It is known that in phase-covariant quantum cloning, the equatorial states on the Bloch sphere can be cloned with a fidelity higher than the optimal bound established for universal quantum cloning. We generalize this concept to include other states on the Bloch sphere with a definite z component of spin. It is shown that once we know the z component, we can always clone a state with a fidelity higher than the universal value and that of equatorial states. We also make a detailed study of the entanglement properties of the output copies and show that the equatorial states are the only states that give rise to a separable density matrix for the outputs.
As physical systems, qubits must evolve from input to output state. We describe a simple scheme in which the effect of a quantum gate is described by the action of an effective Hamiltonian acting for some characteristic time. This model shows that the action of common unary gates is to induce Bloch sphere trajectories along lines of latitude relative to an eigenvector of the gate. Such trajectories would immediately move a `rebit', initially confined to a line of latitude, off this line and acquire a complex phase. The role of the complex phase in bringing about the entanglement of two qubits is also highlighted. It is then asked whether such dynamics could be modelled using real QM. It is shown that the continuous evolution required for such dynamics can only be provided by members of the special orthogonal group of the vector space. Since the matrices representing many quantum gates of interest have determinant -1, no real special orthogonal operators can model their evolution if the dimension of the real vector space is the same as that of the complex space. Next we look at the mapping from a complex vector space of dimension
N to a real space of dimension
2N that is often used to construct `real' QM. It is shown that this is just an isomorphic mapping from the scalar representation of complex numbers to their
2×2 matrix equivalents, so that the resulting matrices are actually represent complex matrices. Finally, we investigate the endomorphism of real vector spaces of dimension
N=2n, where
n∈Z+, suitable for the modelling of
n−1 qubits. We confirm that the mapping from
C2n−1→R2n only maps elements of
End(C2n−1) to a restricted subspace of
End(R2n) that reproduces the `real' representation of complex matrices.
After introducing the no-cloning theorem and the most common forms of approximate quantum cloning, universal quantum cloning is considered in detail. The connections it has with universal NOT-gate, quantum cryptography and state estimation are presented and briefly discussed. The state estimation connection is used to show that the amount of extractable classical information and total Bloch vector length are conserved in universal quantum cloning. The 1 2 qubit cloner is also shown to obey a complementarity relation between local and nonlocal information. These are interpreted to be a consequence of the conservation of total information in cloning. Finally, the performance of the 1 M cloning network discovered by Bužek, Hillery and Knight is studied in the presence of decoherence using the Barenco et al. approach where random phase fluctuations are attached to 2-qubit gates. The expression for average fidelity is calculated for three cases and it is found to depend on the optimal fidelity and the average of the phase fluctuations in a specific way. It is conjectured to be the form of the average fidelity in the general case. While the cloning network is found to be rather robust, it is nevertheless argued that the scalability of the quantum network implementation is poor by studying the effect of decoherence during the preparation of the initial state of the cloning machine in the 1 ! 2 case and observing that the loss in average fidelity can be large. This affirms the result by Maruyama and Knight, who reached the same conclusion in a slightly different manner.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.
We find an optimal quantum cloning machine, which clones qubits of arbitrary symmetrical distribution around the Bloch vector with the highest fidelity. The process is referred to as phase-independent cloning in contrast to the standard phase-covariant cloning for which an input qubit state is a priori better known. We assume that the information about the input state is encoded in an arbitrary axisymmetric distribution (phase function) on the Bloch sphere of the cloned qubits. We find analytical expressions describing the optimal cloning transformation and fidelity of the clones. As an illustration, we analyze cloning of qubit state described by the von Mises--Fisher and Brosseau distributions. Moreover, we show that the optimal phase-independent cloning machine can be implemented by modifying the mirror phase-covariant cloning machine for which quantum circuits are known.