Amplifying hybrid entangled states and superpositions of coherent states

We compare two amplification schemes, photon addition and then subtraction (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mover accent="true"><a:mi>a</a:mi><a:mo>̂</a:mo></a:mover><a:msup><a:mover accent="true"><a:mi>a</a:mi><a:mo>̂</a:mo></a:mover><a:mo>†</a:mo></a:msup></a:mrow></a:math>) and successive photon addition (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msup><d:mover accent="true"><d:mi>a</d:mi><d:mo>̂</d:mo></d:mover><d:mo>†</d:mo></d:msup><d:msup><d:mrow/><d:mn>2</d:mn></d:msup></d:mrow></d:math>), applied to hybrid entangled states (HESs) and superpositions of coherent states (SCSs). We show that the fidelity and gain of the amplification schemes for HESs are the same as those of coherent states. However, SCSs show quite nontrivial behavior by the amplification schemes, depending on the amplitudes of coherent states, the number of coherent-state components, and the relative phases between the components. This implies that appropriate amplification schemes for SCSs should be chosen depending on the tasks and specific forms of the states. To investigate the quality of amplified states, we calculate the quantum Fisher information, a measure of quantum phase estimation. In terms of the quantum Fisher information, the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mover accent="true"><f:mi>a</f:mi><f:mo>̂</f:mo></f:mover><f:msup><f:mover accent="true"><f:mi>a</f:mi><f:mo>̂</f:mo></f:mover><f:mo>†</f:mo></f:msup></f:mrow></f:math> scheme tends to show better performance for relatively small amplitudes (<i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>α</i:mi><i:mo>≲</i:mo><i:mn>0.9</i:mn></i:mrow></i:math>), while the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:msup><j:mover accent="true"><j:mi>a</j:mi><j:mo>̂</j:mo></j:mover><j:mo>†</j:mo></j:msup><j:msup><j:mrow/><j:mn>2</j:mn></j:msup></j:mrow></j:math> scheme is better in the larger amplitude regime. The performance of the two schemes becomes indistinguishable as the amplitude grows sufficiently large.