We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a "maximal entropy principle in DV systems." We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a "second law of thermodynamics for quantum convolutions." We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the "magic gap," which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier.
We introduce a framework to study discrete-variable (DV) quantum systems based on qudits. It relies on notions of a mean state (MS), a minimal stabilizer-projection state (MSPS), and a new convolution. Some interesting consequences are: The MS is the closest MSPS to a given state with respect to the relative entropy; the MS is extremal with respect to the von Neumann entropy, demonstrating a "maximal entropy principle in DV systems." We obtain a series of inequalities for quantum entropies and for Fisher information based on convolution, giving a "second law of thermodynamics for quantum convolutions." We show that the convolution of two stabilizer states is a stabilizer state. We establish a central limit theorem, based on iterating the convolution of a zero-mean quantum state, and show this converges to its MS. The rate of convergence is characterized by the "magic gap," which we define in terms of the support of the characteristic function of the state. We elaborate on two examples: the DV beam splitter and the DV amplifier.
We introduce a quantum convolution and a conceptual framework to study states in discrete-variable (DV) quantum systems. All our results suggest that stabilizer states play a role in DV quantum systems similar to the role that Gaussian states play in continuous-variable systems. Hence we suggest the name ``discrete quantum Gaussians'' for stabilizer states. For example, we prove that the convolution of two stabilizer states is another stabilizer state, and that stabilizer states extremize both quantum entropy and Fisher information. We establish a ``maximal entropy principle,'' a ``second law of thermodynamics for quantum convolution,'' and a ``quantum central limit theorem.'' The latter is based on iterating the convolution of a zero-mean quantum state, which we prove converges to a stabilizer state. We bound the exponential rate of convergence of the quantum central limit theorem by the ``magic gap,'' defined in terms of the support of the characteristic function of the state. We elaborate on our general results with a discussion of some examples, as well as extending many of them to quantum channels.
In this work, we investigate the behavior of quantum entropy under quantum convolution and its application in quantifying magic. We first establish an entropic, quantum central limit theorem (q-CLT), where the rate of convergence is bounded by the magic gap. We also introduce a new quantum divergence based on quantum convolution, called the quantum Ruzsa divergence, to study the stabilizer structure of quantum states. We conjecture a “convolutional strong subadditivity” inequality, which leads to the triangle inequality for the quantum Ruzsa divergence. In addition, we propose two new magic measures, the quantum Ruzsa divergence of magic and quantum-doubling constant, to quantify the amount of magic in quantum states. Finally, by using the quantum convolution, we extend the classical, inverse sumset theory to the quantum case. These results shed new insight into the study of the stabilizer and magic states in quantum information theory.
We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all n-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.
Quantum convolution provides a discrete-variable analogue of classical convolution, with the mean state capturing the stabilizer structure preserved under repeated convolution. We establish a finite-step entropy hierarchy generated by compatible stabilizer dephasings. Along every compatible isotropic flag, the entropy increases toward the mean-state entropy ceiling, while the relative-entropy distance to the mean state decomposes exactly into successive coherence losses and a terminal classical nonuniformity. Optimizing over compatible subspaces yields an intrinsic entropy profile of the state. For quantum convolutional channels, Weyl covariance reduces the one-shot classical communication problem to minimal output entropy. A spectral-transfer argument shows that suitable stabilizer inputs reproduce stabilizer-measurement distributions of the environment as channel-output spectra. This gives a computable Holevo lower bound over all complete stabilizer measurements; its compatible restriction is characterized by the entropy hierarchy and refines the previous mean-state bound of Bu, Gu, and Jaffe. The bound is exact for stabilizer-diagonal environments, for which the Holevo capacity is strongly additive, and yields a single-letter formula for a nonstabilizer qutrit family. The same family also exhibits a coexistence region with simultaneously positive classical and quantum communication rates
Classically hard to simulate quantum states, or "magic states", are prerequisites to quantum advantage, highlighting an apparent separation between classically and quantumly tractable problems. Classically simulable states such as Clifford circuits on stabilizer states, free bosonic states, free fermions, and matchgate circuits are all in some sense Gaussian. While free bosons and fermions arise from quadratic Hamiltonians, recent works have demonstrated that bosonic and qudit systems converge to Gaussians and stabilizers under convolution. In this work, we similarly identify convolution for fermions and find efficient measures of non-Gaussian magic in pure fermionic states. We demonstrate that three natural notions for the Gaussification of a state, (1) the Gaussian state with the same covariance matrix, (2) the fixed point of convolution, and (3) the closest Gaussian in relative entropy, coincide by proving a central limit theorem for fermionic systems. We then utilize the violation of Wick's theorem and the matchgate identity to quantify non-Gaussian magic in addition to a SWAP test.
Classically hard-to-simulate quantum states, or "magic" states, are prerequisites to quantum advantage, highlighting an apparent separation between classically and quantumly tractable problems. Classically simulable states such as Clifford circuits on stabilizer states, free bosonic states, free fermions, and matchgate circuits are all in some sense Gaussian. While free bosons and fermions arise from quadratic Hamiltonians, recent works have demonstrated that bosonic and qudit systems converge to Gaussians and stabilizers under convolution. In this work, we similarly identify convolution for fermions and find efficient measures of non-Gaussian magic in pure fermionic states. We demonstrate that three natural notions for the Gaussification of a state—(1) the Gaussian state with the same covariance matrix, (2) the fixed point of convolution, and (3) the closest Gaussian in relative entropy—coincide by proving a central limit theorem for fermionic systems. We then utilize the violation of Wick's theorem and the matchgate identity to quantify non-Gaussian magic in addition to a SWAP test.