Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

This paper introduces unit-invariant generalizations of von Neumann entanglement entropy based on the Unit-Invariant Singular Value Decomposition (UISVD), demonstrating their stability and physical relevance across diverse frameworks including biorthogonal quantum mechanics, random matrix theory, and Chern-Simons theory.

The Gist

Imagine you are trying to measure the "connectedness" or "entanglement" between two parts of a quantum system. In the standard way of doing this, physicists use a mathematical tool called Singular Value Decomposition (SVD). Think of SVD as a way to break down a complex relationship into simple, fundamental pieces (like breaking a complex recipe down into its basic ingredients).

However, the authors of this paper discovered a flaw in this standard recipe.

The Problem: The "Unit" Trap

Imagine you have a picture of a cat. If you take a photo of it in inches, the cat looks a certain size. If you take a photo of it in centimeters, the numbers describing its size change, even though the cat is exactly the same.

In quantum physics, the standard SVD method is like that. If you change the "units" or the "scale" of your measurement (for example, deciding to measure one part of the system in "big units" and the other in "small units"), the calculated amount of entanglement changes. This is a problem because the physical reality of the connection hasn't changed; only your ruler has. The standard method confuses the actual quantum connection with the arbitrary choice of how you measure it.

The Solution: The "Self-Balancing" Scale

The authors introduce a new method called Unit-Invariant Singular Value Decomposition (UISVD).

To understand this, imagine you have a messy table with plates of different sizes.

• Standard SVD tries to measure the total weight of the food, but if you swap a small plate for a giant one, the total weight number changes, even if the amount of food is the same.
• UISVD is like a magical table that automatically adjusts the size of every plate so they all look the same size before you weigh them. It "balances" the table first.

Once the table is balanced, the measurement of the food (the entanglement) depends only on the food itself, not on the size of the plates you started with. This new method ensures that your answer is the same whether you measure in inches, centimeters, or any other arbitrary unit.

How They Tested It

The authors didn't just invent this math; they tested it in three very different "playgrounds" to see if it works:

1. Random Chaos (Random Matrices): They threw a huge number of random numbers into their new system. They found that the results were stable and followed a predictable, smooth pattern (like a bell curve), proving the method is robust even when the input is chaotic.
2. Knots and Loops (Chern-Simons Theory): They looked at mathematical knots. In this world, tying two knots together or twisting a knot is like changing the "units" of the system. They showed that their new method correctly ignored these twists and ties, measuring only the true "knot-ness" of the connection, whereas old methods got confused by the twisting.
3. Non-Standard Physics (Biorthogonal Quantum Mechanics): There is a version of quantum mechanics where the rules of "normal" physics (like energy being conserved in a simple way) don't apply perfectly. In this strange world, standard measurements often give weird, impossible results (like negative probabilities). The authors showed that their new UISVD method works perfectly here, giving clear, positive, and stable numbers that make physical sense.

The Big Takeaway

The paper claims that by using this "self-balancing" math (UISVD), scientists can finally measure quantum connections without worrying about arbitrary choices of scale or units. It provides a stable, reliable ruler for measuring entanglement in complex, messy, or non-standard quantum systems, ensuring that what we measure is the physics, not just the math.

Technical Summary

Technical Summary: Generalised Entanglement Entropies from Unit-Invariant Singular Value Decomposition

1. Problem Statement
The paper addresses a fundamental limitation in the quantification of quantum entanglement for non-Hermitian, rectangular, or open quantum systems. Traditional measures of entanglement, such as von Neumann entropy derived from the Schmidt decomposition or SVD entropy derived from transition matrices, rely on standard Singular Value Decomposition (SVD). While standard SVD is invariant under unitary transformations, it is not invariant under diagonal rescalings (changes of basis normalization or physical units).

The authors highlight that in contexts such as Biorthogonal Quantum Mechanics (BQM), Chern-Simons theory, or systems with non-Hermitian operators, the singular values—and consequently the derived entropies—depend on arbitrary choices of scale or normalization. This dependence conflates genuine physical correlations with artifacts of the chosen metric or basis scaling, rendering the measures ambiguous or physically meaningless in these specific frameworks.

2. Methodology
To resolve this, the authors adapt the Unit-Invariant Singular Value Decomposition (UISVD), a mathematical construction originally proposed by Uhlmann, to the context of quantum information theory. The methodology proceeds in three stages:

• Definition of Unit-Invariant Singular Values: For a matrix $A$, the authors define three types of balanced matrices by rescaling rows and/or columns using diagonal matrices ($D_L, D_R, D_{BL}, D_{BR}$) such that the resulting matrix has specific geometric mean properties (e.g., row/column norms equal to 1).

  • Left-Unit-Invariant (LUI): Rescales rows based on Euclidean norms ($A_L = D_L A$).
  • Right-Unit-Invariant (RUI): Rescales columns based on Euclidean norms ($A_R = A D_R$).
  • Bi-Unit-Invariant (BUI): Simultaneously rescales rows and columns to balance the matrix ($A_B = D_{BL} A D_{BR}$).
    The singular values of these balanced matrices ($\sigma^L, \sigma^R, \sigma^B$) are defined as the unit-invariant singular values. These are proven to be invariant under transformations of the form $A \to D A U$, $A \to U A D$, or $A \to D A D'$, where $D, D'$ are non-singular diagonal matrices and $U$ is unitary.

• Construction of Entropies: The authors define new entanglement entropies by applying the standard von Neumann entropy formula to the normalized singular values of these balanced matrices.

  • For pure states $|\psi\rangle$ with coefficient matrix $A$, they construct "balanced states" $|\psi_L\rangle, |\psi_R\rangle, |\psi_B\rangle$ using the scaling operators. The reduced density matrices of these states yield the LUI, RUI, and BUI entanglement entropies.
  • For transition matrices (relevant for pseudo-entropy and pre/post-selected states), they apply the same balancing procedure to the transition operator $\tau$ to define Unit-Invariant SVD entropies.

• Statistical Analysis: The paper employs Random Matrix Theory (RMT) to analyze the distribution of these invariants for Ginibre and Wigner ensembles. The authors prove that the empirical spectral distributions of these unit-invariant singular values converge to a quarter-circle law (or a stretched version thereof for BUI), similar to standard singular values but with specific scaling factors dependent on the ensemble.

3. Key Contributions and Results

• Formalism: The paper provides a rigorous framework for defining entanglement measures that are invariant under local diagonal rescalings. It explicitly constructs the balanced matrices and derives closed-form expressions for $2 \times 2$ systems (Table 1).
• Random Matrix Theory: The authors prove that for large random matrices (Ginibre and Wigner ensembles), the distributions of LUI and RUI singular values follow the standard quarter-circle law on $[0, 2]$. For BUI singular values, they demonstrate a "stretched" quarter-circle law where the support depends on the expected logarithm of the matrix entries ($2e^{-c_\star}$).
• Chern-Simons Theory Applications: The authors apply these measures to link complement states in Chern-Simons theory. They demonstrate that:
  • BUI entropy is invariant under connected sums of knots on either component of a link.
  • LUI and RUI entropies are sensitive to the "side" of the connected sum, distinguishing which component was modified, whereas standard entanglement entropy does not.
  • All measures are invariant under changes of framing (local unitary operations).
• Biorthogonal Quantum Mechanics (BQM): The paper applies UISVD entropies to BQM, a framework where scale transformations are physical symmetries.
  • They show that standard biorthogonal entanglement entropies (RL entropy) are often complex-valued, unbounded, and can take negative values.
  • In contrast, the BUI entropy is always real, non-negative, bounded by $\log(\text{rank})$, and fully invariant under the admissible BQM scale transformations.
  • Numerical examples with a two-qubit PT-symmetric Hamiltonian illustrate that BUI entropy provides a stable, physically meaningful spectrum where standard measures fail.

4. Significance and Claims
The paper claims that UISVD restores "natural physical covariance" to operator-based entropies. By removing dependence on arbitrary normalizations or unit choices, these measures offer a more reliable tool for quantifying correlations in systems where unitarity or standard normalization assumptions fail.

The authors position this work as a "proof of concept," demonstrating that UISVD yields stable, physically meaningful entropic spectra in diverse settings ranging from random matrices to topological field theories and non-Hermitian quantum mechanics. They suggest that these tools could be valuable for classifying phases of mixed states and understanding the geometry of entanglement in (A)dS/CFT contexts, particularly as non-Hermitian quantum mechanics gains prominence. The paper explicitly states that the selection of examples is non-exhaustive and intended to illustrate the utility of the framework rather than to exhaustively explore all potential applications.