2019/12/31 by Raffaele Salvia, Vittorio Giovannetti · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Generalization #Gibbs state #Ground state #Hamiltonian (control theory) #Limit (mathematics) #Passivity #Quantum Information and Cryptography #Quantum Mechanics and Non-Hermitian Physics #Quantum many-body systems #Quantum system #Upper and lower bounds #cond-mat.stat-mech #math-ph #math.MP #quant-ph
paper · pdf · doi:10.22331/q-2020-05-28-274
published as Quantum 4, 274 (2020) · 36 pages, 2 figures. Accepted for publication in Quantum
openalex created_date 2019/12/13 · arxiv created 2020/05/27 · openalex publication_date 2020/05/28 · arxiv updated 2020/06/02 · openalex updated_date 2026/08/05
Passive states are special configurations of a quantum system which exhibit no energy decrement at the end of an arbitrary cyclic driving of the model Hamiltonian. When applied to an increasing number of copies of the initial density matrix, the requirement of passivity induces a hierarchical ordering which, in the asymptotic limit of infinitely many elements, pinpoints ground states and thermal Gibbs states. In particular, for large values of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>the energy content of a<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-passive state which is also structurally stable (i.e. capable to maintain its passivity status under small perturbations of the model Hamiltonian), is expected to be close to the corresponding value of the thermal Gibbs state which has the same entropy. In the present paper we provide a quantitative assessment of this fact, by producing an upper bound for the energy of an arbitrary<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-passive, structurally stable state which only depends on the spectral properties of the Hamiltonian of the system. We also show the condition under which our inequality can be saturated. A generalization of the bound is finally presented that, for sufficiently large<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>, applies to states which are<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math>-passive, but not necessarily structurally stable.