The one-way deficit — the entropy cost of an optimal local measurement on one part of a bipartite state, originating from the thermodynamic work-deficit program of Oppenheim et al. — and its susceptibility with respect to a control parameter were shown by Wang, Zhang and Fan (Quantum Information Processing 18, article 19, 2019) to detect quantum phase transitions of the thermodynamic-limit XY chain and topological (winding-number) transitions of the extended Ising model from the reduced state of two adjacent spins, with detection of topological transition points reported accurate only at low temperature. The paper's conclusion points to extending this diagnostic, via numerical techniques such as DMRG, matrix product states, and exact diagonalization, to finite-temperature and dynamical phase transitions of more quantum many-body models; the finite-temperature scaling of the deficit susceptibility and the quantitative temperature window for accurate topological detection are concrete refinements of that direction, while the relation to other quantum-resource measures and subsystems larger than two adjacent spins is outside the scope of this task. These directions remain open: subsequent work on the same models has turned to other correlation measures — quantum Fisher information, coherence, steered coherence, teleportation fidelity, multipartite nonlocality — while one-way-deficit studies have remained in two-qubit or free-fermion-solvable settings.

Background

Quantum phase transitions (QPTs) are nonanalytic changes of a many-body system's ground state as a Hamiltonian control parameter is varied, conventionally located through an order parameter. Since the early 2000s, information-theoretic quantities have served as order-parameter-free probes: the two-site entanglement of the transverse-field XY spin chain peaks or diverges at the critical fields, and the quantum discord — the part of the mutual information of a bipartite state that is destroyed by a local measurement — signals QPTs in spin chains and survives at finite temperature, unlike entanglement (Dillenschneider 2008; Sarandy 2009).

A closely related, thermodynamically motivated measure is the quantum work deficit introduced by Oppenheim, Horodecki, Horodecki and Horodecki (2002): the loss of extractable work when a global operation on a bipartite state coupled to a heat bath is replaced by local operations assisted by one-way classical communication. Its one-way version, the one-way deficit (also called one-way information deficit; Streltsov, Kampermann and Bruß 2011), for a bipartite state ρAB\rho_{AB} is

Δ→(ρAB)=min⁡{Πk}S(∑kΠkρABΠk)−S(ρAB),\Delta^{\to}(\rho_{AB}) = \min_{\{\Pi_k\}} S\Big(\sum_k \Pi_k \rho_{AB} \Pi_k\Big) - S(\rho_{AB}),

where SS is the von Neumann entropy and the minimum runs over local measurements {Πk}\{\Pi_k\} on one subsystem. For the two-qubit X states (density matrices whose nonzero blocks lie on the diagonal and antidiagonal) that arise as reduced states of spin chains, this minimization has closed forms, and the one-way deficit is never smaller than the quantum discord.

Wang, Zhang and Fan (2019), following their XX-model study (2018), evaluated the one-way deficit Δ\Delta of two adjacent bulk spins in the thermodynamic limit for the XY chain and for the extended Ising model of Zhang and Song (2015; see also Zhang, Li and Song 2017) — a transverse-field Ising chain with competing three-site interactions whose phase diagram contains several symmetry-protected topological phases labeled by the winding number of a characteristic complex function; a topological transition occurs when zeros of that function cross the unit circle in the complex plane. Working from exact Jordan-Wigner free-fermion correlators (a standard mapping of such spin chains to noninteracting fermions), they found that the susceptibility dΔ/dλd\Delta/d\lambda — the derivative of the deficit with respect to the driving parameter λ\lambda — extremizes at the Ising and anisotropy critical points of the XY model and at the winding-number-changing transitions of the extended Ising model, giving a locally accessible diagnostic of both conventional and topological transitions. At nonzero temperature TT (in units of the exchange coupling) the susceptibility extrema shift, and detection of the topological transition points was reported to be accurate only at low temperature, T≪1T \ll 1. Their conclusion explicitly calls for extending the approach with DMRG (density matrix renormalization group, a variational ground-state method based on matrix product states), matrix product states (MPS), and exact diagonalization to finite-temperature and dynamical phase transitions in more complex many-body models. Dynamical quantum phase transitions (DQPTs; Heyl, Polkovnikov and Kehrein 2013) are the real-time analogues of QPTs: after a quench across a critical point, the Loschmidt echo — the squared modulus of the overlap between the initial state and its time-evolved state — develops nonanalyticities at critical times in the thermodynamic limit.

Problem Statement

Establish whether the one-way-deficit susceptibility diagnostic — demonstrated for ground-state quantum and topological phase transitions of the free-fermion-solvable XY and extended Ising chains — extends to finite-temperature and dynamical phase transitions in quantum many-body systems beyond those exactly solvable chains, where free-fermion techniques fail and numerical many-body methods are needed (the source conclusion calls for DMRG, matrix-product-state, and exact-diagonalization techniques). The components of this question jointly support one overarching objective: determining the scope and robustness of a thermodynamically motivated, locally accessible correlation probe beyond ground-state, solvable problems. Each of the four components below is mandatory, and the task is resolved by addressing them jointly; a result on any one component is a partial contribution to the overarching objective. Concretely, refining the direction set out in the source paper's conclusion: (i) characterize the scaling behavior of the one-way-deficit susceptibility near critical points at finite temperature TT, including how its extrema shift, broaden, and round as TT increases; (ii) determine quantitatively the temperature window within which topological transition points are detected accurately, sharpening the qualitative T≪1T \ll 1 statement of the source; the accuracy criterion (for example, a stated tolerance on the deviation of the susceptibility extremum from the exact transition point) must be defined explicitly and the window reported with respect to it; (iii) establish whether the one-way deficit signals dynamical phase transitions in time-dependent many-body systems — any standard notion of dynamical transition qualifies, provided the notion used is stated explicitly; the Loschmidt-echo nonanalyticities (dynamical quantum phase transitions) arising after a quench across a critical point are a prominent candidate; and (iv) demonstrate the diagnostic in more complex models — interacting, frustrated, higher-dimensional, or long-ranged — where free-fermion techniques fail. The class of target systems for component (iv) is left open-ended rather than fixed to a single model, so a demonstration on any one concrete model outside the free-fermion-solvable class suffices to complete that component.

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