Multicopy quantum state teleportation with application to storage and retrieval of quantum programs

This work considers a teleportation task for Alice and Bob in a scenario where Bob cannot perform corrections. In particular, we analyse the task of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext class="MJX-tex-mathit" mathvariant="italic">multicopy state teleportation</mml:mtext> </mml:mrow> </mml:math> , where Alice has <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> identical copies of an arbitrary unknown <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> -dimensional qudit state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> to teleport a single copy of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> to Bob using a maximally entangled two-qudit state shared between Alice and Bob without Bob's correction. Alice may perform a joint measurement on her half of the entangled state and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> copies of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> . We prove that the maximal probability of success for teleporting the exact state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> to Bob is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>&amp;#x2212;</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfrac> </mml:math> and present an explicit protocol to attain this performance. Then, by utilising <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> copies of an arbitrary target state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo fence="false" stretchy="false">|</mml:mo> <mml:mi>&amp;#x03C8;</mml:mi> <mml:mo fence="false" stretchy="false">&amp;#x27E9;</mml:mo> </mml:math> , we show how the multicopy state teleportation protocol can be employed to enhance the success probability of storage and retrieval of quantum programs, which aims to universally retrieve the action of an arbitrary quantum channel that is stored in a state. Our proofs make use of group representation theory methods, which may find applications beyond the problems addressed in this work.