Stabilizer Rényi entropies compress the distribution of squared Pauli expectation values of a pure quantum state into scalar measures of nonstabilizerness. Here we show that, before this compression, the distribution has a remarkably simple information-geometric structure. For any smooth pure-state family of $N$ qubits, its classical Fisher information matrix is exactly twice the quantum Fisher information matrix of the state at every regular point. This information can be accessed directly: by preparing the state together with its complex conjugate and performing pairwise Bell measurements between corresponding qubits, one samples the same Pauli distribution and saturates the quantum Fisher information matrix of the conjugate pair. The readout is fixed and does not depend on the number of estimated parameters or on the particular pure-state sensing model. We show that the origin of this result is more general than Pauli algebra: conjugation identifies the doubled pure state with the vectorized density operator, whose coordinates are real in any Hermitian Hilbert--Schmidt operator basis. The situation changes for mixed states, for which the Bell probabilities no longer coincide with normalized squared Pauli expectation values and the same fixed readout is generally not optimal. We derive an exact positive-semidefinite expression for the missing Fisher information, identifying the components of the state variation that are invisible to Bell sampling. Our results reveal a direct connection between stabilizer Rényi statistics, quantum-state geometry, and fixed multiparameter readout.
Stabilizer Rényi entropies compress the distribution of squared Pauli expectation values of a pure quantum state into scalar measures of nonstabilizerness. Here we show that, before this compression, the distribution has a remarkably simple information-geometric structure. For any smooth pure-state family of N qubits, its classical Fisher information matrix is exactly twice the quantum Fisher information matrix of the state at every regular point. This information can be accessed directly: by preparing the state together with its complex conjugate and performing pairwise Bell measurements between corresponding qubits, one samples the same Pauli distribution and saturates the quantum Fisher information matrix of the conjugate pair. The readout is fixed and does not depend on the number of estimated parameters or on the particular pure-state sensing model. We show that the origin of this result is more general than Pauli algebra: conjugation identifies the doubled pure state with the vectorized density operator, whose coordinates are real in any Hermitian Hilbert--Schmidt operator basis. The situation changes for mixed states, for which the Bell probabilities no longer coincide with normalized squared Pauli expectation values and the same fixed readout is generally not optimal. We derive an exact positive-semidefinite expression for the missing Fisher information, identifying the components of the state variation that are invisible to Bell sampling. Our results reveal a direct connection between stabilizer Rényi statistics, quantum-state geometry, and fixed multiparameter readout.
Fisher information appears in quantum theory in two mathematically distinct settings: as a classical probability functional in variational reconstructions of Schrodinger dynamics and as the quantum Fisher metric of parameterized density operators. Here we examine how far these roles can be connected without identifying the two objects. Starting from a Hamilton-Jacobi ensemble action, we recover the Fisher term that generates the quantum potential and isolate its contribution to the kinetic-energy expectation. We then analyze the quantum Fisher information matrix (QFIM) of bipartite states. For any pure two-qubit state, the optimized magnitude of the off-diagonal QFIM element for normalized local generators equals the concurrence, while a collective phase generator yields FQ = 4 C^2. Extending the optimized pure-state quantity by a convex roof gives an exact identity with Wootters concurrence for arbitrary mixed two-qubit states. This differs from evaluating the QFIM directly on a mixed density operator: for Werner states the raw optimized cross-QFIM is 2 p^2/(1 + p) and remains nonzero inside the separable region. The gap between the raw and convex-roof quantities therefore isolates, for this family, correlation geometry not captured by entanglement. Finally, in a two-path interferometer with a which-way marker, the optimal phase Fisher information available from local output statistics is I_phi^opt = V^2. These results place probability-gradient energy, bipartite correlation geometry, entanglement, and decoherence-induced loss of local phase sensitivity in a common mathematical setting while preserving the distinctions between classical Fisher information, QFIM geometry, Bell nonlocality, and global phase quantization.
This paper provides a systematic study of the operational idea that a quantum “state” is only defined up to what can be distinguished by a chosen family of observables. Concretely, any von Neumann algebra of observables M induces an equivalence relation on pure and mixed states by declaring two preparations indiscernible when they give identical statistics for every observable in M. The corresponding quotient, the Holevo space associated with M, is the effective (relational) state space of the experiment, explicitly dependent on the observer’s available measurements. We analyse the resulting geometry and topology of these quotients, and prove a context-complete classical representation theorem: for every von Neumann algebra M there is a canonical lift a↦â to bounded continuous functions on the Holevo space, reproducing expectation values pointwise. In the commutative case this reduces to ordinary probability theory on the joint spectrum. The framework is illustrated in explicit examples, including position measurements of a free particle and polarisation measurements in the qubit, Einstein–Podolsky–Rosen (EPR), and Bell settings. In particular, in the EPR scenario Charlie’s joint observable defines a simplex of joint outcome distributions, whereas the Alice/Bob marginal viewpoint collapses the effective description to a lower-dimensional space by “forgetting” the correlation parameter. We show that by varying the polariser settings, the indiscernibility classes become conjugated (and generically reshuffled), and different settings are typically incompatible at the level of observable algebras.
Entanglement Distance (ED) was originally proposed as a geometric measure of entanglement derived from the Fubini-Study metric on the projective Hilbert space. Independently, the Meyer-Wallach and Scott measures quantify multipartite entanglement via linear entropy. In this work, we demonstrate that these two seemingly distinct frameworks are mathematically identical for pure states of arbitrary finite dimensions. We prove that ED arises naturally as the trace of the Fubini-Study metric tensor over the local subalgebra of observables. Crucially, this geometric unification yields a direct operational interpretation: the global entanglement of a pure state is exactly proportional to the total Quantum Fisher Information (QFI) available for local unitary estimation. This bridges abstract information geometry with quantum metrology, demonstrating that ED dynamically quantifies resourcefulness for distributed quantum sensing, identifying Heisenberg-limited sensitivity in regimes where standard variance-based witnesses fail.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
We develop an information-geometric framework for describing quantum state-update processes associated with measurement and statistical distinguishability. The approach is based on the quantum relative entropy and the quantum Fisher information metric, which together induce a natural Riemannian geometry on the manifold of quantum states. Using the second-order expansion of relative entropy, we show how the Fisher metric governs the local structure of distinguishability between nearby states and defines a corresponding thermodynamic length. This geometric structure provides an effective description of finite quantum state transitions in terms of fluctuation geometry and information-space distance. The formalism is applied to thermal two-level systems and harmonic oscillator states, illustrating how the Fisher metric encodes susceptibilities, fluctuations, and geometric transition costs. We also discuss the relation between thermodynamic length, dissipation bounds, and optimal paths in state space. Within this framework, wave function collapse is interpreted not as a microscopic dynamical mechanism, but as an effective state-update process that admits a geometric characterization in the manifold of density operators. The resulting perspective unifies concepts from quantum information theory, thermodynamics, and differential geometry within a common operational framework based on statistical distinguishability. Possible connections with quantum speed limits, entanglement geometry, and holographic relations between relative entropy and gravitational dynamics are briefly discussed.
Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.