Single-qubit multi-party transmission using universal symmetric quantum cloning

This study considers the hypothetical quantum network case where Alice wishes to transmit one qubit of information (specifically a pure quantum state) to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> parties, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> is some large number. The remote receivers locally perform single-qubit quantum state tomography on the transmitted qubits in order to compute the quantum state within some error rate (dependent on the tomography technique and the number of transmitted qubits). We show that with the use of an intermediate optimal symmetric universal quantum cloning machine (between Alice and the remote receivers) as a repeater-type node in a hypothetical quantum network, Alice can send significantly fewer qubits compared to direct transmission of the message qubits to each of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> remote receivers. This is possible due to two properties of quantum cloning. The first is that single qubit quantum clones retain the same Bloch angle as the initial quantum state. This means that if the mixed state of the quantum clone can be computed to high enough accuracy, the original pure quantum state can be inferred by extrapolating that vector to the surface of the Bloch sphere. The second property is that the state overlap of approximate quantum clones, with respect to the original pure quantum state, quickly converges (specifically for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mn>1</mml:mn><mml:mo>→</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:math> , the limit of the fidelity as M goes to infinity is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mfrac></mml:mrow></mml:math> ). This means that Alice can prepare a constant number of qubits (which are then passed through the quantum cloning machine) in order to achieve a desired error rate if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> is large enough. Combined, these two properties mean that for a large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> , Alice can prepare many orders of magnitude fewer qubits in order to achieve the same single qubit transmission accuracy compared to the naive direct qubit transmission approach.