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arXiv:quant-ph/0204153

A stronger no-cloning theorem

Richard Jozsa

Published Apr 26, 2002• 14 citations
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Abstract

It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given) copy.

DOI
10.48550/arxiv.quant-ph/0204153

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It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given) copy.
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given NNN copies of an unknown state, any channel producing an (N+1)(N+1)(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+1N\rightarrow N+1N→N+1 cloning: for a desired small error εεε, what NNN suffices? For arbitrary pure states on a Hilbert space HHH, the answer is N≃dim⁡H/εN\simeq \dim H/εN≃dimH/ε - astronomically large for most HHH 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: nnn-mode fermionic Gaussian and Slater states, nnn-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in nnn. 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 NNN. 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.
We investigate the cases where a set SSS of states {∣ψi⟩}\{\ket{ψ_i} \}{∣ψi​⟩} cannot be cloned by itself, but is clonable with the help of another system prepared in state ρ^i\hatρ_iρ^​i​. When SSS is pair-wise nonorthogonal, it is known that one can generate the copy from ρ^i\hatρ_iρ^​i​ alone, with no interaction with the original system. Here we show that a set containing orthogonal pairs exhibits a property forming a striking contrast; For any such set, there is a choice of ρ^i\hatρ_iρ^​i​ that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.
It is known that the classical information like strings of bits can be copied. In 1982, Wootters and Zurek proposed the quantum no-cloning principle. No-cloning principle says that it is impossible to make an identical copy of an arbitrary unknown quantum pure state by using unitary evolution. In this paper, we call U ∣ψ>|ψ>∣ψ>, where U is any unitary operator, a U-copy of the state ∣ψ>|ψ>∣ψ> and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.
It is believed that the more we have {\it a priori} information on input states, the better we can make the quality of clones in quantum cloning machines. This common sense idea was confirmed several years ago by analyzing a situation, where the input state is either one of two non-orthogonal states. If the {\it a priori} information is measured by the Shannon entropy, common sense predicts that the quality of the clone becomes poorer with increasing NNN, where NNN is the number of possible input states. We show, however, that the {\it a priori} information measured by the Shannon entropy does not affect the quality of the clones. Instead the no-cloning theorem and `denseness' of the possible input states play important roles in determining the quality. Specifically, the factor `denseness' plays a more crucial role than the no-cloning theorem when N≥3N \geq 3N≥3.
Any physical transformation that equally distributes quantum information over a large number M of users can be approximated by a classical broadcasting of measurement outcomes. The accuracy of the approximation is at least of the order O(M(-1)). In particular, quantum cloning of pure and mixed states can be approximated via quantum state estimation. As an example, for optimal qubit cloning with 10 output copies, a single user has an error probability p(err) > or = 0.45 in distinguishing classical from quantum output, a value close to the error probability of the random guess.