Calculating trace distances of bosonic states in Krylov subspace

This paper presents an efficient numerical method based on a generalized Lanczos algorithm that utilizes moment information to compute trace distances between pure and mixed Gaussian states in continuous-variable systems, while also providing lower bounds for mixed-mixed comparisons and extending to certain non-Gaussian states.

The Gist

Here is an explanation of the paper, translated into everyday language with some creative analogies.

The Big Picture: Measuring the "Difference" Between Quantum States

Imagine you are a detective trying to tell two very similar-looking suspects apart. In the world of quantum physics, these "suspects" are quantum states (the specific condition of a system, like a beam of light).

Sometimes, the suspects are very distinct (easy to tell apart). Sometimes, they are almost identical twins (very hard to tell apart). Physicists need a way to measure exactly how different two states are. They call this measurement the Trace Distance.

• High Trace Distance: The states are totally different. You can easily tell them apart.
• Low Trace Distance: The states are nearly identical. You might mistake one for the other.

The Problem:
In the world of "Continuous Variable" systems (like light beams with infinite possible energy levels), calculating this distance is a nightmare.

• The Old Way: To measure the difference, scientists used to try to write down a giant spreadsheet (a matrix) representing the quantum state. But because these systems are infinite, the spreadsheet would need infinite rows and columns. To make it work, they had to chop off the bottom of the spreadsheet (a "cutoff").
• The Flaw: If you have just one light beam, the spreadsheet is manageable. But if you have 10 or 20 light beams interacting, the spreadsheet grows so huge that it explodes your computer's memory. It's like trying to count every grain of sand on a beach by writing down a number for each one; it takes too long and runs out of paper.

The Solution: The "Krylov Subspace" Shortcut

The authors, Javier and Nicolás, have invented a clever shortcut. They don't need to write down the whole giant spreadsheet. Instead, they use a technique called the Lanczos Algorithm, which is like a smart flashlight in a dark cave.

The Analogy: The Blind Man and the Cave

Imagine you are in a massive, dark cave (the infinite quantum system) and you need to find a specific treasure (the answer to the distance calculation).

• The Old Method: You try to map out the entire cave, wall by wall, before you can find the treasure. This takes forever.
• The New Method (Lanczos): You shine a flashlight (the algorithm) in a specific direction. You only explore the path the light hits. You bounce the light off the walls, listen to the echoes, and build a small, manageable map of just the area you need. You don't need to know the whole cave to find the treasure; you just need to know the shape of the path right in front of you.

How It Works (The "Moment" Trick)

The magic of this new method is that it doesn't look at the "whole" state. It only looks at moments.

Think of a quantum state like a cloud of smoke.

• The Old Way: You try to photograph every single molecule of smoke.
• The New Way: You just measure the average position of the smoke (the center) and how spread out it is (the shape).

In physics, these are called First Moments (the average) and Covariance (the spread). Gaussian states (the most common type of quantum state) are completely defined by just these two pieces of information.

The authors' algorithm takes these simple "average and spread" numbers and uses them to run the "flashlight" (Lanczos algorithm). It calculates the distance by taking a few steps along a path, rather than mapping the whole universe.

What Can This Do?

1. Pure vs. Mixed States: The method is perfect for comparing a "pure" state (a perfect, ideal laser beam) against a "mixed" state (a laser beam that has been messed up by noise or loss). It calculates the difference incredibly fast, even for systems with many light beams.
2. Non-Gaussian States: It can also handle "weird" states that aren't perfect Gaussian clouds, as long as those weird states are built by mixing together several Gaussian clouds.
3. Lower Bounds: If you try to compare two "messy" (mixed) states, the method can't give you the exact answer yet, but it can give you a guaranteed minimum. It's like saying, "I can't tell you the exact distance between these two cities, but I can prove they are at least 50 miles apart." This is still very useful for checking if a quantum computer is working correctly.

Why Should We Care?

This is a huge deal for the future of quantum technology.

• Quality Control: If you build a quantum computer, you need to know if the state you created is the one you intended to create. This tool lets you check that quickly without needing a supercomputer.
• Scalability: As quantum systems get bigger (more modes, more light beams), the old methods become impossible. This new method stays fast and efficient, scaling up like a well-oiled machine.

Summary

The authors found a way to measure the difference between complex quantum states without getting bogged down in infinite math. Instead of trying to count every grain of sand, they use a smart flashlight to find the path, using simple "average and spread" data to solve a problem that was previously too hard to compute. This makes it much easier to certify and learn about the quantum systems that will power our future technology.

Technical Summary

Here is a detailed technical summary of the paper "Calculating trace distances of bosonic states in Krylov subspace" by Martínez-Cifuentes and Quesada.

1. Problem Statement

The trace distance, $d(\hat{\rho}_1, \hat{\rho}_2) = \frac{1}{2}\|\hat{\rho}_1 - \hat{\rho}_2\|_1$, is the fundamental metric for distinguishing quantum states, directly linked to the optimal success probability of state discrimination via the Holevo-Helstrom theorem. While continuous-variable (CV) Gaussian states are ubiquitous in quantum technologies (e.g., photonic quantum computing, metrology), calculating their trace distance poses significant challenges:

• Lack of Analytical Formula: No closed-form analytical expression exists for the trace distance between arbitrary Gaussian states in terms of their first moments ($r$) and covariance matrices ($V$).
• Numerical Intractability: Standard numerical methods require representing the density operators in a discrete basis (typically the Fock basis). This necessitates truncating the infinite-dimensional Hilbert space at a cutoff $c$. For an $M$-mode system, the matrix size scales as $O(c^M)$, making the diagonalization of the difference operator $\Delta\hat{\rho} = \hat{\rho}_1 - \hat{\rho}_2$ computationally prohibitive for multi-mode systems.

2. Methodology

The authors propose an efficient numerical method based on the Lanczos algorithm, a Krylov subspace technique, to compute the trace distance without explicit matrix representations of the density operators.

Core Theoretical Insight

The method relies on a specific property derived in Theorem 1: If one state, $\hat{\rho}_1 = |\psi\rangle\langle\psi|$, is pure and the other, $\hat{\rho}_2 = \hat{\rho}$, is mixed, the operator $\Delta\hat{\rho} = |\psi\rangle\langle\psi| - \hat{\rho}$ possesses exactly one positive eigenvalue ($\lambda_+$).
Consequently, the trace distance simplifies to:
$$d(|\psi\rangle\langle\psi|, \hat{\rho}) = \lambda_+$$
This reduces the problem from diagonalizing a full infinite-dimensional matrix to finding a single, isolated eigenvalue.

The Algorithm

1. Krylov Subspace Construction: The algorithm constructs a Krylov subspace generated by the operator $\hat{A} = |\psi\rangle\langle\psi| - \hat{\rho}$ and a trial vector $|c\rangle = |\psi\rangle$.
2. Metric Calculation via Bargmann Invariants: Instead of computing matrix elements in the Fock basis, the algorithm computes inner products of the form $\langle \psi | \hat{\rho}^k | \psi \rangle$.
   • For Gaussian states, these terms correspond to Bargmann invariants (multivariate traces) of Gaussian states.
   • The authors utilize a known analytical formula for the trace of products of Gaussian states, which depends only on the first moments and covariance matrices.
   • This allows the construction of the tridiagonal Lanczos matrix $T_\ell$ using only the parameters $(r, V)$, with a computational cost scaling polynomially with the number of modes ($M$) and the number of Lanczos steps ($\ell$).
3. Eigenvalue Approximation: The largest eigenvalue of the small tridiagonal matrix $T_\ell$ converges rapidly to $\lambda_+$, providing the trace distance.

Generalization

The method is extended to non-Gaussian states that can be expressed as linear combinations of outer products of pure Gaussian states (e.g., cat states, Fock states). The metric calculation involves summing Bargmann invariants, maintaining polynomial scaling with respect to the number of terms in the linear combination.

3. Key Contributions

• Efficient Pure-Mixed Distance: A polynomial-time algorithm to compute the exact trace distance between a pure Gaussian state and a mixed Gaussian state, bypassing the exponential cost of Fock basis truncation.
• Non-Gaussian Extension: A framework to handle non-Gaussian states (linear combinations of Gaussian states) by leveraging the linearity of the trace and the efficiency of Gaussian trace formulas.
• Lower Bounds for Mixed-Mixed: A technique to provide rigorous lower bounds on the trace distance between two mixed Gaussian states. While the full distance requires summing all positive eigenvalues (which is hard), the Lanczos algorithm efficiently approximates the largest eigenvalues, yielding a valid lower bound.
• Variational Bound: Derivation of a variational lower bound for the trace distance between a pure and mixed Gaussian state using a specific ansatz for the trial vector.

4. Results

The authors validated their method using a Julia implementation, comparing it against direct diagonalization in the Fock basis (with cutoff $c=100$):

• Single-Mode Gaussian States: The Lanczos method (using only $\ell=10$ steps) matched the exact diagonalization results for the trace distance between a squeezed state and the vacuum, as well as between a squeezed state and a lossy mixed state.
• Multi-Mode Systems: The method successfully computed distances for 10-mode systems where direct diagonalization would be infeasible.
• Non-Gaussian States: The algorithm accurately estimated the trace distance between pure and lossy multi-component cat states, matching Fock basis results.
• Mixed-Mixed Bounds: For two mixed Gaussian states, the method provided lower bounds that were significantly tighter than standard bounds (e.g., fidelity-based or Hilbert-Schmidt bounds) and converged toward the true value as the number of steps increased.

5. Significance

• Scalability: The method enables the certification and learning of high-dimensional CV quantum states, a critical bottleneck in photonic quantum computing and quantum metrology.
• Resource Efficiency: By avoiding the Fock basis, the method eliminates the need for arbitrary cutoffs and the associated exponential memory/compute costs.
• Practical Application: It provides a practical tool for experimentalists to quantify how close an experimentally prepared state is to a target theoretical state, even when the target is mixed or the system is large.
• Future Directions: The work opens avenues for developing more efficient Krylov subspace techniques to approximate the trace of the positive part of Hermitian operators, potentially leading to accurate estimates for mixed-mixed distances in the future.

In summary, this paper introduces a transformative numerical tool that leverages the mathematical structure of Gaussian states and the efficiency of the Lanczos algorithm to solve a previously intractable problem in continuous-variable quantum information.