Nature of the $W$-state in the thermodynamic limit

Consider a matrix product state on $$\mathbb{C}^{d N}$$:

$$\Psi = \sum_{\sigma_1,...\sigma_N} A_1(\sigma_1) ... A_N(\sigma_N) |\sigma_1 ... \sigma_N \rangle \quad \quad (\text{OBC MPS})$$

with some matrices $$D_{i-1} \times D_{i}$$-matrices $$A_i(\sigma_i)$$ and $$D_1 = 1, D_N = 1$$. The bond dimension is then $$D = \max_{i} D_i$$.

Such a state may always be written as a translation invariant matrix product state, that is, in the form

$$\Psi = \sum_{\sigma_1,...,\sigma_N} \text{tr}(B(\sigma_1) \cdots B(\sigma_N) )|\sigma_1 ... \sigma_N \rangle \quad \quad (\text{TI MPS})$$

where the $$B(\sigma)$$ are $$D'\times D'$$-matrices. For general MPS, we have to chose $$D' \in \mathcal{O}(N D)$$. This is even true for TI MPS representations of translation invariant states. To be more explicit: In the cited article they consider the $$W$$-state:

$$W_N := \frac{1}{\sqrt{N}} \left( |10000 \cdots 0 \rangle + |0100 \cdots 0 \rangle + |0010 \cdots \rangle + \cdots |0000 \cdots 1\rangle \right) \ .$$

They then argue that any TI MPS representation of the $$W$$-state needs bond dimension at least of order $$N^\frac{1}{3}$$, although there is a OBC MPS representation with bond dimension $$2$$. They prove this from a conjecture about injectivity of MPS.

I came across a peculiarity of this state which i will describe below; i would be glad for somebody putting this into perspective.

Namely, it seems to me that the thermodynamic limit of the W-state is a mean-field state (a MPS of bond dimension 1). To see this i will use the following facts

1. There is the following Schmidt decomposition

$$W_{k+1} = \sqrt{1 - \frac{1}{k+1}} |0\rangle \otimes W_k + \frac{1}{\sqrt{k+1}} |1\rangle \otimes \Omega_k \ , \quad \Omega_k := |0\rangle^{\otimes k} \ .$$

1. It holds that

$$W_N = ( v_L, B_1 \cdots B_N v_R ) \ ,$$

where

$$v_L = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \ , \quad v_R = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \ , \quad B_k = \begin{pmatrix} \sqrt{1-c_k}|0\rangle & \sqrt{c_k}|1\rangle \\ 0 & |0\rangle \end{pmatrix} \ , \quad c_k := \frac{1}{N-k+1} \ .$$

1. Introduce the linear map on operators on $$\mathbb{C}^2$$, indexed by a $$2\times2$$-matrix $$a$$:

$$\mathbb{B}_{\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}}[k] \begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} = \sum_i B_k(i) x B_k(i)^* a_{ji} = \\ = \begin{pmatrix} a_{11}(1-c_k)x_{11} + a_{22} c_k x_{22} + \sqrt{c_k(1-c_k)}(a_{12} x_{21} + a_{21} x_{12}) & a_{11}\sqrt{1-c_k}x_{12} + a_{12} \sqrt{c_k} x_{22} \\ a_{11}\sqrt{1-c_k}x_{21} + a_{21} \sqrt{c_k} x_{22} & a_{11} x_{22} \end{pmatrix} \ .$$

Those people which are familar with MPS know that this object allows to compute expectation values of observables.

I will use two further facts, with $$\mathbb{B}[k] = \mathbb{B}_1[k]$$:

1. $$\mathbb{B}[k] \circ \cdots \circ \mathbb{B}[N](v_R \otimes v_R^*) = 1 \ .$$
2. $$(v_L, \mathbb{B}[1] \circ \cdots \circ \mathbb{B}[k](X) v_L) =\text{tr}(qX) + (1-k/N) \text{tr}(\sigma_3 X) \ , \quad q = \frac{1 - \sigma_3}{2} \ .$$

This finally allows to compute the expectation value of an observable $$A = a_1 \otimes \cdots a_{2k}$$, supported on sites $$N/2 -k,...N/2+k$$, where we take $$N$$ even for simplicity:

$$\langle W_N, A W_N \rangle = \text{tr}\left(\left[q + \left(\frac{1}{2} + \frac{k-1}{N}\right) \sigma_3 \right] \mathbb{B}[N/2-k]_{a_1} \cdots \mathbb{B}[N/2+k]_{a_{2k}}(1)\right) \ .$$

For $$N \rightarrow \infty$$ with $$k$$ finite:

$$\mathbb{B}_a[N/2+k](x) \rightarrow (v_L,a v_L) x \ ;$$

so that

$$\langle W_N, A W_N \rangle \rightarrow \prod_{l=1}^{2k} (v_L,a_l v_L) \ .$$

Sorry for the long and tedious calculation which also has a lot of steps missing, but i thought the result to be too odd to just state it. The weird thing is that for finite $$N$$, this $$W_N$$-state can be represented as a non-manifestly translationally invariant MPS of bond dimension $$2$$ or as TIMPS of bond dimension growing with $$N$$. But taking the limit $$N\rightarrow \infty$$, we can represent it as a mean-field state (with bond dimension $$1$$).

What is happening here?

Maybe you're asking to resolve the tension between the following facts:

(1) For finite length $$N$$, the $$W$$-state $$|W_N\rangle = \frac{1}{\sqrt{N}}(|10...0\rangle + |010...0\rangle + ... + |00...01\rangle)$$ is not a product state. (In particular, it has up to one bit of entanglement on sub-regions, and an MPS representation requires bond dimension at least two.)

(2) In the thermodynamic limit $$N \to \infty$$, the $$W$$-state is "like a product state."

There's no direct paradox of course, but maybe it will be comforting to explore fact (2) a bit. First note that for any fixed $$k$$ while $$N \to \infty$$, the reduced density matrix $$\rho_A=\textrm{Tr}_{\bar{A}} |W_N\rangle\langle W_N|$$ on a region of size $$k$$ approaches $$(|0\rangle\langle0|)^{\otimes k}$$. So the $$W$$-state looks $$\textit{locally}$$ like the product state $$|0...0\rangle$$ as $$N \to \infty$$, even though the full state is in fact orthogonal to $$|0\rangle^{\otimes N}$$ for any finite $$N$$.

The situation here is actually similar to considering a particle on a line, with wavefunction $$|\psi\rangle \in L^2(\mathbb{R})$$, in the case that $$|\psi\rangle$$ is uniformly spread over the interval $$[-N,N]$$. As $$N \to \infty$$, the probability of the particle being in any fixed finite region tends to zero, so the state becomes locally indistinguishable from the state with no particles. (Well, the state of no particles'' is not actually represented in $$L^2(\mathbb{R})$$.) Likewise, one can think of the $$W$$-state as an excitation (spin flip) with uniformly spread wavefunction.

We can also formalize the sense in which the $$W$$-state becomes equal to the $$|0\rangle^{\otimes N }$$ in the strict $$N=\infty$$ limit, even though the states are orthogonal for all finite $$N$$. The formalization depends on how you choose to define the algebra of observables and the space of states for $$N=\infty$$. (The naïve definitions involving infinite tensor products are a bit tricky, because then inner products and norms can yield infinities.) One common choice is to define the algebra of observables as the so-called “quasi-local” algebra, which is (the completion of) the space of finite range (i.e. local) observables. See e.g. [1]. Then one usually defines “states” as positive linear functionals on the algebra of observables (mapping observables to expectation values). In that case, a state is fully defined by how it assigns expectations to only $$\textit{finite-range}$$ observables. Then a sequence of $$W$$-states for increasing $$N$$ really does approach the $$|0\rangle^{\otimes N }$$ state, because the W-states act identically on finite-range observables in the $$N \to \infty$$ limit . And in fact there is no “analog” of the $$W$$-state for $$N=\infty$$, besides the $$|0\rangle^{\otimes N }$$ state: the former doesn’t really exist except insofar as we choose to define it by the latter.

[1] “Quantum Spin Systems on Inﬁnite Lattices,” Pieter Naaijkens, https://arxiv.org/pdf/1311.2717.pdf