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Phase Co-variant Cloning: Difference between revisions
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* We assume that the transformation is a [[unitary]] transformation acting on the Hilbert space of 2 (or 3 for the with ancilla case) qubits. | * We assume that the transformation is a [[unitary]] transformation acting on the Hilbert space of 2 (or 3 for the with ancilla case) qubits. | ||
==Outline== | ==Outline== | ||
Phase Variant cloner can act effectively only on states lying on the equator of Bloch sphere. These states can be described by an angle factor such as <math>\phi</math> and aim of this protocol is to produce optimal copies of these states such that the fidelity of the copies in independent of <math>\phi</math>. This task can be done with two types of protocol: without any ancilla (test states) and with ancilla. Both methods produce copies with same optimal fidelity | Phase Variant cloner can act effectively only on states lying on the equator of Bloch sphere. These states can be described by an angle factor such as <math>\phi</math> and aim of this protocol is to produce optimal copies of these states such that the fidelity of the copies in independent of <math>\phi</math>. This task can be done with two types of protocol: without any ancilla (test states) and with ancilla. Both methods produce copies with same optimal fidelity. | ||
===Phase-covariant cloning without ancilla=== | |||
In this case a unitary transformation acts on two input states (the original state and a blank state) and produces a two-qubit state where the subsystem of each of the copies can be extracted from it. The unitary transformation depends on the [[shrinking factor]] but not on the [[phase]]. This unitary in general acts asymmetrically but it becomes a symmetric case when shrinking factor is equal to <math>\pi/4</math>. | In this case a unitary transformation acts on two input states (the original state and a blank state) and produces a two-qubit state where the subsystem of each of the copies can be extracted from it. The unitary transformation depends on the [[shrinking factor]] but not on the [[phase]]. This unitary in general acts asymmetrically but it becomes a symmetric case when shrinking factor is equal to <math>\pi/4</math>. | ||
===Phase-covariant cloning with ancilla=== | ===Phase-covariant cloning with ancilla=== | ||
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# Perform a 3-qubit unitary | # Perform a 3-qubit unitary | ||
# Discard the extra state | # Discard the extra state | ||
The fidelity of this protocol is the same as the previous protocol. However, the output states (reduced [[density matrices]] of copies) are different. | |||
==Notations== | ==Notations== | ||
*<math>|\psi(\phi)\rangle = \frac{1}{\sqrt{2}} (|0\rangle + e^{i\phi}|1\rangle):</math> Input equatorial state | *<math>|\psi(\phi)\rangle = \frac{1}{\sqrt{2}} (|0\rangle + e^{i\phi}|1\rangle):</math> Input equatorial state | ||