PHY.3:5 - Archetypal Grounding

PHY.3:5.1 - Bound a proposed engine without designing its mechanism

Question. A proposed cyclic machine takes 100 J from an ideal reservoir at 600 K and delivers 60 J as work. Its only other interaction is heat rejection to an ideal reservoir at 300 K. The machine returns to its initial state. Can a different mechanism make the proposal possible under these same conditions?

The physical preparation includes both reservoirs, the work store and restoration of the machine. It supplies no fuel, depleted auxiliary store or initially available nonthermal resource. The two temperatures are absolute thermodynamic temperatures. Energy balance gives 40 J rejected to the cold reservoir; that balance alone leaves the proposed output admissible.

Use the classical second-law account for these conditions. Its Clausius restriction excludes a composite process whose only net effect is transfer of heat from the colder reservoir to the hotter one. An admitted reversible reference between these reservoirs has eta_ref = 1 - 300/600 = 1/2 and can run backward. This is a theoretical reference; no claim of a finite-power reversible laboratory device is needed.

Choose cancellation of the work exchange. A backward reference consuming the proposed 60 J extracts 60 J from the cold reservoir and delivers 120 J to the hot reservoir. The compatible exchanges combine as follows; positive entries increase the named store.

The net process transfers 20 J from cold to hot with no other change. It violates the selected restriction. The argument uses only the proposed exchanges and cyclic condition, so changing the candidate’s internal mechanism cannot repair it within this class.

For a general positive requested work W, the backward reference requires Q_hot_ref=W/eta_ref. If W > eta_ref*Q_hot, then Q_hot_ref > Q_hot; canceling work again produces the prohibited cold-to-hot transfer. Therefore:

W <= eta_ref*Q_hot = (1 - T_cold/T_hot)*Q_hot.
For the stated inputs: W <= 50 J.

The practical result redirects the design question to work below this bound or to a change in the stated resources. It does not supply the design that attains a chosen value, its power or its operating cost.

Change the allowed physical operation. Suppose the proposed device may also receive 20 J of external work. A forward reversible reference producing 40 J from 80 J of hot-reservoir heat, together with routing those 20 J through the work store, can supply 60 J gross output. The net work produced is 40 J. The denominator and net exchange must now reflect the actual question; this construction supplies no engine producing 60 J net work from the original 100 J alone. If exactly 100 J must still be taken from the hot reservoir, an admitted additional transfer of 20 J from hot to cold accounts for the remainder. The total cold-reservoir gain is 60 J, and both laws permit the resulting exchanges. The changed resource condition opens a construction rather than altering the earlier bound.

For a microscopic proposal with initial correlations or a nonthermal auxiliary, use the physical account appropriate to those resources. A reservoir-only calculation leaves their contribution out. The source comparison in :11 identifies a current treatment; it does not prescribe the same generalized formula for every macroscopic device.

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.