It is a well-known fact that the no-cloning theorem forbids the creation of identical copies of an arbitrary unknown quantum state.In other words,there does not exist a quantum cloning machine that can clone all quantum states.However,it is possible to clone given quantum states under certain conditions,for instance,k distinct pure states|ψ1,|ψ2,...,|ψk can be cloned simultaneously if and only if they are orthogonal.This paper discusses the existence and construction of simultaneous cloning machines for mixed states.It is proved that k distinct mixed statesρ1,ρ2,...,ρkof the n-dimensional quantum system Cncan be cloned simultaneously,that is,there exists a quantum channel Φ on MnMn and a state Σ in Mn,such that Φ(ρiΣ)=ρiρi for all i,if and only ifρiρj=0(i j).Also,the constructing procedure of the desired simultaneous cloning machine is given.
We show that using a fixed cloning machine, whether deterministic inexact or probabilistic exact, one can take an arbitrary blank state. Quantum information cannot be cloned. There cannot exist a machine which can produce two (or more) exact copies of an arbitrary state in a deterministic manner [1]. Replication is not allowed even when the input state is taken at random from a given set of two non-orthogonal states [2]. It has been further shown that probabilistic cloning is not possible if the input state is from a given linearly dependent set [3]. Henceforth we deal explicitly with qubits although all the results can be extended to higher dimensions. Although exact cloning is not possible, one can approximately clone an arbitrary input state [4, 5, 6]. In this scheme a fixed state is taken as the blank state, depending on which and the initial machine state, the cloning operation is constructed. We extend this operation such that any arbitrary (pure or mixed) state taken as the blank copy can do the job. The optimal universal 1 → 2 inexact qubit cloner of Brußet al.[5] takes an arbitrary qubit |ψ 〉 〈ψ | = 1 2 (I + − → s. − → σ) along with a fixed blank qubit |b 〉 and a machine state |M 〉 as input. An entangled state of the three qubits is produced as the output such that the reduced density matrices of the first two qubits are two similar approximate copies ρ = 1 2 (I + η− → s. − → σ) of |ψ 〉 〈ψ | with η = 2 3. The unitary operator realizing this process is defined on the combined Hilbert space of the input qubit, the blank qubit and machine by U ′ √ √
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.
Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.