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Journal of Mathematical Physics

No-cloning of nonorthogonal states does not require inner product preserving

Dafa Li, Xiangrong Li, Hongtao Huang, Xinxin Li

Published Jul 28, 2005• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.1996327

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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.
Deterministically cloning (copying) nonorthogonal states is forbidden in quantum mechanics, but deterministic pseudo-unitary cloning is possible in a nonunitary system. We prove and show that, for any two nonorthogonal qubit states, we can find a linear, invertible Hermitian (metric) operator to make these states mutually orthogonal with respect to the corresponding pseudo-inner product. From this metric operator, we construct a pseudo-Hermitian Hamiltonian and its evolution operator as an ideal, deterministic pseudo-unitary cloner. For applicability, suppose Alice lives in our ordinary universe and Bob lives in a universe that has the desired pseudo-unitary evolution. Alice would send her nonorthogonal states to Bob and Bob would do the cloning deterministically and send back to Alice. The quantum channel will be lossy not due to Bob's universe's properties, but rather due to problems making a channel between the two universes. We experimentally demonstrate deterministic pseudo-unitary two-qubit cloning for a photonic pseudo-unitary two-qubit system. In our universe we have to demonstrate that universe by executing a postselected gate that effects loss-based pseudo-unitary evolution. Furthermore, we introduce an algorithmic method for designing experimental realizations of a generic class of nonunitary operators for pseudo-unitary cloning.
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 MnMn and a state Σ in Mn,such that Φ(ρiΣ)=ρiρi for all i,if and only ifρiρj=0(i j).Also,the constructing procedure of the desired simultaneous cloning machine is given.
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.
We investigate the cases where a set SSS of states {∣ψi⟩}\{\ket{ψ_i} \}{∣ψi​⟩} cannot be cloned by itself, but is clonable with the help of another system prepared in state ρ^i\hatρ_iρ^​i​. When SSS is pair-wise nonorthogonal, it is known that one can generate the copy from ρ^i\hatρ_iρ^​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\hatρ_iρ^​i​ that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.