Heralding probability optimization for nonclassical light generated by photon counting measurements on multimode Gaussian states

This paper presents an efficient method to optimize the heralding probability for generating nonclassical light states from multimode Gaussian resources by formulating the maximization problem as a system of polynomial equations that can incorporate experimental constraints like quadrature squeezing bounds.

The Gist

Imagine you are trying to bake a very specific, delicate cake (a special quantum state of light) in a kitchen where you don't have a strong oven (nonlinear interactions). Instead of baking it directly, you use a clever trick: you mix a bunch of ingredients (a "Gaussian state" of light), and then you peek into the kitchen through a small window to see if a specific number of eggs (photons) landed in a specific bowl. If you see exactly the right number of eggs, you know the cake in the main pan is ready. If you don't, you throw everything away and start over.

This "peeking" is called heralding. The problem is that sometimes you peek, and the eggs don't land where you want. You have to start over, which wastes time and energy. The goal of this paper is to figure out how to arrange your ingredients and your kitchen setup so that the eggs land in the right bowl as often as possible.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The Challenge: The "Unlucky" Kitchen

In the world of quantum light, creating strange, non-standard states (like "Fock states" or "cat states") is hard because light doesn't naturally interact with itself strongly enough to change its shape. Scientists use a workaround: they create a complex mixture of light, measure part of it, and if the measurement is "lucky," the rest of the light transforms into the desired shape.

However, this "lucky" event happens very rarely. As experiments get more complex (trying to catch more photons at once), the odds of success drop even lower. If the success rate is too low, the experiment takes forever. The paper asks: How do we tweak the knobs on our machine to make the "lucky" event happen as often as possible?

2. The Solution: Turning the Problem into a Puzzle

The author, Jaromír Fiurášek, discovered that finding the perfect settings for this machine isn't just a matter of guessing and checking. Instead, it can be turned into a mathematical puzzle.

• The Analogy: Imagine you have a set of dials (parameters) on your machine. You want to find the exact position of every dial to get the highest success rate.
• The Discovery: The author showed that the rules for these dials can be written down as a system of polynomial equations (equations with numbers multiplied by variables, like $x^2 + 3x + 2 = 0$).
• Why this matters: Once you have a system of polynomial equations, you don't need to guess. You can use powerful, pre-existing mathematical tools (like "Gröbner bases" or "homotopy continuation") to solve the puzzle exactly and find the best settings efficiently. It's like having a GPS that tells you the exact route to the destination instead of driving around randomly.

3. The "Squeezing" Limit: Don't Ask for the Impossible

In this quantum kitchen, there is a limit to how much you can "squeeze" the ingredients. "Squeezing" is a way of compressing the uncertainty of light to make it more useful, but current technology has a maximum limit on how much you can do it.

• The Problem: If you just ask the math to find the absolute best settings without limits, it might tell you to squeeze the light infinitely, which is impossible in the real world.
• The Fix: The paper shows how to add a "speed limit" to the math. You can tell the solver, "Find the best settings, but don't squeeze harder than this specific limit." This ensures the solution is not just mathematically perfect, but also experimentally possible with today's technology.

4. The Results: Testing the Recipe

The author tested this method on specific examples:

• Single-mode states: Creating a specific type of light in one channel.
• Two-mode states: Creating entangled light in two channels (like a "quantum handshake" between two beams).

They looked at different ways the "eggs" (photons) could land in the "bowls" (detectors). For example, if you need to detect 3 photons in one bowl and 3 in another, versus 4 in one and 2 in the other, the math tells you which arrangement gives the highest success rate.

Key Finding: The paper found that for some target states, a "symmetric" setup (like 3 and 3) works best, but for others, an "asymmetric" setup (like 4 and 2) is actually superior. The method allows scientists to quickly check all these possibilities and pick the winner.

5. The "Squeezed Cake" Extension

The paper also shows how to make a slightly different kind of cake: a "squeezed superposition." This is like taking the cake and giving it a final, precise twist. The author shows that you can incorporate this final twist into the initial recipe (the input settings) without changing the math's ability to find the best success rate.

Summary

In short, this paper provides a mathematical recipe book for scientists building quantum light experiments. Instead of blindly adjusting their equipment to see what works, they can now use a specific set of equations to calculate the exact settings that will give them the highest chance of success, while respecting the physical limits of their current technology. It turns a difficult, trial-and-error process into a solvable math problem.

Technical Summary

1. Problem Statement

The generation of highly non-classical quantum states of light (e.g., Fock states, Schrödinger cat states, Gottesman-Kitaev-Preskill (GKP) states) is crucial for optical quantum computing, metrology, and error correction. Due to the lack of strong optical nonlinearities, these states are typically prepared using conditional (heralded) schemes. In these schemes, a multimode Gaussian state (usually generated from squeezed vacuum states) is measured via photon counting on a subset of modes (heralding modes). A specific detection pattern heralds the successful preparation of a non-Gaussian state in the remaining signal modes.

The Core Challenge:
While the structure of the generated state can be engineered, the heralding probability (the success rate of the experiment) is often extremely low, especially as the complexity of the target state (number of photons) increases. Maximizing this probability is essential for practical implementation. Existing optimization methods often rely on generic numerical algorithms, which may be inefficient or fail to find global optima. Furthermore, experimental constraints, such as limits on available single-mode squeezing, must be incorporated, which complicates the optimization landscape.

2. Methodology

The author proposes a rigorous mathematical framework to transform the maximization of heralding probability into a solvable algebraic problem.

• Theoretical Framework:

  • The input is modeled as an $(m+1)$-mode pure Gaussian state $|G\rangle$ with vanishing coherent displacements, expressed via the stellar (Bargmann) representation.
  • The state is generated by interfering $m+1$ single-mode squeezed vacuum states in an optical interferometer.
  • The matrix $A$ describing the Gaussian state is decomposed into a diagonal matrix $\Lambda$ (containing damping parameters $x_j$) and a complex symmetric matrix $B$ (containing structural parameters $s_j, \nu_{jk}$).
  • The matrix $B$ determines the shape of the conditionally generated state, while $\Lambda$ (specifically the variables $X_j = x_j^2$) influences the heralding probability without altering the state's form (up to normalization).

• Formulation as Polynomial Equations:

  • The heralding probability $p_S$ is derived as a function of the parameters.
  • The optimization problem (maximizing $p_S$ with respect to $X_j$) is shown to be equivalent to finding the roots of a system of polynomial equations.
  • Specifically, the extremal conditions $\frac{\partial p_S}{\partial X_j} = 0$ lead to a system of $m$ polynomial equations in $m$ variables.

• Handling Constraints:

  • Limited Squeezing: The method incorporates experimental bounds on maximum squeezing ($r_{max}$) by treating the condition $\det(\mu^2 I - AA^\dagger) = 0$ (where $\mu = \tanh r_{max}$) as a constraint. This is solved using Lagrange multipliers, resulting in an extended system of polynomial equations.
  • Coupled Optimization: The framework is extended to cases where the target state parameters are not fully fixed, allowing simultaneous optimization of the state structure and the heralding probability.
  • Squeezed Superpositions: The method is generalized to generate squeezed superpositions of Fock states by incorporating the squeezing operator into the input Gaussian state parameters.

• Solution Techniques:

  • Because the problem reduces to a system of polynomial equations, the author utilizes algebraic geometry tools such as Gröbner basis construction and homotopy continuation to find all solutions (including global maxima) efficiently, rather than relying on gradient-based numerical searches.

3. Key Contributions

1. Algebraic Reformulation: The paper establishes that maximizing heralding probability for Gaussian-based conditional state preparation is equivalent to solving a system of polynomial equations. This allows for the application of powerful algebraic techniques to find exact or high-precision numerical solutions.
2. Constraint Integration: A novel procedure is developed to seamlessly incorporate physical constraints (maximum available squeezing) directly into the polynomial system, ensuring the solutions are experimentally feasible.
3. Generalization to Multimode and Squeezed States: The formalism is demonstrated not only for single-mode target states but also for multimode entangled states (e.g., single-rail Bell states) and squeezed superpositions of Fock states.
4. Handling Parity and Core States: The approach specifically targets Gaussian "core states" (where $A_{00}=0$), which naturally generate states with well-defined photon number parity (even or odd), covering important classes like GKP and cat states.

4. Results

The methodology was applied to specific examples involving two heralding modes ($m=2$):

• Target States:

  • Generation of balanced superpositions of even Fock states up to $N=6$ ($|0\rangle + |2\rangle + |4\rangle + |6\rangle$).
  • Generation of balanced superpositions of odd Fock states up to $N=7$.
  • Generation of a two-mode entangled single-rail Bell state $|\Phi(w)\rangle \propto |11\rangle + w|00\rangle$.

• Optimization Outcomes:

  • For each target state, multiple detection patterns (e.g., $(3,3)$ vs. $(4,2)$ photons) were analyzed.
  • The system identified up to six distinct configurations (sets of parameters $s_1, s_2, \nu$) for a given target state.
  • Global Maxima: The polynomial solver successfully identified the global maximum for the heralding probability $p_S$ for each configuration. For the $N=6$ target, the symmetric pattern $(3,3)$ yielded the highest probability.
  • Squeezing Constraints: The paper plotted $p_S$ as a function of the maximum available squeezing ($\mu^2$). It showed that $p_S$ increases with available squeezing until an optimal point is reached, after which it decreases if the squeezing constraint forces the system away from the optimal unconstrained region.
  • Entangled States: For the Bell state, the analysis proved that setting the diagonal parameters $s_1 = s_2 = 0$ is optimal, and the heralding probability increases monotonically with the entanglement parameter $w$.

5. Significance

• Experimental Relevance: As experiments move toward generating complex states with higher photon numbers, the success rate becomes the bottleneck. This work provides a direct, efficient path to maximize these rates, making complex quantum state engineering more viable.
• Methodological Advancement: By shifting from generic numerical optimization to algebraic polynomial solving, the paper offers a more robust and exhaustive search method that avoids local minima traps common in gradient-based approaches.
• Scalability: The ability to handle constraints and multimode systems suggests this framework can be scaled to design protocols for larger, more complex quantum networks and error-corrected logical qubits (like GKP states).
• Resource Efficiency: The explicit inclusion of squeezing limits ensures that proposed protocols are not just theoretically optimal but practically achievable with current or near-future technology.

In conclusion, Fiurásek presents a powerful mathematical toolkit that transforms the probabilistic nature of optical quantum state engineering into a deterministic algebraic problem, significantly advancing the design and optimization of non-Gaussian state generation schemes.