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arXiv:2606.24522

No-deleting principle for two unitary copies

Dafa Li

quant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations
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Abstract

Pati and Braunstein defined a deleting machine and showed the impossibility of deleting one of two identical copies of an unknown quantum state. So far, no one has defined two non-identical copies of a quantum state, of course no one has discussed the impossibility of deleting one of two non-identical copies of an unknown quantum state. In this paper, we define $u|ψ>$ and $U|ψ>$, where $u$ and $U$ are any unitary operators, as two unitary copies of a quantum state $|ψ>$, and show that it is impossible to delete one of two unitary copies of an unknown state.

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Pati and Braunstein defined a deleting machine and showed the impossibility of deleting one of two identical copies of an unknown quantum state. So far, no one has defined two non-identical copies of a quantum state, of course no one has discussed the impossibility of deleting one of two non-identical copies of an unknown quantum state. In this paper, we define u∣ψ>u|ψ>u∣ψ> and U∣ψ>U|ψ>U∣ψ>, where uuu and UUU are any unitary operators, as two unitary copies of a quantum state ∣ψ>|ψ>∣ψ>, and show that it is impossible to delete one of two unitary copies of an unknown state.
Suppose we are given two identical copies of an unknown quantum state and we wish to delete one copy from among the given two copies. The quantum no-deletion principle restricts us from perfectly deleting a copy but it does not prohibit us from deleting a copy approximately. Here we construct two types of a ``universal quantum deletion machine'' which approximately deletes a copy such that the fidelity of deletion does not depend on the input state. The two types of universal quantum deletion machines are (1) a conventional deletion machine described by one unitary operator and (2) a modified deletion machine described by two unitary operators. Here it is shown that the modified deletion machine deletes a qubit with fidelity 3∕43∕43∕4, which is the maximum limit for deleting an unknown quantum state. In addition to this we also show that the modified deletion machine retains the qubit in the first mode with average fidelity 0.77 (approx.) which is slightly greater than the fidelity of measurement for two given identical states, showing how precisely one can determine its state [S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995)]. We also show that the deletion machine itself is input state independent, i.e., the information is not hidden in the deleting machine, and hence we can delete the information completely from the deletion 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.
It is known that the classical information like strings of bits can be copied. In 1982, Wootters and Zurek proposed the quantum no-cloning principle. No-cloning principle says that it is impossible to make an identical copy of an arbitrary unknown quantum pure state by using unitary evolution. In this paper, we call U ∣ψ>|ψ>∣ψ>, where U is any unitary operator, a U-copy of the state ∣ψ>|ψ>∣ψ> and show it is impossible to make a U-copy of an arbitrary unknown quantum pure state by using unitary evolution.
Unmeasureability of a quantum state has important consequences in practical implementation of quantum computers. Like copying, deleting of an unknown state from among several copies is prohibited. This is called no-deletion prinicple. Here, we present a no deleting principle for qudits. We obtain a bound on NNN-to-MMM deleting and show that the quality of deletion drops exponentially with the number of copies to be deleted. In addition, we investigate conditional, state-dependent and approximate quantum deleting of unknown states. We prove that unitarity does not allow us to delete copies from an alphabet of two non-orthogonal states exactly. Further, we show that no-deleting principle is consistent with no-signalling.
Is there any point of principle that prohibits us from doing one or more forms of quantum information processing? It is now well known that an unknown quantum state can neither be copied nor deleted perfectly. Given a set of states which are not necessarily orthogonal, is it possible to compare any two states from the set, given some reasonable ordering of such states? Is it possible to sort them in some specific order? In the context of quantum computation, it is shown here that there is no quantum circuit implementing a unitary transformation, for comparing and sorting an unrestricted set of quantum states.
One of the fundamental restrictions that quantum mechanics imposes is the "No deletion Theorem" which tells us that given two identical unknown quantum states, it is impossible to delete one of them. But nevertheless if not perfect, people have tried to delete it approximately. In these approximate deleting processes our basic target is to delete one of the two identical copies as much as possible while preserving the other copy. In this brief report, by using the No communication theorem (NCT) (impossibility of sending signal faster than light using a quantum resource) as a guiding principle, we obtain a bound on the sum of the fidelity of deletion and the fidelity of preservation. Our result not only brings out the complementary relation between these two fidelities but also predicts the optimal value of the fidelity of deletion achievable for a given fidelity of preservation under no signaling constraint. This work eventually saturates the quest for finding out the optimal value of deletion within the NCT framework.