Universal Control of Symmetric States Using Spin Squeezing

This paper proposes a universal control scheme for symmetric quantum states using only coherent rotations and spin squeezing, enabling the creation of complex entangled states like Schrödinger cat and GKP states and their subsequent transfer to photonic systems for generating desired quantum light.

The Gist

Imagine you have a massive choir of identical singers. In the world of quantum physics, these singers are "emitters" (like tiny atoms or artificial atoms), and they are all standing in a perfect circle, indistinguishable from one another.

Usually, to create a complex song (a specific quantum state), you would need a conductor to whisper a unique instruction to every single singer individually. If you have 40 singers, that's 40 separate instructions. If you have 100, the number of instructions explodes, making it impossible to do quickly before the singers get tired (lose their "coherence").

The Big Discovery
The researchers at the Technion in Israel found a shortcut. They discovered that you don't need to talk to every singer individually. Instead, you can control the entire choir using just two simple, group-wide actions:

1. The Group Spin (Coherent Rotation): Imagine the conductor waving a baton to tell everyone to turn their heads left or right at the exact same time.
2. The Group Squeeze (Spin Squeezing): Imagine the conductor telling the choir to huddle closer together in a specific pattern, tightening their formation in one direction while stretching it in another.

The paper proves that by mixing these two simple group moves—spinning the whole group and squeezing the whole group—you can create any possible song the choir can sing. You don't need to micromanage individuals. You just need the right sequence of group commands.

The Magic Trick: Turning Singers into Light
Here is the most magical part of the paper. When these singers (the quantum emitters) finish their song and stop, they naturally "sing out" a burst of light (a photon).

Usually, when things emit light randomly, it's like a chaotic crowd shouting; the information gets lost in the noise. But because this choir is perfectly synchronized (symmetric), when they all emit light at once, that chaotic noise disappears. Instead, their collective song is perfectly transferred into a single, pure beam of light.

Think of it like this: The choir doesn't just make noise; they act as a quantum printer. By arranging the choir using just the "spin" and "squeeze" moves, they can print out specific, complex shapes of light that are usually very hard to make.

What They Actually Built
The researchers didn't just theorize this; they wrote a computer program to figure out the exact sequence of "spins" and "squeezes" needed to print specific, famous shapes of light. They successfully designed sequences to create:

• Schrödinger's Cat States: Imagine a light beam that is simultaneously in two different states (like being both "on" and "off" at the same time). They created versions with two "legs" (two states) and four "legs."
• GKP States: These are complex, grid-like patterns of light (shaped like squares or hexagons) that are highly valuable for protecting quantum information from errors.

The Results
Using their method, they found that with a choir of about 40 singers, they could create these complex light shapes with very high accuracy (over 94% to 98% fidelity).

Why This Matters (According to the Paper)

• Efficiency: Instead of needing millions of steps to control a large group, they only need a number of steps that grows slowly (polynomially) as the group gets bigger.
• Simplicity: You don't need complex, individual controls for every particle. Just global "spin" and "squeeze" commands are enough.
• New Light Sources: This offers a new way to create special types of light (non-Gaussian states) that are difficult to make with current laser technology, which usually relies on weak, inefficient tricks.

In short, the paper claims that by treating a group of quantum particles as a single, synchronized unit and using simple "spin" and "squeeze" commands, we can universally control them to create any desired quantum state, which can then be instantly converted into high-quality, specialized beams of light.

Technical Summary

Technical Summary: Universal Control of Symmetric States Using Spin Squeezing

Problem Statement
The manipulation of quantum many-body systems remains a frontier challenge. While universal control in conventional qubit-based systems is theoretically achievable via single-qubit gates and two-qubit entangling gates, the practical creation of most quantum states requires a sequential application of gates that grows exponentially with the number of qubits. This exponential scaling renders the approach impractical for modest system sizes (e.g., 40 qubits) and is incompatible with current quantum architectures limited by short coherence times.

Concurrently, there is growing interest in symmetric states—quantum states invariant under the permutation of constituent qubits. These states naturally arise in diverse platforms including nitrogen-vacancy centers, nuclear magnetic resonance systems, superconducting circuits, trapped ions, neutral atoms, and quantum dots. While coherent rotations and spin squeezing have been demonstrated in these systems, it remained an open question whether these operations alone could constitute a universal control set capable of generating any arbitrary symmetric state. Furthermore, the potential to transfer these static symmetric states into high-fidelity, single-mode traveling photonic states (quantum light) via spontaneous emission had not been fully explored as a mechanism for creating non-Gaussian quantum light.

Methodology
The authors propose a scheme for universal control relying solely on two types of operations acting symmetrically on the entire population of $N$ non-interacting two-level emitters:

1. Coherent Rotations: Simultaneous rotations acting on every particle ($R(\theta, \hat{n}) = \exp(i\theta \vec{S} \cdot \hat{n})$).
2. Spin Squeezing: Non-linear operations acting on the combined state ($S_x^2(\alpha)$ and $S_y^2(\beta)$).

The theoretical foundation rests on the Dicke ladder basis, where the Hilbert space of symmetric states is spanned by polynomials of the collective spin operators $S_x, S_y, S_z$. The authors provide a proof of universality by demonstrating that the algebra generated by the set of Hamiltonians $\{S_x, S_y, S_x^2, S_y^2\}$ spans the entire symmetric Hilbert space. Through commutator relations, they show that these operations can construct polynomials of arbitrary order (e.g., deriving $S_y S_z$ from $[S_x, S_y^2]$ and subsequently higher-order terms via induction), thereby enabling the construction of any unitary in the symmetric subspace.

To translate this theory into practice, the authors developed an optimization protocol. Starting from the ground state $|0\rangle$, the protocol searches for a sequence of $M$ steps, where each step consists of a coherent rotation followed by a combination of $x$ and $y$ squeezing operations. The parameters ($\theta, \hat{n}, \alpha, \beta$) for each step are optimized using a random search algorithm followed by the Nelder-Mead method to maximize the fidelity between the final state and a target symmetric state.

Key Contributions and Results
The paper demonstrates that the combination of coherent rotations and spin squeezing is sufficient for universal control over symmetric states, requiring only a polynomial number of steps rather than the exponential number required by conventional gate-based approaches.

Using the optimization protocol, the authors successfully generated specific high-fidelity symmetric states with $N=40$ emitters:

• Schrödinger's Cat States: Two-legged and four-legged cat states were created with fidelities of 97% and 94%, respectively.
• Gottesman-Kitaev-Preskill (GKP) States: Square and hexagonal GKP states with 10 dB squeezing were generated with fidelities of 95% and 94%, respectively. A specific 11-step sequence achieved a square GKP state with 98.37% fidelity.

A critical contribution is the demonstration of the direct transfer of these symmetric states to photonic states. The authors leverage the property that spontaneous emission from symmetric-state superpositions maps precisely onto a single quantum-optical mode. Consequently, the optimized symmetric states of the emitters are transferred to traveling photonic states (flying qubits) with high fidelity. The results show that this method can produce non-Gaussian quantum light (cat and GKP states) without the need for strong higher-order nonlinearities (like Kerr nonlinearity) or conditional post-selection, which are typically inefficient in optical ranges.

Significance
The paper claims that this work establishes a powerful mechanism for the creation of desired quantum light states, addressing a major bottleneck in photonic quantum computation and communication. By utilizing the intrinsic non-linearity of emitters at high excitation regimes, the method allows for the creation of arbitrary symmetric states and their subsequent transfer to light pulses without addressing emitters individually.

The authors emphasize that this approach is relevant to state-of-the-art platforms such as trapped ions, superconducting circuits, and waveguide/cavity QED systems. The findings suggest that low-order nonlinearities (spin squeezing) are sufficient for universal control, offering a scalable path to generating multipartite entangled states and non-Gaussian quantum light resources (such as Schrödinger's cat and GKP states) essential for continuous-variable quantum information processing.