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arXiv:quant-ph/0604009

Quantum cloning with nonlocal assistance: Complement of Jozsa's stronger no-cloning theorem

Koji Azuma, Masato Koashi, Hosho Katsura, Nobuyuki Imoto

Published Apr 3, 2006• 0 citations
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Abstract

We investigate the cases where a set $S$ of states $\{\ket{ψ_i} \}$ cannot be cloned by itself, but is clonable with the help of another system prepared in state $\hatρ_i$. When $S$ is pair-wise nonorthogonal, it is known that one can generate the copy from $\hatρ_i$ 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 $\hatρ_i$ that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.

DOI
10.48550/arxiv.quant-ph/0604009

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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 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.
In classical information theory one can, in principle, produce a perfect copy of any input state. In quantum information theory, the no cloning theorem prohibits exact copying of nonorthogonal states. Moreover, if we wish to copy multiparticle entangled states and can perform only local operations and classical communication (LOCC), then further restrictions apply. We investigate the problem of copying orthogonal quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D. We show that if some of the states to be copied are non-maximally entangled, then novel LOCC copying procedures based on entanglement catalysis are possible. We then study in detail the LOCC copying problem where both the blank state and at least one of the states to be copied are maximally entangled. For this to be possible, we find that all the states to be copied must be maximally entangled. We obtain a necessary and sufficient condition for LOCC copying under these conditions. For two orthogonal, maximally entangled states, we provide the general solution to this condition. We use it to show that for D=2,3, any pair of orthogonal, maximally entangled states can be locally copied using a maximally entangled blank state. However, we also show that for any D which is not prime, one can construct pairs of such states for which this is impossible.
The no-cloning principle tells us that nonorthogonal quantum states cannot be cloned, but it does not tell us that orthogonal states can always be cloned. We suggest a situation where the cloning transformations are restricted, leading to a novel type of no-cloning principle. In the case of a composite system made of two subsystems: if the subsystems are only available one after the other then there are various cases when orthogonal states cannot be cloned. Surprising examples are given, which give a radically better insight regarding the basic concepts of quantum cryptography.
Quantum information is nonlocal in the sense that local measurements on a composite quantum system, prepared in one of many mutually orthogonal states, may not reveal in which state the system was prepared. It is shown that in the many copy limit this kind of nonlocality is fundamentally different for pure and mixed quantum states. In particular, orthogonal mixed states may not be distinguishable by local operations and classical communication, no matter how many copies are supplied, whereas any set of N orthogonal pure states can be perfectly discriminated with m copies, where m<N. Thus mixed quantum states can exhibit a new kind of nonlocality absent in pure states. We also argue that a set of orthogonal quantum states may be said to be maximally indistinguishable iff the set is not conclusively locally distinguishable with multiple copies.
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.