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arXiv:2610.00187

Entanglement cost of quantum depolarization

Kun Fang

quant-phcs.ITSubmitted Sep 17, 2026• Announced Sep 18, 2026• 0 citations
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Abstract

Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing $2\to 3$ norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most $2\%$ of $\log d$ that vanishes as $d$ grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.

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Entanglement cost is the asymptotic rate of Bell pairs required to prepare a quantum state. Its regularized definition has made exact evaluation difficult, even for isotropic states (i.e., depolarized maximally entangled states). In this work, we determine the entanglement cost of every qubit isotropic state and, more generally, every two-qubit Bell-diagonal state. Our proof uses a family of suitably tuned qubit semigroups to transform a general log-Sobolev entropy bound into supporting lines of Wootters' function. A recent exact tensorization theorem of Dong et al. [arXiv:2606.17729] extends these bounds to arbitrarily many copies, yielding a general lower bound on entanglement cost. Matching this bound with the entanglement of formation also determines the exact cost for a broader class of two-qubit states. For qudit isotropic states, we derive another general lower bound from the full depolarizing 2→32\to 32→3 norm. This bound substantially improves on the PPT-relative entropy of entanglement and nearly matches the entanglement of formation, leaving a gap of at most 2%2\%2% of log⁡d\log dlogd that vanishes as ddd grows. Finally, the exact qubit results and qudit bounds extend to the entanglement cost of preparing quantum depolarizing channels under both parallel and sequential strategies.
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Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability ppp, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2)h_2((1+\sqrt p)/2)h2​((1+p​)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1p<1p<1, and construct a binary Kraus representation attaining the uniform formation bound.
Quantifying the minimum entanglement needed to prepare quantum states and implement quantum processes is a key challenge in quantum information theory. In this Letter, we develop computable and faithful lower bounds on the entanglement cost under quantum operations that completely preserve the positivity of partial transpose (PPT operations), by introducing the generalized divergence of k-negativity, a generalization of logarithmic negativity. Our bounds are efficiently computable via semidefinite programming and provide nontrivial values for all states that are non-PPT (NPT), establishing their faithfulness for the resource theory of NPT entanglement. Notably, we find and affirm the irreversibility of asymptotic entanglement manipulation under PPT operations for full-rank entangled states. Furthermore, we extend our methodology to derive lower bounds on the entanglement cost of both point-to-point and bipartite quantum channels. Our bound demonstrates improvements over previously known computable bounds for a wide range of quantum states and channels. These findings push the boundaries of understanding the structure of entanglement and the fundamental limits of entanglement manipulation.
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We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein--Podolsky--Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.