The quantum no cloning theorem is an essential result in quantum information theory. Following this idea, we give a physically natural definition of cloning in the context of classical mechanics using symplectic geometry, building on work of A. Fenyes, J. Math. Phys. 53, 012902 (2012). We observe, following Fenyes, any system with phase space (R2N,dxi∧dyi) can be cloned in our definition. However, we show that if (M, ω) can be cloned in our definition, then M must be contractible. For instance, this shows the simple pendulum cannot be cloned in Hamiltonian mechanics. We further formulate a robust notion of approximate cloning, and show that if (M, ω) can be approximately cloned, then M is contractible. We give interpretations of our results and in some special cases reconcile our no cloning theorems with the general experience that classical information is clonable. Finally we point to new directions of research, including a connection of our result with the classical measurement problem.
The fact that you can’t clone a quantum system is closely related to the fact that the tensor product in the category of Hilbert spaces is non-Cartesian. At the end of his 2008 classical mechanics course, John Baez pointed out that the tensor product in the category of Poisson manifolds is also non-Cartesian, which should mean there’s a classical analogue of the no-cloning theorem [4]! This turns out to be true in symplectic mechanics, and I don’t see any reason it shouldn’t be true in more generalized settings as well. 1.2 Symplectic mechanics In symplectic mechanics, physical systems are represented by symplectic manifolds, and physical processes are represented by symplectomorphisms. If M and N are symplectic manifolds with symplectic forms ω and σ, respectively, the product manifold M × N has a natural symplectic form Ω given by Ω[(u, s), (v, t)] = ω(u, v) + σ(s, t), where u and v are tangent vectors at the same point in M, and s and t are tangent vectors at the same point in N [3, page 7]. I’ll denote the resulting symplectic manifold M ⊗ N to emphasize that it’s not a Cartesian product in the category of symplectic manifolds. I’ll often write a point in M ⊗ N as an ordered pair of points in M and N, just as I previously wrote a tangent vector on M ⊗ N as an ordered pair of tangent vectors on M and N. 2
The no-cloning theorem is a basic limitative result for quantum mechanics, with particular significance for quantum information. It says that there is no unitary operation that makes perfect copies of an unknown (pure) quantum state. We re-examine this foundational result from the perspective of the categorical formulation of quantum mechanics recently introduced by the author and Bob Coecke. We formulate and prove a novel version of the result, as an incompatibility between having a “natural” copying operation and the structural features required for modeling quantum entanglement coexisting in the same category. This formulation is strikingly similar to a well-known limitative result in categorical logic, Joyal's lemma, which shows that a “Boolean cartesian closed category” trivializes and hence provides a major roadblock to the computational interpretation of classical logic. This shows a heretofore unsuspected connection between limitative results in proof theory and no-go theorems in quantum mechanics. The argument is of a robust, topological character and uses the graphical calculus for monoidal categories to advantage. 1.1 Introduction The no-cloning theorem (Dieks 1982; Wootters and Zurek 1982) is a basic limitative result for quantum mechanics, with particular significance for quantum information. It says that there is no unitary operation that makes perfect copies of an unknown (pure) quantum state. A stronger form of this result is the no-broadcasting theorem (Barnum et al. 1996), which applies to mixed states. There is also a no-deleting theorem (Pati and Braunstein 2000).
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.
The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus. This is, on the one hand, an extension of the classical probability calculus and, on the other hand, a mathematical generalization of the Lueders - von Neumann quantum measurement process. In the non-classical case, a very special type of conditional probability emerges, describing the probability for the transition from a past event to a future event independently of any underlying state. This probability results from the algebraic structure of the quantum logic only and is invariant under algebraic morphisms, which is used to prove the generalized no-cloning theorem in a rather abstract, though simple and basic fashion without relying on a tensor product construction as required in other approaches.