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.
We show that, there are physical means for cloning two non-orthogonal pure states which are secretly chosen from a certain set
. The states are cloned through a unitary evolution together with a measurement. The cloning efficiency can not attain 100%. With some negative measurement results, the cloning fails.
We consider the problem of approximate cloning of quantum states: given
n copies of an unknown state
ρ∈Cd×d, prepare an
(n+k)-copy state with high fidelity to
ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that
n=Θ(kd/ε) copies are necessary and sufficient to clone
k additional copies of an unknown pure state to fidelity
1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given
n copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using
n=O(krd/ε) copies to clone rank-
r states. Can one do any better? We show that the answer is no: one must use
n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state
ρ is promised to be of the form
P/r, where
P is a rank-
r orthogonal projector.
As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert
ρ⊗n to a
k-copy state with high fidelity to
(ρT)⊗k. Here, we again show
n=Θ(krd/ε) copies are necessary and sufficient for this task.
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.