100

It is well known (Geometry of quantum states by Bengtsson and Życzkowski) that the set of $N$-dimensional density matrices is stratified by the adjoint action of $U(N)$, where each stratum corresponds to orbits with a fixed type of degeneracy structure of the density matrix spectrum.

These orbits are flag manifolds. For example the orbit of density matrices of spectrum $(1, 0, 0, 0)$ (pure states) is $\mathbb{C}P^3$ and the orbit of density matrices of spectrum $(0.5, 0.5, 0, 0)$ (and also $(0.4, 0.4, 0.1, 0.1)$) is the complex Grassmannian $Gr(4,2, \mathbb{C})$.

These spaces - being coadjoint orbits - are known to be Kahler-Einstein.

An observation by Ingemar Bengtsson which was stated and proved in this article, asserts that in the composite system of two $N-$ state quantum systems $\mathcal{H}^N \otimes \mathcal{H}^N$, whose orbit of pure states is $\mathbb{C}P^{N^2-1}$, the orbit of maximally entangled pure states is $\mathbb{R}P^{N^2-1}$, which is a minimal Lagrangian submanifold.

My questions:

  1. Does this result generalize to non-pure states? For example, the orbit of maximally entangled states in $Gr(N^2,2, \mathbb{C})$ (the orbit of density matrices of the type $(0.5, 0.5, 0, 0, . . .)$ in $\mathcal{H}^N \otimes \mathcal{H}^N$) isomorphic to $Gr(N^2,2, \mathbb{R})$.

  2. Is there a physical interpretation of this result? (this is also a question left open by Bengtsson).

Update:

This is a clarification following Peter Shor's remark:

Among the biparticle pure states, the maximally entangled states have the property that their partial trace with respect to one system is mixed (see the discussion following equation 22 in the second reference). As a generalization, I wish to know the local orbits of the states within a fixed biparticle density matrix orbit whose partial trace has the maximal von Neumann entropy relative to all other states in the same biparticle orbit. My motivation is that if the local orbits will happen to be the real flag Lagrangian submanifolds of the complex flag manifolds defining the biparticle state orbits (which is the case for the pure states), these manifolds have well known geometries, and this can contribute to our understanding of mixed state entanglement.

Sahil Muhammed
  • 37
David Bar Moshe
  • 31,499

1 Answers1

1

I’m not an expert, but here’s how I’ve pieced it together from Bengtsson & Życzkowski and other discussions:

1) Pure vs. Mixed “Maximally Entangled” States

  • Pure States (Easy Case):
    For a bipartite system $H^{N} \otimes H^{N}$, we say a pure state $ \rho = \left| \psi \rangle \langle \psi \right|$ is maximally entangled if its reduced density matrix (after tracing out one subsystem) is the maximally mixed state (i.e. proportional to the identity). In that situation, the partial trace $\mathrm{Tr}_{2} \!\left( \rho \right)$ has full rank $\mathbf{N}$ and is $\tfrac{1}{N} I_{N}$. Geometrically, the entire set of pure states is $\mathrm{CP}^{N^{2} - 1}$, but the subset of maximally entangled pure states forms an $\mathrm{RP}^{\,N^{2} - 1}$ (real projective space) sitting inside there, and it also has a “minimal Lagrangian” property. That’s the result Bengtsson points out: these real projective submanifolds are special “slices” of the larger complex projective space.

  • Mixed States (Harder Case):
    Once we allow $\rho$ to be mixed (i.e. it has rank $> 1$ overall), there’s no single “universal” measure of entanglement anymore. Different entanglement criteria can disagree. But Bengtsson’s approach suggests the following idea:

    Within a given orbit of density matrices (i.e., all states with the same global spectrum), pick out those states whose partial trace has maximal von Neumann entropy.
    Those are the “most entangled” states within that orbit, in the sense that the subsystem is as “mixed” as it can possibly get if you only look at states in that same global orbit.

2) Geometry of Density Matrix Orbits

  • The set of all density matrices of size $N$ is stratified by their eigenvalue multiplicities. Each “stratum” (fixed spectrum type) is called a coadjoint orbit of $U \!\left( N \right)$. For two-qubit or two-$N$-level systems, we similarly look at density matrices in $H^{N} \otimes H^{N}$ and group actions of $U \!\left( N^{2} \right)$.
  • A typical example:
    • The orbit of pure states with spectrum $ \left( 1, 0, 0, \dots \right)$ is $\mathrm{CP}^{N^{2}-1}$.
    • The orbit of states with spectrum $ \left( 0.5, 0.5, 0, \dots \right)$ might be something like $\mathrm{Gr} \!\left( N^{2}, 2, \mathbb{C} \right)$ (a complex Grassmannian) because it has rank $2$ with each eigenvalue $0.5$.

When we say “orbit of maximally entangled states” in the mixed-state case, we’re talking about some submanifold inside that orbit—namely, those states whose reduced density matrix is “as mixed as possible” among that orbit. The question is whether that submanifold is itself some real form (like $\mathrm{Gr} \!\left( N^{2}, 2, \mathbb{R} \right)$) or “real Lagrangian” slice, mirroring how pure-state maximal entanglement gave us an $\mathrm{RP}^{N^{2}-1}$ slice.

3) Does the “Real Lagrangian” Property Generalize?

  • Short Answer: It’s partially open. There are indications (including in Bengtsson’s article) that for certain specific orbits, the subset of “maximally entangled” states (in the sense of largest reduced-entropy within that orbit) may indeed form a real (and possibly minimal) submanifold akin to the pure-state case. But proving this in general for all ranks and all $N$ is non-trivial.
  • The geometry gets more complicated because:
    1. Mixed states can have different shapes of spectra (different ranks, different eigenvalue multiplicities).
    2. One must choose which entanglement criterion. (Here, we’re going with “maximize subsystem von Neumann entropy inside the orbit.”)
    3. Even if you do define it that way, showing that submanifold is “real Lagrangian” (or something similarly nice) is not always straightforward.

In other words, pure states gave us a clean story: the orbit was a complex projective space, and the “maximally entangled submanifold” was a real projective space inside it. For mixed states, each orbit is a higher-dimensional complex flag or Grassmannian manifold, and the “maximally entangled slice” might or might not be a neat real submanifold in every case. We suspect (and some special cases confirm) that real-flag or real-Grassmann structures can appear, but a general classification remains a more advanced research question.

4) Physical Interpretation

  • In the pure-state scenario, “maximally entangled” means the bipartite system looks completely mixed to either subsystem alone. That gives you an $\mathrm{RP}^{\, N^{2} -1}$ inside $\mathrm{CP}^{N^{2} -1}$.
  • In the mixed-state scenario, we can only talk about “maximally entangled” within an orbit with fixed global eigenvalues. The partial trace is as mixed as possible (highest von Neumann entropy) among that set of states. Physically, you’re saying: “Given I can’t change the overall spectrum (which might reflect constraints like total energy, coupling to environment, etc.), which arrangement of the bipartite system yields the ‘most entanglement’ for the subsystems?”
  • It’s still an open question how to interpret these sets in experiments—maybe they correspond to states that, within certain thermodynamic or environmental constraints, have the largest bipartite correlations. Researchers are interested because if these submanifolds turn out to be special geometric objects (like real Grassmannians or real flag manifolds), that might provide a deeper understanding of how entanglement “fits” into quantum state space.

In a Nutshell

  • Yes, there is a proposed generalization to mixed states, but it’s more subtle because you have to fix an orbit (i.e. fix the global density matrix’s eigenvalue structure) and then look for states in that orbit whose partial trace is “maximally mixed.”
  • Geometrically, we hope these “maximally entangled submanifolds” might be something like real forms (real flag manifolds) inside the full complex orbit, mirroring the pure-state situation.
  • Physically, it means “within all states that share the same global eigenvalues, pick the ones that have the largest subsystem entropy.” Those are the “most entangled” you can get in that orbit, so they’re the natural generalization of pure maximally entangled states—albeit with a more complicated structure.

Whether or not they always turn out to be “nice” submanifolds (like $\mathrm{Gr} \!\left(N^{2}, 2, \mathbb{R} \right)$) in every example is still an open area of research. So that’s where the question stands—and why folks like Bengtsson highlight it as something we don’t yet fully understand.

ZaellixA
  • 1,824
Aazad Suryan
  • 51