Translation symmetry-enforced long-range entanglement in mixed states

This paper demonstrates through a counting argument that translation symmetry enforces long-range entanglement in mixed states, specifically showing that the strong-to-weak spontaneous symmetry breaking fixed point cannot be represented as a mixture of short-range entangled states despite the existence of symmetric short-range entangled eigenstates.

The Gist

The Big Idea: A Crowd That Can't Fit in a Small Room

Imagine you have a very long line of people (a quantum system) standing in a circle. This line has a special rule: Translation Symmetry. This means if you ask everyone to take one step to the right, the line looks exactly the same as it did before.

In the world of quantum physics, scientists are interested in how "entangled" these people are.

• Short-Range Entangled (SRE): Think of this as people only holding hands with their immediate neighbors. It's easy to prepare this state; you just tell everyone to grab the hand of the person next to them.
• Long-Range Entangled (LRE): This is like a complex, invisible web connecting everyone in the circle, no matter how far apart they are. You can't just tell neighbors to hold hands to create this; it requires a much more complex, global coordination.

The Paper's Discovery:
The authors prove a surprising fact: Even though it seems like you could describe a perfectly balanced, symmetrical line of people using only simple "neighbor-holding" (SRE) states, you actually can't.

There aren't enough simple "neighbor-holding" states to fill up the entire "zero momentum" room (the specific state where the line looks perfectly symmetrical). Because the simple states run out, the remaining state must be the complex, long-range entangled kind.

The "Counting" Argument: Why Simple States Fail

The paper uses a "counting argument" to prove this. Here is the analogy:

Imagine you are trying to paint a massive, complex mural (the full space of symmetrical states) using only a limited set of simple stamps (the simple SRE states).

1. The Mural is Huge: The number of possible symmetrical patterns grows exponentially as the line gets longer. It's like a library with infinite books.
2. The Stamps are Limited: The number of patterns you can make with simple "neighbor-holding" rules grows much slower. It's like having a small box of crayons.
3. The Mismatch: The authors calculated that no matter how many times you mix and match your simple stamps, you simply don't have enough of them to cover the whole mural. There is a huge gap between what you can make with simple rules and what exists in the symmetrical world.

Because you can't cover the whole picture with simple stamps, the final picture (the "Maximally Mixed State" of translation symmetry) must contain a hidden, complex structure that simple stamps can't replicate. This hidden structure is Long-Range Entanglement.

The "Strong-to-Weak" Breaking: A Broken Clock

The paper discusses a concept called "Strong-to-Weak Spontaneous Symmetry Breaking" (SWSSB).

• Strong Symmetry: Imagine a clock that ticks perfectly. Every tick is identical.
• Weak Symmetry: Imagine the clock is broken, but on average, it still looks like it's ticking.

The paper shows that when a translation symmetry "breaks" from strong to weak (like that broken clock), the resulting state is not just a messy mix of simple, broken clocks. It is a specific, complex state that is inherently entangled.

You might think, "If I just mix a bunch of simple, non-entangled states together, I should get a simple mixed state." The paper says: No. If you try to mix simple states to create this specific symmetrical outcome, you will fail. The math proves that the only way to get this specific result is if the mixture itself is fundamentally complex (entangled).

The "Invisible" Entanglement

Here is the most subtle part of the discovery.

Usually, when we say something is "long-range entangled," we expect to see it by looking at how far-away parts of the system talk to each other (correlation functions). It's like seeing two people miles apart whispering to each other.

The Twist:
The authors show that this specific type of entanglement is invisible to standard tests.

• If you look at the line and ask, "Are people far apart whispering?" the answer is No.
• If you look at the line and ask, "Is the whole system a simple mix of neighbors holding hands?" the answer is No.

It is a "ghost" entanglement. It exists because of the sheer mathematical impossibility of filling the space with simple states, not because of any obvious long-distance signal. It's like a puzzle where the pieces fit together in a way that looks random from the outside, but is actually a rigid, unbreakable structure on the inside.

The "Time" Cost

The paper also mentions that creating this state is hard.
If you wanted to build this symmetrical state starting from a simple line of people, you would need to run a complex process. The authors prove that the time it takes to build this state grows with the square root of the size of the system.

Think of it like trying to organize a massive parade. If you only tell people to talk to their neighbors, it takes a while for the order to spread. But to get this specific "perfectly symmetrical" order, you can't just rely on neighbors; you need a global coordination that takes a long time to propagate across the whole line.

Summary

1. The Problem: Can we describe a perfectly symmetrical quantum system using only simple, local connections?
2. The Answer: No. There are too many symmetrical possibilities and not enough simple connections to cover them all.
3. The Consequence: The symmetrical state must be "Long-Range Entangled."
4. The Catch: This entanglement is "subtle." You can't detect it by looking for long-distance signals; you only know it's there because the math proves simple states aren't enough to build it.

In short: Nature forces a complex, invisible web of connections in symmetrical systems, even when you try to build them out of simple, local parts.

Technical Summary

Technical Summary: Translation Symmetry-Enforced Long-Range Entanglement in Mixed States

Problem Statement
Recent research into open quantum systems has revealed that mixed states exhibit entanglement patterns distinct from pure states. A mixed state is defined as long-range entangled (LRE) if it cannot be approximated as a mixture of "easy-to-prepare" short-range entangled (SRE) states. While symmetries are known to enforce entanglement in mixed states, the specific role of translation symmetry remains nuanced.

For on-site symmetries (where the symmetry operator acts locally on each site), the maximally mixed state in a symmetry sector (MMIS) can typically be spanned by symmetric product states, rendering it SRE. Conversely, for anomalous symmetries, strong symmetry constraints force the MMIS to be LRE. Translation symmetry occupies an intermediate position: it admits symmetric product states (in the zero momentum sector), yet states with non-zero momentum must be LRE. The central question addressed by this work is whether the zero momentum sector of translation symmetry can be spanned by SRE states. Specifically, does the fixed-point state representing strong-to-weak spontaneous symmetry breaking (SWSSB) of translation symmetry exhibit LRE, and if so, is this entanglement detectable via standard correlation functions?

Methodology
The authors employ a counting argument to compare the dimensionality of the space of all translation-invariant states against the dimensionality of the subspace spanned by translation-invariant SRE states (specifically, those preparable by depth-$d$ quantum circuits).

1. Definitions and Setup: The study focuses on 1D spin chains of length $n$ with local dimension $q$ and periodic boundary conditions. The object of study is $\rho_{TI}$, the maximally mixed state of translation-invariant (zero momentum) states.
2. Approximation via Circuit Depth: The authors utilize the fact that time-$\tau$ evolution under a bounded, geometrically local Hamiltonian can be well-approximated by a quantum circuit of depth $d \sim O(\tau \text{polylog}(n))$. To prove $\rho_{TI}$ is LRE, it suffices to show it is not well-approximated by "depth-$d$ states" for any polynomially small $d$.
3. Dimensional Analysis of SRE States:
   • The authors first analyze Translation-Invariant Matrix Product States (TIMPS). They demonstrate that the dimension of the span of TIMPS with bond dimension $d$ grows polynomially with $n$ (specifically as a degree-$n$ homogeneous polynomial in the tensor parameters).
   • They then generalize this to translation-invariant depth-$d$ circuits. Using a "cutting argument," they decompose a translation-invariant state generated by a depth-$d$ circuit into blocks of size $2d+1$. By treating these blocks as effective physical sites, they map the state to a TIMPS structure.
   • Crucially, they derive an upper bound on the dimension of the span of these translation-invariant depth-$d$ states, denoted $D_{SRE}(n, d, q)$. This bound scales roughly as a polynomial in $n$ of degree proportional to $d^2$.
4. Comparison with Total Space: The dimension of the full space of translation-invariant states, $D_{TI}(n, q)$, grows exponentially with $n$ (related to the number of $q$-ary necklaces).
5. Trace Distance Argument: Using a "Tails Lemma," the authors relate the trace distance between $\rho_{TI}$ and any mixture of SRE states to the projection of $\rho_{TI}$ onto the subspace of depth-$d$ states. Since the dimension of the SRE subspace is exponentially smaller than the total space for $d \ll \sqrt{n}$, the projection is exponentially small, implying a large trace distance.

Key Contributions and Results

• Theorem 1 (Main Result): The maximally mixed state of translation-invariant states, $\rho_{TI}$, is not well-approximated by any mixture of pure states preparable by time-$\tau$ evolution from a product state via a range-$r$ Hamiltonian, unless $\tau = \Omega(\sqrt{n}/\text{polylog}(n))$.
• Mechanism of Entanglement: The paper establishes a new mechanism for enforcing LRE in mixed states: a "mismatch" between the volume of the space of zero-momentum states and the volume of the space spanned by SRE zero-momentum states. Even though symmetric product states exist, they are insufficient in number to span the zero-momentum sector.
• Subtlety of Entanglement: The authors demonstrate that this LRE is "subtle." Unlike other forms of LRE, it is not detected by long-range connected correlation functions. The paper proves that $\rho_{TI}$ has exponentially decaying connected correlation functions for any local operators, meaning standard diagnostics for LRE (like non-vanishing long-range correlations) fail to detect this entanglement.
• Thermalization Context: The authors identify that these SWSSB states arise naturally as steady states of non-local quantum channels, specifically in the context of thermalizing Floquet dynamics and dual-unitary circuits, where the steady states are spanned by temporal shift operators.

Significance and Claims
The paper claims to identify a distinct class of LRE mixed states enforced solely by translation symmetry, which is anomaly-free and admits symmetric product states. This challenges the intuition that the existence of symmetric product states implies the symmetry sector can be spanned by them.

The significance lies in:

1. Redefining LRE in Mixed States: It shows that LRE can exist without strong anomalies or non-abelian symmetries, arising instead from the geometric constraints of translation symmetry on the Hilbert space dimension.
2. Undetectability: It highlights a form of entanglement that is invisible to local correlation functions, suggesting that current diagnostic tools for mixed-state entanglement may be insufficient for translation-invariant systems.
3. Connection to SWSSB: It provides a rigorous characterization of the fixed-point state for strong-to-weak spontaneous symmetry breaking of translation symmetry, proving it is inherently long-range entangled.

The authors remain modest regarding broader implications, noting that while their proofs are in 1D, they generalize to higher dimensions. They suggest future work could explore the connection between this counting argument and the "on-siteability" of non-invertible symmetries, but do not propose specific experimental realizations beyond the theoretical context of Floquet thermalization.