No Cloning of Quantum Ensembles

Abstract: Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.

• The paper proves that cloning ensemble members or estimating second-moment properties generally requires exponentially many purification copies, with lower bounds of 2^{Ω(n_B)} in key settings.
• It shows that bounded circuit complexity enables sample-efficient reconstruction, but classical post-processing can remain intractable when the preparation circuit is not efficiently simulable.
• Under QPRF, QPRP, and LWE assumptions, estimating nonlinear ensemble properties remains super-polynomially hard—even when the state-preparation circuit is fully known—limiting access to measurement-induced entanglement and deep thermalization.

The paper "No Cloning of Quantum Ensembles" (2606.27756) establishes a set of information-theoretic and computational no-go results for cloning the individual members of a quantum ensemble and for estimating nonlinear properties of ensembles, even when the algorithm is granted access to many copies of the ensemble's purification or to a complete classical description of the state-preparation circuit. These results formalize a barrier that directly affects the experimental accessibility of measurement-induced phenomena such as deep thermalization and measurement-induced entanglement.

Setting and task definitions

A quantum ensemble $\mathcal{E} = \{(p(\bm z), \ket{\psi_{\bm z})\}$ is realized operationally as the projected ensemble of a purification $\ket{\psi}_{AB}$: computational-basis measurements on the environment $B$ of size $n_B$ produce states on system $A$ of size $n_A$ with probabilities $p_\psi(\bm z)$. The paper assumes the modern experimental setting in which copies of the purification are available, consistent with high-repetition-rate platforms used to study deep thermalization and measurement-induced phase transitions.

Two tasks are studied. The cloning task requires preparing the classical–quantum state $\sigma^{(2)}_\psi = \sum_{\bm z} p(\bm z)\, \psi_{\bm z}^{\otimes 2} \otimes \ketbra{\bm z}$, i.e., producing two copies of the same post-selected trajectory with the correct outcome label. This is strictly stronger than resampling two independent trajectories, which is trivial given two copies of $\psi_{AB}$. The estimation task requires outputting $o^{(2)}_\psi = \tr[O\, \rho^{(2)}_\psi]$ for an observable $\ket{\psi}_{AB}$0 on $\ket{\psi}_{AB}$1, where $\ket{\psi}_{AB}$2 is the second moment state. Estimation is the natural framework for measurement-induced phenomena: ensemble-averaged subsystem purity corresponds to $\ket{\psi}_{AB}$3 with the swap operator, measurement-induced magic to a fourth moment with a Pauli-sum observable, and certifying deep thermalization amounts to showing $\ket{\psi}_{AB}$4 for all $\ket{\psi}_{AB}$5. Since estimation hardness implies cloning hardness, results for estimation are the stronger statements.

Information-theoretic hardness

The first main result is a no-cloning theorem for ensembles: any quantum algorithm with access to $\ket{\psi}_{AB}$6 copies of an unknown purification $\ket{\psi}_{AB}$7 requires $\ket{\psi}_{AB}$8 copies to clone ensemble members with small constant error (average case over Haar-random $\ket{\psi}_{AB}$9) or to estimate $B$0 for observables with $B$1 (worst case). The proof of the average-case bound proceeds in four steps: restriction to collision-free measurement outcomes (collisions are exponentially unlikely for Haar-random states when $B$2), dephasing between outcome sequences not related by permutations, reduction via symmetry twirling to a select-and-clone task on $B$3 independent Haar-random states, and finally a Choi–Jamiołkowski argument showing no channel clones a Haar-random state with fidelity above $B$4.

Two features distinguish this from the conventional no-cloning theorem. First, the hardness persists even with sub-exponential access to the purification and arbitrary entangling operations across copies, a strictly stronger access model than in prior learning separations. Second, the mechanism is not the indistinguishability of non-orthogonal states but a tradeoff between measurements on $B$5 and sample complexity: the theorem quantifies an explicit lower bound involving $B$6, and the complexity can be exponential even for a single-qubit ensemble if $B$7 is large.

For estimation, the worst-case hardness is established via a distinguishing game between random subset states $B$8 and Haar-random states. These are statistically indistinguishable from $B$9 copies unless $n_B$0, yet their projected ensembles differ drastically: subset-state projections are computational-basis product states, while Haar projections are highly entangled. Any estimator of $n_B$1 to constant additive accuracy would solve this distinguishing problem. A corollary extends this to any property with a constant gap between computational-basis ensembles and Haar-projected ensembles, yielding worst-case exponential sample complexity for observing measurement-induced entanglement, magic, circuit complexity, and for verifying emergent state designs (deep thermalization) in the regime $n_B$2. The authors note an important caveat: for Haar-typical states the estimation task is trivial on average, since $n_B$3 concentrates at the thermal value; the hardness is therefore inherently a worst-case statement, relevant to ensembles such as monitored circuits whose purifications are not Haar-typical.

Sample-efficient algorithm from structural prior knowledge

The information-theoretic barrier is bypassed when the ensemble has bounded circuit complexity. If the purification is prepared by a $n_B$4-gate circuit, the bounded-complexity tomography protocol of Zhao et al. learns $n_B$5 to trace distance $n_B$6 using $n_B$7 copies, and a lemma shows that $n_B$8th-order moment states are $n_B$9-Lipschitz in the purification trace distance. Consequently $A$0 is learnable to error $A$1 with polynomially many copies. This explains why measurement-induced phenomena can be observed with few samples in current experiments. The limitation is explicit: the classical post-processing needed to construct the cloned ensemble or evaluate nonlinear quantities is generally intractable once the circuit is no longer classically simulable.

Computational hardness with full prior knowledge

The central result is that the estimation task remains computationally hard even when the preparation circuit is fully specified, in sharp contrast to the conventional no-cloning theorem, which fails once the state is known. Two statements are proven. First, assuming quantum-secure pseudorandom functions and permutations (QPRF/QPRP), any algorithm that estimates $A$2 for efficiently measurable $A$3, given copy access to an efficiently preparable unknown state with $A$4, requires super-polynomial time. Second, assuming the quantum hardness of LWE with standard parameters, hardness holds even when the algorithm receives the full classical description of a polynomial-size preparation circuit, for $A$5. Under sub-exponentially secure pseudorandom primitives, the bounds strengthen to $A$6 time and $A$7.

The proofs construct pairs of efficiently preparable ensembles that are computationally indistinguishable (via hybrid arguments through subset states to Haar-random states) yet have maximally separated values of $A$8: the QPRF-based ensemble has $A$9 for every member, while its Hadamard-transformed counterpart has $n_A$0. For the circuit-description setting, an LWE instance $n_A$1 is encoded into a state whose second-moment value is $n_A$2 for LWE samples and $n_A$3 for uniform samples, so an efficient estimator solves decisional LWE. A corollary shows that estimating the ensemble-averaged probability of entanglement across a single-qubit cut is at least as hard as LWE under the same conditions, establishing a concrete computational barrier for observing measurement-induced entanglement in the state-aware setting.

Implications and limitations

The results carry three immediate implications. Measurement-induced randomness cannot derandomize circuits that require reusing a single random instance, since one sampled trajectory cannot be cloned; this is a concrete drawback of measurement-based derandomization relative to controllable unitary designs. Post-selection-based experimental probes of ensemble properties are intrinsically hard, explaining why existing demonstrations are confined to classically simulable or specially structured regimes. And the paper articulates a dilemma: nonlinear ensemble phenomena are either computationally hard to access even for quantum computers, or accessible only where classical methods already suffice.

The paper is explicit about the scope of its claims. The information-theoretic hardness for cloning is average-case, while estimation hardness is worst-case; average-case estimation is trivial for Haar-typical states. The computational results rest on standard but unproven cryptographic assumptions (QPRF/QPRP existence, LWE hardness), and the lower bounds apply to universal algorithms — structured dynamics such as spacetime-dual circuits admit efficient purity estimation, though these typically sacrifice criticality and analytic tractability. The corollary on deep thermalization applies to verifying emergent state designs in the regime $n_A$4, and the authors anticipate, but do not prove, analogous barriers for magic, complexity, and higher moments. An open question they identify is whether access to $n_A$5th moments for $n_A$6 yields computational power approaching post-selection (the class PostBQP), which would establish a hierarchy over moment order.

Conclusion

This work establishes that cloning quantum ensembles and estimating their nonlinear properties are barred by two distinct mechanisms: an information-theoretic tradeoff between environmental measurements and sample complexity, and a computational barrier that persists even with complete knowledge of an efficiently preparable purification. Together these results delineate a new boundary on quantum information processing, constrain measurement-based derandomization proposals, and formalize why experimental signatures of measurement-induced phenomena have so far been limited to classically tractable regimes.