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There's no cloning in symplectic mechanics

Aaron Fenyes

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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

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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
Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracket on the phase space with the commutator for the operators. Mathematically the phase space is associated with some symplectic manifold MMM and the non-degenerate 2-form ωωω, which comes with MMM. Geometric prequantization is a process by which one does this in a mathematically "rigorous" manner and we shall attempt to just introduce the methods here. We do this by exploring this contruction for (R2n,∑i=1ndpi∧dqi\mathbb{R}^{2n}, \sum_{i=1}^n dp_i \wedge dq_iR2n,∑i=1n​dpi​∧dqi​) which avoids using complex line bundles in any non-trivial way. One should note however that the Hilbert space one obtains is in fact too "big", in the sense that it has too many functions in order to correspond with actual physically significant Hilbert spaces. Geometric quantization remedies this situation but it should be noted that not all manifolds are prequantizable. We shall not discuss either of these issues however.
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 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 geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the language of symplectic manifolds and avoid linear structure of Hilbert space in such a way that the results can be stated for an arbitrary symplectic manifold.
Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.