State Duplication in Quantum and Classical Systems

• State Duplication Problem is the study of when and how information-bearing states, both quantum and classical, can be duplicated under the constraints of physical laws.
• Recent frameworks introduce virtual cloning operations, error-tolerant redundant encoding, and circuit-level approximate cloning, quantified through semidefinite programming.
• The implications span quantum information theory, DNA sequencing, and noisy communication channels, linking simulation cost with state distinguishability.

The state duplication problem encompasses the question of when and how an information-bearing state—quantum or classical—can be “duplicated” under the constraints of physical laws. In quantum mechanics, the no-cloning theorem fundamentally prohibits the perfect copying of unknown quantum states. Nonetheless, the state duplication problem is central across quantum information, classical information theory, and practical contexts such as DNA sequencing, where state duplication appears as a channel distortion. Recent advances have led to systematic frameworks describing both the fundamental limits and achievable mechanisms for state duplication, including virtual cloning operations, error-tolerant redundant encoding, circuit-level approximate cloning, and information-theoretic decomposition for noisy duplication channels.

1. Quantum No-Cloning and the Foundations of the State Duplication Problem

The quantum no-cloning theorem states that there exists no universal unitary operation $U$ and fixed “blank” state $\ket{0}$ such that, for all pure quantum states $\ket{\psi}$,

$$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$

This rule is an immediate consequence of the linearity of quantum mechanics: for nonorthogonal states $\ket{\psi_1}, \ket{\psi_2}$, assuming a universal cloner leads to a contradiction unless $\langle \psi_1 | \psi_2 \rangle = 0$ or a trivial superposition (Wang et al., 2018). This result is foundational, sharply distinguishing quantum information—the fundamentally uncopyable nature of generic quantum states—from classical information, which is duplicable without restriction.

2. Virtual Cloning Operations and Their Existence Criterion

Despite the no-cloning prohibition for physical (completely positive trace-preserving, CPTP) maps, it is possible to formally extend the class of allowed transformations to Hermitian-preserving trace-preserving (HPTP) maps, which need not be physically implementable. Given a set $\{\rho_i\}_{i=1}^m$ of density operators on $\mathcal{H}$, a virtual-cloning operation is a linear map

$$\mathcal{N} : \mathcal{L}(\mathcal{H}) \to \mathcal{L}(\mathcal{H}\otimes\mathcal{H}),$$

that is HPTP and satisfies $\mathcal{N}(\rho_i) = \rho_i \otimes \rho_i$ for all $\ket{0}$0.

The necessary and sufficient condition for such a map to exist is the linear independence of the states $\ket{0}$1 in the space of Hermitian operators. Explicitly, for a $\ket{0}$2-dimensional Hilbert space, no more than $\ket{0}$3 states can be virtually cloned simultaneously, but any two distinct states—even nonorthogonal—are always virtually clonable under this extended framework (Bi et al., 23 Jul 2025). This linear-independence criterion is proven by Vandermonde arguments on operator traces and implies that virtual cloning dramatically enlarges the set of duplicable states compared to CPTP constraints.

3. Simulation Cost, Semidefinite Programming, and Bounds

Although virtual cloning operations are generally not physically implementable, they can be formally simulated via randomized mixtures of CPTP maps. Any HPTP map $\ket{0}$4 admits a decomposition

$\ket{0}$5

where $\ket{0}$6 are CPTP, $\ket{0}$7, and $\ket{0}$8. The simulation cost $\ket{0}$9 quantifies the overhead in physically realizing measurement statistics for $\ket{\psi}$0 via rescaling outcomes. For a given set $\ket{\psi}$1, the minimal simulation cost for perfect state duplication is the solution to a convex semidefinite program (SDP) formulated via the Choi–Jamiołkowski isomorphism: $\ket{\psi}$2 (Bi et al., 23 Jul 2025). The dual SDP offers a witness-operator interpretation: observables $\ket{\psi}$3 on the output, constraining the correspondence between the input set and cloned output.

Lower and upper bounds for the optimal simulation cost $\ket{\psi}$4 are expressible in terms of quantum state distinguishability. For $\ket{\psi}$5 cloning of two distinct states $\ket{\psi}$6: $\ket{\psi}$7 and for pure states $\ket{\psi}$8, $\ket{\psi}$9,

$$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$0

(Bi et al., 23 Jul 2025).

4. From Quantum Cloning to Classical Duplication

Classical information can be robustly duplicated by encoding a logical bit in a large number $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$1 of nearly identical quantum states, with local measurement errors tolerated up to a threshold $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$2. In the classical limit, with majority-vote decoding, the block fidelity approaches unity as $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$3 provided the single-copy fidelity $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$4: $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$5 (Wang et al., 2018). Thus, although individual quantum states cannot be cloned perfectly, collective redundancy, error tolerance, and environmental decoherence enable the emergence of perfect classical duplication in the macroscopic regime, reconciling the quantum no-cloning theorem with classical copyability.

5. Quantum Circuit Approaches to Approximate State Duplication

Practical circuit constructions approximate state duplication for restricted families of states or via probabilistic outputs. The “nesting doubled qubits” protocol prepares an entangled 3-qubit register using CNOT, Hadamard, and permutation gates:

• Initialize the unknown state on $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$6 and ancillas $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$7, $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$8 in $$U\left(\ket{\psi}\otimes\ket{0}\right) = \ket{\psi}\otimes\ket{\psi}.$$9
• Apply CNOT$\ket{\psi_1}, \ket{\psi_2}$0, $\ket{\psi_1}, \ket{\psi_2}$1, CNOT$\ket{\psi_1}, \ket{\psi_2}$2, and two-Toffoli “2-XOR” permutation
• Upon measuring $\ket{\psi_1}, \ket{\psi_2}$3, the remaining two qubits encode the amplitudes from the input state in either computational branch

This scheme does not yield perfect copies except for basis states, with average fidelity $\ket{\psi_1}, \ket{\psi_2}$4, which drops to $\ket{\psi_1}, \ket{\psi_2}$5 for maximally superposed states. The implementation is experimentally feasible with standard gates and mid-circuit measurement on NISQ devices, subject to trade-offs in gate fidelity and quantum depth (Grigoryan et al., 2022).

6. State Duplication in Noisy Duplication Channels

In classical information theory, state duplication arises as a channel feature, notably in noisy duplication channels such as those modeling nanopore DNA sequencing. Here, the observed output results from block duplications (variable sample repetition) and intersymbol interference (ISI). The channel is modeled by Markovian input, geometric duplication ($\ket{\psi_1}, \ket{\psi_2}$6), and additive Gaussian noise: $\ket{\psi_1}, \ket{\psi_2}$7 with $\ket{\psi_1}, \ket{\psi_2}$8 the duplicated-state Markov process.

The information capacity decomposes into two additive terms: $\ket{\psi_1}, \ket{\psi_2}$9 with $\langle \psi_1 | \psi_2 \rangle = 0$0 the intrinsic ISI channel rate, and $\langle \psi_1 | \psi_2 \rangle = 0$1 the penalty for segmentation uncertainty caused by duplications. The latter is tightly quantified via a soft-alignment rate computed through a Soft-DTW-type functional. Lower bounds and operational coding theorems follow from strong AEP properties of this decomposition. Achievable rates and their scaling are dictated by the jump matrix of the underlying state-level transitions, and optimal code design decouples ISI coding from duplication synchronization (McBain et al., 5 Jun 2026).

7. Connections, Limitations, and Broader Implications

The state duplication problem establishes rigorous boundaries between physical duplicability, operational simulation cost, and emergent classical copyability, with implications spanning quantum foundations, circuit design, coding theory, and biotechnology. Key limitations remain: virtual-cloning maps are unphysical except as statistical constructs, and real-world channels often present burst lengths and noise models beyond the geometric/AWGN setting. Nonetheless, the decomposition and simulation-cost frameworks are adaptable to broader contexts, such as insertion–deletion channels, semi-Markov processes, and multidimensional duplication phenomena. The interplay between duplication and distinguishability—quantified via trace-norms and state-discrimination POVMs—emerges as a unifying principle across quantum and classical information domains (Bi et al., 23 Jul 2025, Wang et al., 2018, McBain et al., 5 Jun 2026).