The AdS/CFT correspondence, proposed by Juan Maldacena in the late 1990s, asserts that a gravitational theory in a volume of Anti-de Sitter spacetime is exactly equivalent, or dual, to a conformal field theory living on the boundary of that space. It is a kind of holographic dictionary: every black hole in the bulk corresponds to a hot quantum fluid on the boundary, and thermodynamic quantities such as mass, charge, temperature and entropy map onto energy, electric potential and the central charge of the field theory. The most famous example is the Hawking-Page transition, in which a black hole becomes thermodynamically favored over empty AdS space, interpreted holographically as a confinement-deconfinement transition in the boundary gauge theory. The new work, by Abhishek Baruah, Amijit Bhattacharjee and Prabwal Phukon, takes a deliberately restricted but sharp question: when a black hole is built from nonlinear electromagnetism and its entropy is deformed by generalized statistics, does the pattern of critical points on the gravity side still match the pattern on the field theory side?

The team examined three distinct classes of charged AdS black holes arising from nonlinear electrodynamics, a family of theories that modify Maxwell’s equations in strong electromagnetic fields. The first is the ModMax black hole, based on a theory remarkable for preserving both conformal invariance and electromagnetic duality while introducing a single dimensionless deformation parameter gamma. The second is a generic nonlinear electrodynamics model whose Lagrangian is suppressed by hyperbolic functions of the field invariant, producing magnetically charged black holes with a finite nonlinear field energy. The third, and richest, is the Euler-Heisenberg black hole, rooted in the genuine quantum correction to electrodynamics first calculated by Heisenberg and Euler in 1936, which captures vacuum polarization effects that become important near the horizon. In each case the nonlinear corrections alter the metric function of the black hole, and therefore reshape the temperature-entropy relation and the response functions that govern stability.

On top of these electromagnetic deformations, the authors layered a second axis of modification: the choice of entropy functional. Alongside the standard Bekenstein-Hawking area law, they employed two generalized entropies drawn from non-extensive statistical mechanics. The Renyi entropy, a one-parameter generalization of the von Neumann entropy familiar from quantum information theory, arises naturally in replica methods and probes the full entanglement spectrum. The Kaniadakis entropy, by contrast, emerges from a kappa-deformed relativistic statistical mechanics built on generalized logarithms and exponentials, providing a consistent non-Gibbsian extension of equilibrium thermodynamics. Both reduce to the Bekenstein-Hawking entropy in appropriate limits, but their deformation parameters can substantially alter the phase structure, shifting or even multiplying the critical points at which a black hole changes stability.

The analytical machinery that ties these threads together is geometrothermodynamics, a framework developed by Hernando Quevedo that applies differential geometry to the space of thermodynamic equilibrium states. Earlier metrics proposed by Weinhold and Ruppeiner suffered from a crucial flaw: they were not invariant under Legendre transformations, so different choices of thermodynamic potential yielded different curvatures. The geometrothermodynamic construction repairs this by building a Legendre-invariant metric on an extended phase space whose contact structure encodes the first law of thermodynamics. The payoff is a scalar curvature, R_GTD, with a clean physical interpretation: flat geometry corresponds to a non-interacting system such as an ideal gas, while curvature singularities mark phase transitions. For black holes, the promise is that the curvature diverges exactly where the heat capacity diverges and where the temperature-entropy curve reaches an extremum, providing a geometric, potential-independent diagnostic of criticality.

The results are strikingly consistent. For the ModMax AdS black hole with Bekenstein-Hawking entropy, the temperature profile shows two extrema at entropy values of 16.454 and 88.265, the specific heat diverges at precisely those points, and the GTD curvature develops singularities at exactly the same locations. Under Renyi entropy the two critical points persist at shifted positions, while Kaniadakis entropy introduces a third critical point that is absent in the other two descriptions. The same triple-diagnostic pattern, temperature extrema coinciding with heat capacity divergences and curvature singularities, repeats for the generic NED black hole and, most dramatically, for the Euler-Heisenberg black hole, which exhibits three critical points under Bekenstein-Hawking entropy, three under Renyi, and four under Kaniadakis statistics. The Euler-Heisenberg case is the richest of the three models, a fact the authors attribute to the higher-order QED vacuum polarization corrections that reorganize the competition between gravitational attraction and electromagnetic repulsion near the horizon.

The crucial test comes when the bulk black holes are mapped to their holographic duals. Using the restricted phase space thermodynamics framework, in which the cosmological constant is held fixed and the thermodynamic variables are reconstructed in terms of boundary quantities such as the central charge C and the CFT volume, the authors computed the energy, temperature and heat capacity of the dual field theories. The mapping is implemented through a conformal scaling factor relating bulk mass, temperature and charge to their boundary counterparts, with the central charge tied to the AdS radius and Newton’s constant. Remarkably, the number, ordering and structure of the critical points are preserved across the holographic dictionary. The ModMax black hole with two critical points maps to a CFT with two critical points; the Euler-Heisenberg system with three maps to a boundary theory with three; and the Kaniadakis-deformed descriptions carry their extra critical point from bulk to boundary in every case.

The authors are careful about what this does and does not prove. The numerical entropy values of the critical points do not coincide between bulk and boundary, and they are not expected to, since the generalized entropies are nonlinear functions of the underlying variables. What is preserved is the qualitative critical organization: the same number of thermodynamic branches, the same ordering of instabilities, and the same one-to-one correspondence between curvature singularities and heat capacity divergences. The extra Kaniadakis critical point, they argue, is statistical rather than gravitational in origin, arising because the kappa-deformed entropy functional modifies the derivatives that enter the temperature and heat capacity, rather than from any new bulk field or conserved charge. The analysis is therefore a nontrivial consistency check of bulk-boundary thermodynamics under simultaneous deformations of the gauge sector and the entropy prescription, not a full dynamical proof of AdS/CFT.

The study leaves open a tantalizing next step: extending the comparison from the locations of critical points to the near-critical scaling behavior itself, including critical exponents and universality classes. That undertaking is considerably harder within restricted phase space thermodynamics, which lacks the Van der Waals-style equation of state that makes exponent extraction straightforward in the extended phase space framework, and it would need to be performed consistently on both sides of the duality. For now, the message of this work is quietly profound. Even when the electromagnetic self-interaction is deformed by quantum vacuum polarization and the entropy law is rewritten by non-extensive statistics, the holographic mirror does not crack. The phase structure of a black hole, encoded in the curvature of an abstract thermodynamic manifold, is faithfully reflected in the thermodynamics of a quantum field theory it never touches, a consistency that continues to justify the strange and beautiful conviction that spacetime itself may be a hologram.

Subject of Research: Thermodynamic geometry and holographic consistency checks of nonlinear electrodynamics AdS black holes with generalized entropies

Article Title: Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies

Article References: Baruah, A., Bhattacharjee, A., & Phukon, P. (2026). Testing the AdS/CFT correspondence through thermodynamic geometry of nonlinear electrodynamics AdS black holes with generalized entropies. The European Physical Journal C, 86(10), Article 1144. https://doi.org/10.1140/epjc/s10052-026-16330-7

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16330-7

Keywords: AdS/CFT correspondence, black hole thermodynamics, geometrothermodynamics, nonlinear electrodynamics, ModMax, Euler-Heisenberg, Renyi entropy, Kaniadakis entropy, phase transitions, holography, conformal field theory, critical points

To appreciate why this matters, it helps to understand what these quantities measure. Divergences are the workhorses of information theory: they quantify how distinguishable two probability distributions, or in the quantum setting two quantum states, are from one another. The famous quantum relative entropy, the Rényi divergences used in cryptography and quantum channel discrimination, and the fidelity-like measures that certify quantum memories are all members of the f-divergence family. The maximal quantum f-divergence, in particular, is defined through an optimization over measurements and captures the ultimate statistical distinguishability of two quantum states. Computing it exactly is generally intractable on classical machines because it involves functions of matrices whose dimensions grow exponentially with the number of qubits. The Kubo–Ando means, introduced by Fumio Kubo and Tsuyoshi Ando in 1980, form the complete family of two-variable matrix means that respect the right symmetries and monotonicity properties, generalizing the arithmetic, geometric, and harmonic means from numbers to matrices.

The technical engine behind the new algorithms is the block-encoding framework, which has become one of the most powerful unifying tools in quantum computing. A block encoding embeds an arbitrary matrix A into a larger unitary operator, the only kind of operation a quantum computer can natively perform. Formally, a unitary U acting on the system qubits plus a set of ancilla qubits is called an (α, a, ε)-block encoding of A if the top-left block of U, after subtracting out the ancilla register, equals A scaled by a normalization factor α, up to error ε. Once a matrix is encoded this way, an entire toolbox of quantum algorithms can act on it with provable complexity guarantees. Matrix inversion, Hamiltonian simulation, eigenvalue estimation, and principal component analysis can all be expressed in this language, which is precisely why the authors describe block encoding as a foundational bridge that transforms mathematical objects into quantum operations.

Sitting on top of block encoding is the quantum singular value transformation, or QSVT, developed by András Gilyén and collaborators in 2019. QSVT takes a block-encoded matrix and applies a polynomial transformation to its singular values: given a polynomial p of degree d bounded by one on the interval from minus one to one, a quantum circuit of length proportional to d, built from alternating applications of the block-encoding unitary and carefully chosen single-qubit phase rotations, implements a block encoding of p applied to the matrix. The consequence is striking. Smooth analytic functions such as the exponential can be approximated with circuit depth growing only logarithmically in the inverse precision, while even discontinuous functions like the sign function require depth scaling as one over the precision. For the quantities in this paper, the relevant functions are the operator monotone functions, a special class that includes the square root, the logarithm, and the power functions x raised to any exponent between zero and one.

Here the authors deploy a piece of classical mathematics that dates back to the early twentieth century. Operator monotone functions on the positive real line admit a representation, due to Löwner and Pick and Nevanlinna, as an integral of simple rational functions of the form x divided by x plus λ, weighted by a positive measure. This structure means that such functions extend analytically to Stieltjes transforms, and a classical theorem on the convergence of Padé approximants guarantees that rational functions of this form converge uniformly to the target function on any compact interval away from the singularities. The upshot, formalized in the paper as a positive rational approximation lemma, is that any operator monotone function can be approximated on a relevant interval by a sum of positive rational terms with an error that shrinks exponentially in the number of terms. Each rational term can itself be implemented efficiently, because it requires only the matrix A and the resolvent-like operation of adding a scaled identity, both of which are straightforward within the block-encoding paradigm.

With these ingredients assembled, the construction of the algorithms becomes transparent, at least in outline. For the Kubo–Ando means, the key observation is that every such mean of two positive definite matrices A and B can be written using an operator monotone function applied to a ratio-like combination of the two matrices. The authors first build block encodings of the individual matrices, then use composition lemmas to form products and linear combinations of block-encoded operators, and finally apply the QSVT machinery with the rational approximation of the relevant operator monotone function. Along the way, they rely on established results for handling matrix powers: negative powers, which amount to matrix inversion, cost complexity proportional to the condition number times the block-encoding cost times the logarithm of the inverse precision, while fractional positive powers between zero and one carry a similar logarithmic dependence on precision. The end result is a quantum circuit that block-encodes the desired Kubo–Ando mean with complexity scaling polynomially in the condition number and logarithmically in the target accuracy.

The computation of maximal quantum f-divergences follows a parallel logic. Because these divergences are built from operator monotone functions of the ratio between two density matrices, the same rational-approximation-plus-QSVT pipeline applies. The generality of the framework is one of its most attractive features: since the Rényi entropies and the standard matrix means appear as special cases, the paper’s results immediately subsume and generalize a string of earlier algorithms, including quantum algorithms for estimating Rényi entropies of quantum states, algorithms for estimating quantum entropies more broadly, and a 2025 npj Quantum Information paper on quantum algorithms for matrix geometric means. Rather than treating each of these quantities with a dedicated procedure, the new work folds them into a single modular architecture, in the spirit of how block encoding itself unified quantum linear algebra.

The complexity analysis reveals both the power and the honest limitations of the approach. The algorithms inherit the standard caveats of block-encoding-based methods: they require coherent access to the input matrices through efficient block-encoding oracles, and their running times depend on normalization factors and condition numbers that can in the worst case be large. The dependence on precision, however, is only logarithmic for the smooth functions at the heart of the construction, which is exponentially better than naive approaches based on Taylor series or direct sampling. This logarithmic dependence traces directly back to the exponentially convergent Padé and Gauss-type quadrature approximations of Stieltjes functions, a beautiful example of classical approximation theory doing heavy lifting inside a quantum circuit. For practitioners, the message is that whenever the relevant matrices admit efficient block encodings, which is the case for many structured families arising in quantum information and physics, these divergences and means can be estimated to high accuracy with resources that scale gently in the desired precision.

Beyond the immediate results, the paper signals a broader trend in which quantum algorithms are being extended from computational tasks like solving linear systems to the evaluation of the mathematical quantities that populate theoretical physics and information theory. Recent preprints by the same authors on estimating nonlinear physical quantities by measuring ancillas and on refined algorithms for principal component analysis suggest a sustained research program along these lines. As quantum hardware matures, tools of this kind could become standard instruments for characterizing entanglement, benchmarking quantum channels, and computing geometric distances between quantum states, including the Bures–Wasserstein distance that has attracted attention in both quantum information and machine learning. By showing that entire families of divergences and means, not just isolated examples, can be handled within one framework, Dinh and Nghiem have provided a template that future algorithm designers can instantiate again and again, turning some of the most elegant structures of matrix analysis into runnable quantum circuits.

Subject of Research: Quantum algorithms for computing maximal quantum f-divergences and Kubo–Ando operator means

Article Title: Quantum algorithms for computing maximal quantum f-divergence and Kubo–Ando means

Article References: Dinh, T. H., & Nghiem, N. A. (2026). Quantum algorithms for computing maximal quantum f-divergence and Kubo–Ando means. Quantum Information Processing, 25(10), Article 322. https://doi.org/10.1007/s11128-026-05309-8

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05309-8

Keywords: quantum algorithms, quantum information, f-divergence, Kubo–Ando means, block encodings, quantum singular value transformation, Rényi entropy, matrix means, operator monotone functions, quantum computing, entropy estimation, Padé approximation