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arXiv:quant-ph/9609010 · Modern Physics Letters B

Quantum Signal Splitting that Avoids Initialization of the Targets

Dima Mozyrsky, Vladimir Privman A

Published Dec 20, 1997• 1 citation
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Abstract

The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system (qubit, spin) into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t+Δt. In essence, the restriction on the initial target states is exchanged for uncertainty as to which two of the three qubits retain copies of the initial source state.

DOI
10.1142/s0217984997001523

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•Oct 2, 2026•0 citations
The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system (qubit, spin) into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t+Δt. In essence, the restriction on the initial target states is exchanged for uncertainty as to which two of the three qubits retain copies of the initial source state.
The classical signal splitting and copying are not possible in quantum mechanics. Specifically, one cannot copy the basis up and down states of the input (I) two-state system into the copy (C) and duplicate-copy (D) two-state systems if the latter systems are initially in an arbitrary state. We consider instead a quantum evolution in which the basis states of I at time t are duplicated in at least two of the systems I, C, D, at time t + \\Delta t. For possible applications in quantum computing, we derive an explicit Hamiltonian to accomplish this process; it turns out to involve only three-spin x,y-component interactions.
We derive an explicit Hamiltonian for copying the basis up and down states of a quantum two-state system—a qubit—onto n “copy ” qubits (n ≥ 1) initially all prepared in the down state. In terms of spin components, for spin- 1 2 particle spin states, the resulting Hamiltonian involves n- and (n + 1)-spin interactions. The case n = 1 also corresponds to a quantum-computing controlled-NOT gate. PACS numbers: 03.65.Bz, 85.30.St. – 1 – Interest in quantum computing [1-27] has boosted studies of quantum mechanics of two-state systems such as the spin states of spin- 1 2 particles. We will use “spin ” to indicate a two-state system in this context. The “binary ” up and down spin states are of particular significance and
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.
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.
The concept of infinity long ago led to an understanding that proper subsets of an infinite set can be put in a one-to-one correspondence with the original set. This mechanism of ‘parts’ of a set ‘duplicating’ the original set took on deeper significance when in 1914 Hausdorff showed that the three-dimensional sphere S2 can be decomposed into a small finite number of pairwise disjoint subsets which may then be transformed by simple rotations to form two identical copies of S2. Further, the basic Hausdorff result may be extended to the case of the formation of any finite number of identical copies, and even to the case of countably infinite copies, if the restriction to finitely many initial pairwise disjoint pieces is relaxed to allow countably many such pieces. We demonstrate the non-causal mechanics of state space multiplicity by the use of a free group of an independent pair of rotations acting on the state space of a qubit, S2, i.e. rotations from the group SO(3). We go on to speculate on the ontological implications of these rotational actions, namely that quantum state space is the ontological basis of a quantum system, and admits non-causal non-classical structure mappings which may provide a framework for superposition basis state co-existence, and even for a frequentist probability theory.