A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics (2208.04134v4)

Abstract: In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator $\rho_S$. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of $\rho_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from $\rho_S$. Using examples where the SOI are generated using representations of $SU(2)$, $SO(3)$, and $SO(N)$, we show two features of the CG. (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grainied space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to maximum entanglement between system and environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function w.r.t the purity of $\rho_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG.

• The paper introduces a novel differential-geometric method to quantify missing quantum information by computing the volume of the 'Surface of Ignorance'.
• It demonstrates that the SOI volume mirrors entropic properties, behaving similarly to von Neumann entropy and concaving with respect to state purity.
• The study uses unitary parameterizations (SU(2), SO(3), SO(N)) to connect geometric multiplicity of quantum states with typicality in statistical mechanics.

This paper introduces a differential-geometric method to quantify the "ignorance" or missing information associated with a quantum mixed state $\rho_S$. This ignorance arises when $\rho_S$ is viewed as a reduced density operator obtained by tracing out an environment (E) from a larger pure state $|\psi_{ES}\rangle$. The core idea is to calculate the volume of the manifold containing all possible pure states (purifications) in the combined system-environment space ($\mathcal{H}_{ES}$) that could result in the observed $\rho_S$. This manifold is termed the "Surface of Ignorance" (SOI).

Methodology: Calculating the SOI Volume

1. Purification: Any mixed state $$\rho_S = \sum_{i=1}^{N} \lambda^i |\lambda^i_S \rangle \langle \lambda^i_S|$$ (where $\lambda^i$ are eigenvalues and $|\lambda^i_S \rangle$ eigenvectors) can be "purified" into a pure state in a larger Hilbert space $$\mathcal{H}_{ES} = \mathcal{H}_E \otimes \mathcal{H}_S$$. A canonical choice is $$|\phi^{\rho_S}_{ES} \rangle = \sum_{i=1}^{N} \sqrt{\lambda^i} |\lambda^i_E\rangle |\lambda^i_S\rangle$$, where $\{|\lambda^i_E\rangle\}$ is an orthonormal basis for the environment $\mathcal{H}_E$, often taken as a copy of $\mathcal{H}_S$.
2. Generating All Purifications: All other purifications $|\bar{\Gamma}_{ES}^{\rho_S}(\vec{\xi})\rangle$ that trace down to the same $\rho_S$ can be generated by applying unitary transformations $U_E(\vec{\xi})$ acting only on the environment space to the canonical purification: $$|\bar{\Gamma}_{ES}^{\rho_S}(\vec{\xi})\rangle = (U_E(\vec{\xi}) \otimes \hat{1}_S)|\phi^{\rho_S}_{ES} \rangle$$. The parameters $\vec{\xi}$ define the specific unitary transformation within a chosen group (e.g., SU(N), SO(N)).
3. Metric Tensor: The set of all purifications $$F^{\rho_S} = \{ |\bar{\Gamma}_{ES}^{\rho_S}(\vec{\xi}) \rangle \}$$ forms a manifold (the SOI). A distance measure on this manifold is defined using the first fundamental form, $ds^2 = \langle d\bar{\Gamma}|d\bar{\Gamma}\rangle$, where $|d\bar{\Gamma}\rangle$ represents an infinitesimal change in the purification state corresponding to a change $d\vec{\xi}$ in the parameters. The components of the metric tensor $\mathbf{g}$ are given by:

   $$g_{ij}(\vec{\xi}) = \langle \partial_{\xi_i} \bar{\Gamma} | \partial_{\xi_j} \bar{\Gamma} \rangle$$

   where $\partial_{\xi_i} \bar{\Gamma}$ is the partial derivative of the purification state with respect to the parameter $\xi_i$.

4. Volume Calculation: The volume of the SOI is computed by integrating the volume element over the entire parameter space of $U_E(\vec{\xi})$:

   $$V = \int \sqrt{\det[\mathbf{g}(\vec{\xi})]} d\xi_1 d\xi_2 \dots d\xi_n$$

   where $n$ is the number of parameters in $\vec{\xi}$.

Entanglement Coarse-Graining (ECG) Framework

The paper interprets this volume within an "Entanglement Coarse-Graining" (ECG) framework:

• Microstates: The individual purifications $|\bar{\Gamma}_{ES}^{\rho_S}(\vec{\xi})\rangle$.
• Macrostates: The reduced density operators $\rho_S$, characterized by their eigenvalue spectrum $\vec{\lambda}$. Each $\rho_S$ corresponds uniquely to an SOI.
• Volume as Multiplicity: The volume $V$ of the SOI associated with $\rho_S$ acts as the multiplicity of the macrostate, analogous to the number of microstates corresponding to a macrostate in classical statistical mechanics. A larger volume signifies greater ignorance, as more underlying pure states (microstates) are consistent with the observed $\rho_S$ (macrostate).

Key Findings and Analogies to Statistical Mechanics

The paper demonstrates through examples ($SU(2)$, $SO(3)$, $SO(N)$) that this volume exhibits properties analogous to those underlying Boltzmann's statistical mechanics:

1. (Feature 1) Monotonicity/Concavity: The SOI volume behaves similarly to the von Neumann entropy $S_{VN}(\rho_S)$.
   • It is zero for pure states ($\rho_S$ has one eigenvalue equal to 1, $S_{VN}=0$).
   • It is maximal for the maximally mixed state ($\rho_S = \mathbf{1}/N$, all $\lambda^i = 1/N$, $S_{VN}$ is maximal).
   • It is a concave function with respect to the purity $Tr[\rho_S^2]$ (Figs. 3 & 5).
   • Practical Implication: This means that processes increasing the entanglement between the system and environment (leading to a more mixed $\rho_S$) correspond to evolving towards macrostates (SOIs) with strictly larger volumes. The volume tracks the increase in entanglement/missing information.
2. (Feature 2) Typicality: For systems where the environment dimension is large (studied via $SO(N)$ for large N), the macrostate corresponding to maximum entanglement (maximally mixed $\rho_S$) occupies the vast majority of the total "coarse-grained space".
   • This is shown by demonstrating that the volume distribution becomes sharply peaked around the maximally mixed state as N increases (Fig. 8).
   • Furthermore, the average von Neumann entropy of the states occupying >99.99% of the total volume approaches the maximum possible entropy as N increases (Fig. 9).
   • Practical Implication: This reinforces the concept of typicality in quantum statistical mechanics: if a large composite system is in a randomly chosen pure state, the reduced state of a subsystem is overwhelmingly likely to be very close to maximally mixed (i.e., highly entangled with the rest of the system).

Implementation Examples

• General Unitary Parameterization: The paper uses the standard parameterization of $U(N)$ based on successive 2-dimensional rotations (Sec III.A) to define the parameters $\vec{\xi}$ for the unitaries $U_E(\vec{\xi})$.
• SU(2) Example (N=2):
  • $U_E(\phi, \psi, \chi)$ is the standard SU(2) matrix (Eq. 20).
  • The metric components $g_{ij}$ are calculated explicitly.
  • The volume is found analytically: $$V_{SU(2)} = 4\pi^2 \sqrt{\lambda^1(1-\lambda^1)} = 4\pi^2 \sqrt{S_L/2}$$, where $S_L = 1 - Tr[\rho_S^2]$ is the linear entropy (Eq. 22). This calculation demonstrates Feature 1.
• SO(3) Example (N=3):
  • $U_E(\phi_{12}, \phi_{13}, \phi_{23})$ is the SO(3) rotation matrix (Eq. 24).
  • Metric components $g_{ij}$ are calculated explicitly (Eqs. 27-31).
  • The volume is found analytically: $$V_{SO(3)} = (\pi^2/4)\sqrt{(\lambda^1+\lambda^2)(\lambda^1+\lambda^3)(\lambda^2+\lambda^3)}$$ (Eq. 33).
  • Feature 1 is demonstrated by comparing the normalized volume to $S_{VN}$ and $S_L$ (Fig. 5).
  • Feature 2 is demonstrated by:
  • Discretizing the space of possible $\rho_S$ (the probability simplex $\mathcal{S}$) using a uniform sampling method (Eqs. 36-38, Fig. 6).
  • Grouping the discrete $\rho_l$ based on their SOI volume.
  • Calculating the fraction of $\mathcal{S}$ belonging to each volume group and the average $S_{VN}$ within each group (Fig. 7). This shows that the largest fraction of states corresponds to high volume and high $S_{VN}$.
• SO(N) Example (Large N):
  • Calculating $\det[\mathbf{g}]$ becomes hard. The paper assumes the volume element separates into $\lambda$-dependent and $\xi$-dependent parts. By setting $\vec{\xi}=0$, they infer the $\lambda$-dependent part, justified numerically for N=4 (Fig. 8).
  • Inferred Volume: $$V_{SO(N)} \propto \prod_{i<j} \sqrt{\lambda^i + \lambda^j}$$ (Eq. 46, ignoring constant factors).
  • Analyzes a specific family of states $\rho_S(\lambda^1)$ (Eq. 43) mixing a pure state with a maximally mixed state of dimension N-1.
  • The normalized volume $V^{\textrm{norm}}_{SO(N)}(\lambda^1)$ is plotted (Fig. 9), showing concentration near the maximally mixed state ($\lambda^1=1/N$) for large N.
  • The average $S_{VN}$ for states within the region containing >99.99% of the total volume is calculated (Eq. 48) and shown to approach 1 (maximal normalized entropy) as N increases (Fig. 10), demonstrating Feature 2 in the large N limit.

Generalization and Potential Applications

• The framework is generalized (Sec IV) to include unitary transformations $U_S$ on the system space $\mathcal{H}_S$, allowing comparison between SOIs associated with different eigenbases (Fig. 10).
• This connects the SOI concept to the Uhlmann-Jozsa fidelity $\mathcal{F}_{UJ}$, which measures the similarity between two quantum states $\rho$ and $\sigma$. One definition of $\mathcal{F}_{UJ}$ involves maximizing the overlap between purifications from the SOIs of $\rho$ and $\sigma$ (Eq. 53). The SOI volume and geometry could thus provide new insights into quantum state distinguishability.
• Potential Uses:
  • Provide a geometric measure of information missing in a reduced density operator.
  • Analyze the dynamics of entanglement generation from a geometric perspective.
  • Offer an alternative framework for understanding typicality arguments in quantum statistical mechanics.

Implementation Considerations

• Computational Complexity: Calculating the metric tensor requires computing derivatives of state vectors with respect to unitary parameters. Finding the determinant $\det[\mathbf{g}]$ and performing the multi-dimensional integral for the volume $V$ can be computationally expensive, especially for large system dimensions N or complex unitary groups with many parameters. Symbolic computation becomes intractable quickly (e.g., noted for SU(3)).
• Numerical Methods: For complex cases, numerical integration (like Monte Carlo, used for the SO(4) check in Fig. 8) might be necessary to estimate the volume.
• Approximations/Simplifications: The paper relies on specific Lie groups (SU(N), SO(N)) and symmetries. The SO(N) analysis uses an approximation (setting $\vec{\xi}=0$ to infer the $\lambda$-dependence) justified numerically. Analyzing specific, simpler families of states (like $\rho_S(\lambda^1)$ in the SO(N) case) can make large-N analysis feasible.
• Choice of Environment/Unitaries: The calculated volume depends on the dimension of the environment $\mathcal{H}_E$ (assumed to be N) and the group of unitaries $U_E$ used to generate the purifications. This choice reflects assumptions about the environment's symmetries or the nature of the system-environment interaction generating the entanglement.