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A cooling transition with no ground-space thermal realization

Abstract

A four-level cooling-map transition cannot be realized by any finite ancilla in its normalized ground-space state and any energy-conserving unitary.

Definition 1.1 (Cooling Kraus operators).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.IsCoolingKraus (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

In the increasing energy basis, an allowed Kraus operator is diagonal or a complex multiple of the matrix unit with row j and column k, where j < k. The latter sends the kth energy coordinate to the lower jth coordinate. Zero operators are allowed. This expresses the two classes in LTT_arXiv.tex, arXiv:1409.7740v3, lines 693–700.

Definition 1.2 (Transitions through cooling maps).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.IsCoolingMap (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

There is a finite Kraus family K, each member of which has one of the two allowed forms, whose adjoint products sum to the identity and whose channel sends ρ to σ. This is the state-transition reading of the cooling-map definition in LTT_arXiv.tex, lines 693–700. The completeness identity makes the Kraus channel trace preserving; its Kraus form makes it completely positive. D5/S3/Quantum/Foundation/FiniteKrausChannel.of_kraus applies the two families by summing K i times ρ times the adjoint of K i. The two family arguments are both K.

Definition 1.3 (All ground-energy coordinates).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.groundSet (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

For the diagonal ancilla Hamiltonian with real energies e, take every coordinate a whose energy is no greater than the energy of any coordinate b. All degeneracies are retained. This describes the ground eigenspace used in LTT_arXiv.tex, lines 137–141. If m is positive this finite set is nonempty; for m = 0 it is empty.

Definition 1.4 (The normalized ground-space projector).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.groundState (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

The diagonal entry at a ground coordinate is the inverse of the number of ground coordinates, explicitly cast from a natural number to a complex number; every other entry is zero. For positive m this is the normalized projector onto the entire ground eigenspace. It encodes the idealized state in LTT_arXiv.tex, lines 137–141; line 648 fixes the paper’s subsequent low-temperature convention. The definition also permits m = 0, when the matrix has no entries.

Definition 1.5 (Commutation with total energy).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.IsEnergyConserving (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

The total Hamiltonian is diagonal on pairs x, with energy E x.1 + e x.2. Each real sum is included in the complex scalars. Energy conservation means that U commutes with this matrix, as in LTT_arXiv.tex, lines 121–134. The predicate accepts an arbitrary complex matrix; unitarity is required separately in the transition assertion.

Definition 1.6 (The ground-space thermal output).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.thermalOutput (✓ std3).

Citation. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

Tensor ρ with the ancilla’s normalized ground-space projector, conjugate by U, and trace out the ancilla. This combines LTT_arXiv.tex, lines 126–128 and 137–141, with the idealized ground-space convention of line 648. D5/S3/Quantum/Information/PartialTraceMutualInformation.partialTraceRight retains the system coordinate and sums the diagonal ancilla coordinates. The definition itself imposes neither unitarity nor energy conservation.

Definition 1.7 (The state-transition direction of Conjecture 1).

Formalization. D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.claim (✓ std3).

Source. Repository-derived.

Acknowledgement. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

Conjecture 1 states: “Cooling maps are equivalent to low-temperature thermal operations, with regard to the feasibility of state transitions.” LTT_arXiv.tex, lines 924–927, explicitly distinguishes this assertion from equality of the two sets of channels. The formula expresses its cooling-to-thermal direction for every finite system with strictly increasing energies and distinct positive gaps, the assumptions in lines 109–111. Both input and output satisfy D5/S3/Quantum/Information/SLDSpectralEnergy.IsDensity, meaning positive semidefiniteness and trace one. The ancilla has positive finite dimension, arbitrary real diagonal energies and its full normalized ground-space state. The unitary is explicitly coerced to a matrix in both occurrences. A change of ancilla energy basis can be absorbed into the unitary. The assertion uses the exact idealized state; it makes no assertion about positive-temperature or approximate transitions.

Theorem 1.8 (A four-level counterexample).

Proof. Machine-checked in Lean as D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.result (✓ std3). ∎

Resolves. Problems/narasimhachar-gour-2015-cooling-map-thermal-operation-refutation (refuted) by D5/S3/QuantumChannels/CoolingMapThermalOperationRefutation.result.

Source. Repository-derived.

Acknowledgement. Varun Narasimhachar, Gilad Gour (2015). Low-temperature thermodynamics with quantum coherence. DOI: 10.1038/ncomms8689. URL: https://arxiv.org/abs/1409.7740v3.

Commentary.

Take d = 4 and E = (0, 1, 3, 7), with coordinates numbered 0 through 3. The six positive gaps are 1, 2, 3, 4, 6 and 7, all distinct. Let V have rows (1, 0), (0, 1), (3/5, 4/5) and (3/5, 4i/5), where i is the imaginary unit. Put σ = VV†/4 and let K 0 and K 1 be the diagonal matrices formed from the two columns of V. Every row of V has squared norm one, so (K 0)†K 0 + (K 1)†K 1 = I and every diagonal entry of σ is 1/4.

Use the matrix D5/S3/Quantum/Information/NiceErrorBasisNonNormalStabilizer.Pm as ρ: every entry is 1/4. It is vv† for v = (1/2, 1/2, 1/2, 1/2), hence positive semidefinite with trace one. The same properties hold for σ, since VV† is positive semidefinite and σ has four diagonal entries 1/4. The two diagonal Kraus operators send ρ to σ by direct multiplication, proving that the transition is a cooling transition.

For any positive finite ancilla dimension m and energy function e, let G be the set of its minimum-energy coordinates and g its cardinality. A finite minimum exists, so g > 0. If U commutes with the total Hamiltonian, its entry U((j,b),(k,a)) vanishes whenever E j + e b differs from E k + e a. For a in G and k < j, minimality of e a and strict increase of E force this inequality of total energies. Thus a ground-space input cannot raise the system energy.

Expanding the partial trace of U(ρ ⊗ groundState e)U† gives its (i,j) entry as 1/(4g) times the sum of U((i,b),(k,a)) conjugate(U((j,b),(l,a))) over b, l, a in G and k. For a diagonal output entry, energy conservation and injectivity of E eliminate the terms with k ≠ l. The real population at j is therefore 1/(4g) times the sum of |U((j,b),(k,a))|² over k, a in G and b.

Define the real matrix T by T(j,k) = ∑ over a in G and b of |U((j,b),(k,a))|². All entries are nonnegative. If the output is σ, its uniform populations give row sums g. Unitarity gives column sums g, while the downward-energy restriction makes T(j,k) zero for k < j. Comparing row 0 with column 0 forces T(0,1), T(0,2) and T(0,3) to vanish. Comparing row 1 with column 1 then forces T(1,2) and T(1,3) to vanish, and the corresponding comparison at 2 forces T(2,3) to vanish. Thus every off-diagonal entry of T is zero.

A zero sum of these nonnegative squared moduli forces every summand to vanish. Hence for a in G, U((j,b),(k,a)) is zero when j ≠ k. When j = k but b is outside G, equality of total energies would imply e b = e a, placing b in G; energy conservation therefore forces these entries to vanish as well. For each k, restrict U to the coordinates (k,a) with a in G and call the resulting square matrix A k. The support just proved reduces U†U = I to (A k)†A k = I on G, so each A k is unitary.

Return to the full output-entry expansion. The same support restrictions leave only system indices k = i and l = j, and ancilla indices in G. The remaining sum is trace((A i)(A j)†), so equality of the output with σ gives trace((A i)(A j)†) = g(VV†)(i,j) for all i and j. Expand the squared Hilbert–Schmidt norms of A 2 − (3/5)A 0 − (4/5)A 1 and A 3 − (3/5)A 0 − (4i/5)A 1. Substitution of these trace identities and the displayed rows of V makes both squared norms zero. Consequently A 2 = (3/5)A 0 + (4/5)A 1 and A 3 = (3/5)A 0 + (4i/5)A 1.

Set X = (A 0)(A 1)†. Expand the unitary identities (A 2)(A 2)† = I and (A 3)(A 3)† = I using those two linear relations and the unitarity of A 0 and A 1. The first gives X + X† = 0; the second gives X − X† = 0. Thus X = 0. But X is a product of unitaries on the nonempty space indexed by G, hence invertible and nonzero, a contradiction.

Finally, applying the asserted transition principle to E, ρ and σ would provide precisely such a positive finite ancilla, energy-conserving unitary and thermal output. The contradiction refutes the stated direction of Conjecture 1. It concerns exact finite ground-space thermal operations and does not assert a result for their approximate or positive-temperature variants.

References