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.
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.
We investigate the cases where a set S of states {∣ψi⟩} cannot be cloned by itself, but is clonable with the help of another system prepared in state ρ^i. When S is pair-wise nonorthogonal, it is known that one can generate the copy from ρ^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 that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.
The no-cloning theorem says there is no quantum copy machine which can copy any one-qubit state. Inner product preserving was always used to prove the no-cloning of nonorthogonal states. In this paper we show that the no-cloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states ∣ψ⟩=a∣0⟩+b∣1⟩ and ∣φ⟩=c∣0⟩+d∣1⟩, assume that a linear operator L can copy them, that is, L(∣ψ,0⟩)=∣ψ,ψ⟩ and L(∣φ,0⟩)=∣φ,φ⟩. Then the two states are orthogonal if and only if L(∣0,0⟩) and L(∣1,0⟩) are unit length states. Thus we only need linearity and that L(∣0,0⟩) and L(∣1,0⟩) are unit length states to prove the no-cloning of nonorthogonal states. It implies that inner product preserving is not necessary for the no-cloning of nonorthogonal states.
All existing quantum cryptosystems use nonorthogonal states as the carriers of information. Nonorthogonal states cannot be cloned (duplicated) by an eavesdropper. As a result, any eavesdropping attempt must introduce errors in the transmission, and, therefore, can be detected by the legal users of the communication channel. Orthogonal states are not used in quantum cryptography, since they can be faithfully cloned without altering the transmitted data. We present a cryptographic scheme based on orthogonal states, which also assures the detection of any eavesdropper.
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.
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 non-orthogonal 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, entangled quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D . We show that if all 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.