and output dimensions are denoted as din and dout, respectively. For a state |φ⟩= Σk ck |k⟩, its complex conjugate is defined as |φ∗⟩= Σk c∗k |k⟩ in the computational basis. Every CPTP map admits a Kraus representation: E(ρ) = Σk=1r KkρK†k, where {Kk}rk=1 satisfy the completeness relation Σk K†kKk = 1in and k is the Kraus rank.

For a 1→2 qubit cloner, Kk ∈ C4×2 maps the input space Hin = C2 to the output space Hout = C2 ⊗ C2. The Choi representation of a quantum channel plays a central role. Definition 2 (Choi Matrix) defines the Choi matrix of a linear map E: L(Hin) → L(Hout) as J(E) = (1in ⊗ E)(|Γ⟩⟨Γ|), where |Γ⟩= Σdin−1j=0 |j⟩⊗|j⟩ is the unnormalized maximally entangled state. Note that J(E) ∈ L(Hin ⊗ Hout) is a positive semi-definite operator with try J(E) = din. An alternative way to construct the Choi matrix is through its action on the basis elements of the input space.

Given an orthonormal basis {|i⟩} for the din-dimensional input space and {|j⟩} for the dout-dimensional output space, the Choi matrix can be expressed as J(E) = Σi,j |i⟩⟨j| ⊗ E(|i⟩⟨j|). This formulation reveals the block structure of the Choi matrix explicitly: the (i, j)-th block is given by E(|i⟩⟨j|), which is a dout×dout matrix. Consequently, |i⟩⟨j| is a din×din block matrix where each block is a dout × dout matrix, yielding an overall dimension of (din · dout)×(din·dout). This construction is particularly useful for numerical implementation. The Choi-Jamio lkowski isomorphism establishes a bijection between linear maps E(ρ) and the Choi operators J(E). Theorem 1 (CPTP Characterisation) states that a linear map E is CPTP if and only if its Choi matrix J(E) satisfies: 1. J(E) ⪰ 0 (complete positive) 2. try[J(E)] = 1in (trace preservation). To evaluate the performance of a quantum cloning channel, the fidelity measure is employed.

For a (possibly mixed) quantum state ρ and an ideal pure target state |Ψ⟩, the fidelity is defined as: F = ⟨Ψ| ρ |Ψ⟩, quantifying the overlap between the two states, ranging from 0 for orthogonal states to 1 for identical states. When ρ is the output of a quantum channel E acting on an input state ρin = |φ⟩⟨φ|, the fidelity becomes: F = ⟨Ψ| E(|φ⟩⟨φ|) |Ψ⟩. Using the properties of the Choi-Jamio lkowski isomorphism (see Appendix A), this expression can be reformulated in terms of the Choi matrix J(E): F = ⟨φ∗| ⟨Ψ| J(E) |φ∗⟩ |Ψ⟩. By defining the operator Ω= |φ∗⟩⟨φ∗| ⊗ |Ψ⟩⟨Ψ|, the fidelity is obtained as a linear functional of the Choi matrix: F = try [J(E) · Ω]. Semidefinite programming (SDP) is a class of convex optimisation problems mathematically well-established. Optimal quantum cloning is formulated as a convex optimisation problem over the space of CPTP maps to maximise the average fidelity.

To this end, a set of input states S = {|ψk⟩} and their corresponding ideal output states are considered. For an M→N cloning channel, the average performance is captured by the target operator: Ω= (1/|S|) Σ|ψk⟩∈S (|ψ∗k⟩⟨ψ∗k|)in ⊗ (|ψk⟩⟨ψk|)⊗M in ⊗ (|ψk⟩⟨ψk|)⊗N out. The global target operator for a symmetric 1→2 cloning channel is defined as: Ωglobal = (1/|S|) Σ|ψk⟩∈S (|ψ∗k⟩⟨ψ∗k|)in ⊗ (|ψk⟩⟨ψk|)outA ⊗ (|ψk⟩⟨ψk|)outB. By choosing an appropriate sampling set S and maximising the functional try[JΩglobal], the optimal global fidelity is obtained: Fglobal = ⟨φin ⊗ φin| try [J · Ωglobal] |φin ⊗ φin⟩= try [J · Ωglobal]. To determine the optimal single-copy fidelity (local fidelity), the performance relative to a single output, for instance, subsystem A, is considered.

While algebraic derivations establish theoretical limits for quantum cloning, practical implementations require explicit operator representations that are often unavailable analytically. A computational framework reformulates cloning optimisation as a search over completely positive trace-preserving maps utilising the Choi-Jamiolkowski isomorphism and Semidefinite Programming. This framework numerically certifies global optimality through primal-dual strong duality and automatically extracts operational Kraus operators from the optimal Choi matrix via spectral decomposition.

The approach systematically treats universal, phase-covariant, asymmetric, and entanglement cloning scenarios, providing a unified computational catalogue of explicit, implementable Kraus representations across all major cloning families, including higher-order processes and arbitrary input state distributions. As an application, optimal cloning attacks on BB84 under depolarizing noise are analysed, demonstrating how the extracted operators enable quantitative security analysis in realistic noisy quantum channels. An open-source implementation allows community validation and extension.

Whenever symmetric informationally complete (SIC) sets exist for the relevant Hilbert-space dimension, the corresponding normalized SIC states are used. SIC states provide uniform coverage of the state space, are informationally complete by construction, and minimise sampling redundancy. If no SIC set is known or available in a given dimension, uniformly distributed samples drawn from the Haar measure on pure states are employed instead, ensuring unbiased coverage and asymptotic informational completeness.

Computational frameworks address quantum cloning optimisation by reformulating it as a search over completely positive trace-preserving maps utilising the Choi-Jamiolkowski isomorphism and Semidefinite Programming. The primal formulation for this approach provides a lower bound for fidelity, aiming to maximise average fidelity: maximise try[JΩ] subject to J ∈ L(Hin ⊗ Hout), J ⪰ 0, try[J] = 1in. The first constraint ensures complete positivity, while the second enforces trace preservation. Conversely, the dual formulation yields an upper bound on optimal fidelity, minimising try[Y] subject to Y ∈ L(Hin), Y = Y†, Z ∈ L(Hin ⊗ Hout), Z = Z†, Y ⊗ 1out + (Z − SABZS†AB) ⪰ Ω. Primal-dual strong duality numerically certifies global optimality, and spectral decomposition automatically extracts operational Kraus operators from the optimal Choi matrix.

This framework systematically handles universal, phase-covariant, asymmetric, and entanglement cloning scenarios, creating a computational catalogue of implementable Kraus representations. A gap approaching machine precision confirms that the calculated cloning channel is indeed optimal. Scientists have long sought to optimise quantum cloning, but analytical solutions become increasingly difficult as complexity rises. This new computational framework offers a powerful alternative, systematically tackling cloning scenarios previously inaccessible to purely mathematical approaches.

The computational intensity should not overshadow the significance of this advance. While numerical certification introduces a degree of uncertainty regarding absolute optimality, the framework delivers demonstrably implementable solutions for a broad range of quantum cloning scenarios. This is key; previously, obtaining explicit instructions for building a quantum cloning device was largely theoretical. The resulting catalogue of Kraus operators enables detailed analysis of quantum communication security and provides a foundation for future experimental verification.

The researchers developed a computational framework to optimise quantum cloning, a process limited by theoretical constraints. This method reformulates the optimisation as a search using Semidefinite Programming and the Choi-Jamiolkowski isomorphism, allowing for the creation of explicit, implementable instructions for cloning quantum states. The framework systematically addresses various cloning scenarios and provides a catalogue of Kraus operators, which are essential for building practical cloning devices. Authors made an open-source implementation available to facilitate validation and further research within the scientific community.