We construct a unique optimal quantum device for turning a finite number of
d-level quantum systems in the same unknown pure state \ensuremath{\sigma} into
M systems of the same kind, in an approximation of the
M-fold tensor product of the state \ensuremath{\sigma}.
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.
In this project, quantum cloning machines are analyzed that take in N quantum systems in the same unknown pure state and output M quantum systems with M > N, such that the output best resembles the ideal, but impossible output of an M-fold tensor product of the pure input state. The proof of Keyl and Werner is reviewed in which a unique optimal solution is constructed, both in the case where the quality of the cloning is determined by the entire output, and in the case where the quality is determined by measurements on a single clone. Furthermore, it is shown that previous work on qubit cloning machines is a special case of the presented optimal solution.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for
d-dimensional states of rank at most
r: producing
M additional copies with worst-case fidelity at least
1−ε requires and is achievable with
N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For
M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
We analyze quantum algorithms for cloning of a quantum measurement. Our aim is to mimic two uses of a device performing an unknown von Neumann measurement with a single use of the device. When the unknown device has to be used before the bipartite state to be measured is available we talk about $1\ensuremath{\rightarrow}2$ learning of the measurement, otherwise the task is called $1\ensuremath{\rightarrow}2$ cloning of a measurement. We perform the optimization for both learning and cloning for arbitrary dimension
d of the Hilbert space. For $1\ensuremath{\rightarrow}2$ cloning we also propose a simple quantum network that achieves the optimal fidelity. The optimal fidelity for $1\ensuremath{\rightarrow}2$ learning just slightly outperforms the estimate and prepare strategy in which one first estimates the unknown measurement and depending on the result suitably prepares the duplicate.
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.