We present a nonlinear generalization of quantum mechanics where one can copy arbitrary pure states with fidelities higher than those of the quantum cloners. Weinberg’s nonlinear quantum mechanics has been criticized for its possible implications for superluminal communication. We demonstrate that universal symmetric cloners can make better copies than standard quantum ones and yet they do not violate the no-signalling principle. If linearity constraint is added, we show that for a given fidelity there is a unique cloner which can be written down explicitly.
We investigate the class of quantum cloning machines that equally duplicate all real states in a Hilbert space of arbitrary dimension. By using the no-signaling condition, namely, that cloning cannot make superluminal communication possible, we derive an upper bound on the fidelity of this class of quantum cloning machines. Then, for each dimension d, we construct an optimal symmetric cloner whose fidelity saturates this bound. Similar calculations can also be performed in order to recover the fidelity of the optimal universal cloner in d dimensions.
We study the role of interference in the process of quantum cloning. We show that in order to achieve better than classical cloning of a qubit no interference is needed. In particular, a large class of symmetric universal $1\ensuremath{\rightarrow}2$ qubit cloners exists which achieve the optimal average fidelity
5∕6 for such machines, without using any interference in the propagation of the original and copied qubits. We also obtain optimal average fidelities for interference-free cloning in asymmetric situations, and discuss the relation of the quantum cloners found to the Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery quantum cloner.
If black holes were able to clone quantum states, a number of paradoxes in black hole physics would disappear.However, the linearity of quantum mechanics forbids exact cloning of quantum states.Here we show that black holes indeed clone incoming quantum states with a fidelity that depends on the black hole's absorption coefficient, without violating the no-cloning theorem because the clones are only approximate.Perfectly reflecting black holes are optimal universal 'quantum cloning machines' and operate on the principle of stimulated emission, exactly as their quantum optical counterparts.In the limit of perfect absorption, the fidelity of clones is only equal to what can be obtained via quantum state estimation methods.But for any absorption probability less than one, the cloning fidelity is nearly optimal as long as ω ⩾ T 10, a common parameter for modest-sized black holes.
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given
N copies of an unknown state, any channel producing an
(N+1)-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate
N→N+1 cloning: for a desired small error
ε, what
N suffices? For arbitrary pure states on a Hilbert space
H, the answer is
N≃dimH/ε - astronomically large for most
H of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information:
n-mode fermionic Gaussian and Slater states,
n-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in
n. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high
N. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.