PHY.3:5.2 - Test copying across inputs while allowing auxiliary outputs

Question. Can one device take one input qubit in an unknown pure state and produce two perfect copies on every run? The device can use an auxiliary prepared independently of the input, and its final auxiliary state may depend on that input. Discarding an auxiliary is allowed.

Use the standard quantum description of a deterministic operation. Include the device’s environment and any measurement records in the description of the complete process. A fixed mixed auxiliary preparation can be purified by adding a reference system. The total evolution can then be represented by one isometry V, which preserves inner products. This includes deterministic operations obtained by interaction and later discarding part of the system; it does not impose that the visible two-qubit map itself be unitary. If both required output copies are pure, their joint output factors from the remaining pure total state.

Choose two distinct nonorthogonal input states |a> and |b>. Let |0> be the blank second qubit and |e> the fixed initial auxiliary. Perfect copying would require:

V(|a>|0>|e>) = |a>|a>|e_a>
V(|b>|0>|e>) = |b>|b>|e_b>.

The final auxiliary states are deliberately allowed to differ. Define s=abs(<a|b>), with 0<s<1, and r=abs(<e_a|e_b>), with 0<=r<=1. Taking inner-product magnitudes before and after the same isometry gives:

s = s*s*r.
Dividing by s>0 gives 1=s*r.
But s*r <= s < 1.

The requirements are inconsistent. For the concrete pair |a>=|0> and |b>=(|0>+|1>)/sqrt(2), s=1/sqrt(2) would require r=sqrt(2). That exceeds the allowed overlap of normalized auxiliary states. More unobserved auxiliary output cannot make this deterministic perfect copier possible under the stated account.

This is a physical restriction on one operation across its input family. The mathematical step is preservation of an inner product; the physical work is establishing why the candidate devices admit that common description with the stated preparation. Showing failure of one guessed gate arrangement would leave this broader question unanswered.

Change the input family. Restrict it to the computational-basis states |0> and |1>, with a blank second qubit |0>. Controlled-NOT gives |0>|0> -> |0>|0> and |1>|0> -> |1>|1>. The same arrangement therefore copies every input in that restricted family. A different known orthogonal pair can first be mapped to that basis, copied, and mapped back on both outputs.

Change the performance instead. If failed runs may be discarded, the accepted operation is conditioned on an outcome and the all-runs argument no longer directly characterizes its normalized successful output. One must specify the allowed input set, success probability and failure output, then derive their restrictions. Allowing imperfect copies likewise changes the output relation and requires an accuracy question. The present result identifies why either revised problem differs; it supplies no unexamined claim that a desired success rate or accuracy is achievable.