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Role of quantum state texture in probing resource theories and quantum phase transition

Ayan Patra, Tanoy Kanti Konar, Pritam Halder, Aditi Sen De

Abstract

Building on the recently developed quantum state texture resource theory, we exhibit that the difference between maximum and minimum textures is a valid purity monotone in any dimension and provide a lower bound for existing purity measures. We introduce a texture-based resource monotone applicable across general convex resource theories, encompassing quantum coherence, non-stabilizerness, and entanglement. In particular, we propose the notion of non-local texture, which corresponds to the geometric measure of bipartite and multipartite entanglement in pure states. Furthermore, we demonstrate that the texture of the entire ground state or its subsystems can effectively signal quantum phase transitions in the Ising chain under both transverse and longitudinal magnetic fields, offering a powerful tool for characterizing quantum criticality.

Role of quantum state texture in probing resource theories and quantum phase transition

Abstract

Building on the recently developed quantum state texture resource theory, we exhibit that the difference between maximum and minimum textures is a valid purity monotone in any dimension and provide a lower bound for existing purity measures. We introduce a texture-based resource monotone applicable across general convex resource theories, encompassing quantum coherence, non-stabilizerness, and entanglement. In particular, we propose the notion of non-local texture, which corresponds to the geometric measure of bipartite and multipartite entanglement in pure states. Furthermore, we demonstrate that the texture of the entire ground state or its subsystems can effectively signal quantum phase transitions in the Ising chain under both transverse and longitudinal magnetic fields, offering a powerful tool for characterizing quantum criticality.

Paper Structure

This paper contains 10 sections, 5 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Resource theory of quantum texture
  3. Quantification of purity through texture
  4. Construction of monotones for convex resource theories based on texture
  5. Construction of non-stabilizerness and coherence monotones for pure states
  6. Construction of entanglement monotone
  7. Characterizing phase transitions using texture-based metrics
  8. Ising spin chain with longitudinal magnetic field
  9. Conclusion
  10. Ising-spin chain model diagonalization

Key Result

Lemma 1

For a given state $\rho$, the max-texture and min-texture are given by $\mathcal{T}^{\max}(\rho) = 1 - \lambda_d^\downarrow$ and $\mathcal{T}^{\min}(\rho) = 1 - \lambda_1^\downarrow$, respectively. Here $\lambda_1^\downarrow$ and $\lambda_d^\downarrow$ denote the largest and smallest eigenvalues of

Figures (3)

  • Figure 1: Connecting quantum state texture to other resource theories. Real parts of all elements of a four-dimensional density matrix $\left(\text{Re}[\rho_{ij}]\right)$ are displayed. In the case of a textured state, the elements exhibit varying colors, indicating underlying structure in the density matrix. Conversely, the texture-less state would have a uniform color across all elements, reflecting a lack of structure. In a given basis, each element of the texture-less state's density matrix takes the value $1/d$, where $d$ is the dimension of the system (here, $d=4$). We connect texture resource theory to purity and other resource theories $\mathcal{R}$, which consists of free pure states that are interconnected by free unitaries of $\mathcal{R}$, e.g., non-stabilizerness, coherence, and entanglement.
  • Figure 2: Normalized rugosity of the ground state against the external magnetic field, $h$. Normalized value of the texture (ordinate) of the ground state (panel (a)) and its reduced two-party density matrix (panel (b)) for the Ising chain, as functions of the transverse field $h$ (abscissa). The plots exhibit symmetry about $h = 0$ and display distinct curvature changes near the phase transition points at $h = \pm 1$, highlighted by the derivative $\frac{d\mathfrak{R}}{dh}$ shown in the insets. Here $g = 0$ and $N = 512$. All axes are dimensionless.
  • Figure 3: Rugosity ($\mathfrak{R}$) (ordinate) of the ground state vs the longitudinal magnetic field, $g$ (abscissa). We present the texture of the ground state for an Ising chain subjected to a longitudinal and transverse magnetic field with different system sizes (normalized rugosity is plotted in the inset). The texture remains zero for $g < 0$ and increases with $g$ for $g \geq 0$, capturing the phase transition from the paramagnetic to the ferromagnetic phase at the critical point $g = 0$. We set $h = 0.5$. All axes are dimensionless.

Theorems & Definitions (11)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Definition 2
  • ...and 1 more