[Paper Review] Manifold of density matrices

Aik-meng Kuah, E. C. G. Sudarshan|ArXiv.org|Jul 30, 2003

Quantum Information and Cryptography1 references3 citations

TL;DR

This paper establishes that the manifold of N×N density matrices arises as the quotient space $\mathbb{C}P^{N^2-1}/SU(N)$, derived via the purification of mixed states into $N^2$-dimensional pure states. The key contribution is identifying the geometry of density matrices as a stratified space of orbits under $SU(N)$, with strata indexed by rank $\mu$, and showing that for $N=2$, this yields the familiar Bloch sphere.

ABSTRACT

We show that the manifold of density matrices can be derived from CP^{N^2-1} by the action of SU(N). We give some preliminary observations on the structure of this manifold.

Motivation & Objective

• To characterize the mathematical structure of the set of $N \times N$ density matrices as a geometric manifold.
• To address the lack of a well-defined geometric framework for mixed quantum states compared to pure states.
• To establish a correspondence between density matrices and orbits of $SU(N)$ acting on $\mathbb{C}P^{N^2-1}$ via purification.
• To analyze the stratified geometry of the manifold based on the rank of density matrices.
• To demonstrate that for $N=2$, the manifold reduces to the well-known Bloch sphere, validating the framework.

Proposed method

• Using the purification procedure to map $N \times N$ density matrices to pure states in an $N^2$-dimensional Hilbert space.
• Applying the Schmidt decomposition to represent purified states as $|\psi\rangle = \sum_{i} \sqrt{\rho_i} |i^A\rangle \otimes |i^B\rangle$, where $\rho$ is the density matrix.
• Proving that two pure states yield the same reduced density matrix if and only if they are related by a unitary transformation $I^A \otimes U^B$ with $U \in SU(N)$.
• Identifying the manifold of density matrices as the orbit space $\mathbb{C}P^{N^2-1}/SU(N)$, where orbits correspond to density matrices.
• Classifying the stabilizer subgroups of states under $SU(N)$ action, showing they are isomorphic to $U(N-\mu)$ for Schmidt number $\mu$.
• Analyzing the dimensionality of strata as $\mu(2N - \mu) - 1$, with $\mu$ ranging from 1 to $N$, and identifying the pure state manifold as $\mathbb{C}P^{N-1}$.

Experimental results

Research questions

• RQ1 How can the set of $N \times N$ density matrices be endowed with a well-defined geometric manifold structure?
• RQ2 What is the relationship between the manifold of density matrices and the space of pure states $\mathbb{C}P^{N^2-1}$?
• RQ3 How does the $SU(N)$ group action on $\mathbb{C}P^{N^2-1}$ classify density matrices via orbit equivalence?
• RQ4 What is the stratified geometric structure of the manifold of density matrices, and how does it depend on the rank of the density matrix?
• RQ5 Does the framework reproduce known results, such as the Bloch sphere for qubits ($N=2$)?

Key findings

• The manifold of $N \times N$ density matrices is diffeomorphic to the quotient space $\mathbb{C}P^{N^2-1}/SU(N)$, with orbits of $SU(N)$ corresponding to unique density matrices.
• The dimension of the manifold of density matrices is $N^2 - 1$, consistent with the expected dimension of the space of $N \times N$ density matrices.
• For a density matrix of rank $\mu$, the corresponding stratum in the manifold has dimension $\mu(2N - \mu) - 1$, with the lowest-dimensional stratum ($\mu = 1$) being $2N - 2$-dimensional and corresponding to pure states.
• The stratum of pure states ($\mu = 1$) is isomorphic to $\mathbb{C}P^{N-1}$, and forms the convex hull of all density matrices.
• For $N=2$, the manifold of density matrices is a 3-dimensional sphere, confirming the standard Bloch sphere representation.
• The geometry is stratified with decreasing dimensionality as rank decreases, and each stratum of rank $\mu$ is a convex covering for strata of lower rank.