Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Quantum superimposing multiple cloning-cum-complementing machine

Yan‐Ling Li, Jian Feng, Bao‐Long Liang, Xiang‐Guo Meng

Published Aug 1, 2007• 0 citations
Publisher

Abstract

We prove that the nonorthogonal states randomly selected from a set can evolve into a linear superposition of multiple original and orthogonal complementing states with failure branch if and only if the input states are linearly independent. The results for a single-input state are also generalized into the case of several copies of an input state.

DOI
10.1088/1751-8113/40/33/021

Related papers

Quantum Superposition of Multiple Clones and the Novel Cloning Machine

Arun Kumar Pati

We envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch can exist in a superposition of composite states independent of the input state. We prove that unknown nonorthogonal states chosen from a set $S$ can evolve into a linear superposition of multiple clones and failure branches by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the deterministic and probabilistic clonings are special cases of our novel cloning machine.

arXiv:quant-ph/9903038•Physical Review Letters•Oct 4, 1999•74 citations

Some analogies between quantum cloning and quantum deleting

Daowen Qiu

We further verify the impossibility of deleting an arbitrary unknown quantum state, and also show it is impossible to delete two nonorthogonal quantum states as a consequence of unitarity of quantum mechanics. A quantum approximate (deterministic) deleting machine and a probabilistic (exact) deleting machine are constructed. The estimation for the global fidelity characterizing the efficiency of the quantum approximate deleting is given. We then demonstrate that unknown nonorthogonal states chosen from a set with their multiple copies can evolve into a linear superposition of multiple deletions and failure branches by a unitary process if and only if the states are linearly independent. It is notable that the proof for necessity is somewhat different from Pati's [Phys. Rev. Lett. 83, 2849 (1999)]. Another deleting machine for the input states that are unnecessarily linearly independent is also presented. The bounds on the success probabilities of these deleting machines are derived. So we expound some preliminary analogies between quantum cloning and deleting.

Physical Review A•Apr 15, 2002•15 citations

Probabilistic cloning and deleting of quantum states

Yuan Feng, Shengyu Zhang, Mingsheng Ying

We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.

Physical Review A•Apr 10, 2002•13 citations

Quantum cloning with nonlocal assistance: Complement of Jozsa's stronger no-cloning theorem

Koji Azuma, Masato Koashi, Hosho Katsura +1 more

We investigate the cases where a set $S$ of states $\{\ket{ψ_i} \}$ cannot be cloned by itself, but is clonable with the help of another system prepared in state $\hatρ_i$. When $S$ is pair-wise nonorthogonal, it is known that one can generate the copy from $\hatρ_i$ 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 $\hatρ_i$ that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.

arXiv:quant-ph/0604009•Apr 3, 2006•0 citations

Probabilistic Cloning and Identification of Linearly Independent Quantum States

L.-M. Duan, Guang‐Can Guo

We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},|{\ensuremath{\Psi}}_{n}〉}$ can be probabilistically cloned if and only if $|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},\mathrm{and}|{\ensuremath{\Psi}}_{n}〉$ are linearly independent. We derive the best possible cloning efficiencies. Probabilistic cloning has a close connection with the problem of identification of a set of states, which is a type of $n+1$ outcome measurement on $n$ linearly independent states. The optimal efficiencies for this type of measurement are obtained.

arXiv:quant-ph/9804064•Physical Review Letters•Jun 1, 1998•474 citations

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

Dafa Li, Xiangrong Li, Hongtao Huang +1 more

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.

Journal of Mathematical Physics•Jul 28, 2005•2 citations
We prove that the nonorthogonal states randomly selected from a set can evolve into a linear superposition of multiple original and orthogonal complementing states with failure branch if and only if the input states are linearly independent. The results for a single-input state are also generalized into the case of several copies of an input state.
We envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch can exist in a superposition of composite states independent of the input state. We prove that unknown nonorthogonal states chosen from a set SSS can evolve into a linear superposition of multiple clones and failure branches by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the deterministic and probabilistic clonings are special cases of our novel cloning machine.
We further verify the impossibility of deleting an arbitrary unknown quantum state, and also show it is impossible to delete two nonorthogonal quantum states as a consequence of unitarity of quantum mechanics. A quantum approximate (deterministic) deleting machine and a probabilistic (exact) deleting machine are constructed. The estimation for the global fidelity characterizing the efficiency of the quantum approximate deleting is given. We then demonstrate that unknown nonorthogonal states chosen from a set with their multiple copies can evolve into a linear superposition of multiple deletions and failure branches by a unitary process if and only if the states are linearly independent. It is notable that the proof for necessity is somewhat different from Pati's [Phys. Rev. Lett. 83, 2849 (1999)]. Another deleting machine for the input states that are unnecessarily linearly independent is also presented. The bounds on the success probabilities of these deleting machines are derived. So we expound some preliminary analogies between quantum cloning and deleting.
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.
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.
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},|{\ensuremath{\Psi}}_{n}〉}$ can be probabilistically cloned if and only if $|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},\mathrm{and}|{\ensuremath{\Psi}}_{n}〉$ are linearly independent. We derive the best possible cloning efficiencies. Probabilistic cloning has a close connection with the problem of identification of a set of states, which is a type of n+1n+1n+1 outcome measurement on nnn linearly independent states. The optimal efficiencies for this type of measurement are obtained.
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.