A fixed-size calculation for the exact PPT entanglement cost of two qutrits

What kind of entanglement cost?

Entanglement cost measures the entanglement resource consumed per copy when preparing many copies of a shared quantum state. The relevant rate is defined in the limit of increasingly large batches. For mixed states, which describe statistical mixtures rather than a single pure quantum state, that rate need not follow directly from the cost of preparing one copy.

Wei Song, Xiao-Lan Zong and Ming Yang study exact preparation: at every sufficiently large batch size, the prepared output must equal the desired state exactly. This differs from allowing a preparation error that merely approaches zero as the batch grows.

Their allowed operations preserve positive partial transpose, abbreviated PPT. Partial transposition transposes the matrix indices belonging to one of the two subsystems. A state is PPT if its matrix remains positive semidefinite after this operation, meaning it has no negative eigenvalues. The paper uses a precise channel condition that preserves this property even when additional systems are included.

This is not the same operational setting as restricting the parties to local operations and classical communication. The result concerns PPT operations specifically; it does not determine the entanglement cost under that more restrictive setting.

When successive bounds stop changing

Previous work expressed exact PPT entanglement cost through sequences of lower and upper bounds calculated by semidefinite programming. An SDP optimizes a linear objective subject to matrix positivity constraints. Increasing the hierarchy level generally introduces more matrix variables and tighter bounds.

The preprint proves that, for every state of two qutrits, the lower-bound hierarchy stops changing at its second level. The second and third levels agree, and that equality forces every subsequent level to agree too. The shared value determines the cost.

Concretely, the calculation uses three Hermitian matrices, each measuring nine by nine. Hermitian matrices equal their conjugate transpose. Starting with the target state's partial transpose, successive constraints require each optimization matrix to bound both the positive and negative versions of the preceding partially transposed matrix. The objective is to minimize the trace—the sum of diagonal entries—of the final matrix. Taking the base-two logarithm of that optimum gives the entanglement cost.

The mathematical formula is exact, but a numerical implementation still has optimization error. What disappears is uncertainty caused by stopping the hierarchy too early.

How the proof covers every state

The central construction shortens a feasible third-level solution of the dual optimization problem to a second-level solution while preserving the operator that determines its objective value. A dual problem is a companion optimization that supplies bounds on the original problem. Preserving this operator makes the construction work for any input state, rather than selected examples.

Spectral decomposition reduces the difficult part to a vector with three Schmidt coefficients. These nonnegative coefficients describe how a bipartite pure vector decomposes into paired local basis states.

The authors split the resulting parameter domain into two regions. Away from the boundary where the smallest coefficient vanishes, they use 126 groups of finite certificates. These certify matrix positivity across whole parameter regions, not merely sampled points.

According to the paper, the certificates were verified using exact rational arithmetic. Their verification explicitly bounds residual errors in coefficient identities and checks that the regions cover the required domain. It does not rely on floating-point eigenvalue calculations or SDP solver output as proof.

Close to the boundary, the authors instead derive analytic matrix estimates. This argument meets the certificate region at a smallest coefficient of 1/640. Together with the simpler cases, the two approaches cover all two-qutrit states, including complex matrices and states without full rank.

Scope and remaining questions

The paper illustrates the formula with two limits: PPT states have zero cost under the allowed operations, while a maximally entangled two-qutrit state costs log₂3 units of maximally entangled qubit pairs per copy.

The dimension restriction matters. The source discusses higher-dimensional examples whose hierarchy levels keep differing at arbitrarily high stages as dimension increases. The new proof therefore does not establish a universal stopping level across all dimensions, nor does it extend to every system pairing a qutrit with a larger subsystem.

This is a theoretical preprint, not an experimental result or an independently verified computational replication. It establishes second-level sufficiency but leaves open whether the first level already suffices for every two-qutrit state.