Reeh–Schlieder property in a separable Hilbert space

As has been discussed here before, the Reeh–Schlieder theorem is an initially confusing property of the vacuum in quantum field theory. It is difficult to find an illuminating discussion of it in the literature, whether in the context of algebraic QFT (from which it originated) or the more modern QFT grounded in RG and effective theories. I expect this to change once more field theorists get trained in quantum information.

The Reeh–Schlieder theorem states that the vacuum \vert 0 \rangle is cyclic with respect to the algebra \mathcal{A}(\mathcal{O}) of observables localized in some subset \mathcal{O} of Minkowski space. (For a single field \phi(x), the algebra \mathcal{A}(\mathcal{O}) is defined to be generated by all finite smearings \phi_f = \int\! dx\, f(x)\phi(x) for f(x) with support in \mathcal{O}.) Here, “cyclic” means that the subspace \mathcal{H}^{\mathcal{O}} \equiv \mathcal{A}(\mathcal{O})\vert 0 \rangle is dense in \mathcal{H}, i.e., any state \vert \chi \rangle \in \mathcal{H} can be arbitrarily well approximated by a state of the form A \vert 0 \rangle with A \in \mathcal{A}(\mathcal{O}). This is initially surprising because \vert \chi \rangle could be a state with particle excitations localized (essentially) to a region far from \mathcal{O} and that looks (essentially) like the vacuum everywhere else. The resolution derives from the fact the vacuum is highly entangled, such that the every region is entangled with every other region by an exponentially small amount.

One mistake that’s easy to make is to be fooled into thinking that this property can only be found in systems, like a field theory, with an infinite number of degrees of freedom. So let me exhibitMost likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code…a   a quantum state with the Reeh–Schlieder property that lives in the tensor product of a finite number of separable Hilbert spaces:

    \[\mathcal{H} = \bigotimes_{n=1}^N \mathcal{H}_n, \qquad \mathcal{H}_n = \mathrm{span}\left\{ \vert s \rangle_n \right\}_{s=1}^\infty\]

As emphasized above, a separable Hilbert space is one that has a countable orthonormal basis, and is therefore isomorphic to L^2(\mathbb{R}), the space of square-normalizable functions. Thus \mathcal{H} could be, for instance, the Hilbert space of a finite number of oscillators in a chain.

First, consider the normalized, Bell-ish state

    \[\vert \eta \rangle = \frac{1}{\sqrt{2}}\vert 1,1 \rangle + \frac{1}{\sqrt{4}} \vert 2,2 \rangle + \frac{1}{\sqrt{8}} \vert 3,3 \rangle +\cdots = \sum_{s=1}^\infty \frac{1}{\sqrt{2^{s}}} \vert s,s \rangle\]

in \mathcal{H}_1 \otimes \mathcal{H}_2. It’s easy to see that any basis state \vert s\rangle_2 \in \mathcal{H}_2 can be obtained by projecting \vert \eta \rangle onto the corresponding state in \mathcal{H}_1 as so: (2^{s/2} \,{}_1\langle s \vert ) \vert \eta \rangle = \vert s \rangle_2. More generally, we can get any state \vert \chi \rangle = \sum_{s,s'} c_{s,s'}\vert s,s'\rangle \in \mathcal{H}_1 \otimes \mathcal{H}_2 by actingNote that this sort of “acting” on \vert \eta \rangle is possible mathematically but not physically. That is, it’s tempting to think that a local agent could “act” on the first system \mathcal{H}_1 with a local operator like A^{(s)} \otimes I = {\sqrt{2^{s}} \vert s \rangle_1 \langle s \vert \otimes I to ensure the creation of any (normalized) state \vert s \rangle_2 on the second system \mathcal{H}_2. But this operator simply does not correspond to a physically realizable action that can be taken by a local agent! Indeed, such an ability would allow superluminal signaling. Rather, the agent on the first system can only perform unitaries or make (POVM/PVM) measurements. For instance, the agent could applying a local unitary that evolves the state \vert s=2 \rangle_1 to \vert s=99 \rangle_1, but now if you got outcome 99 from a measurement of s in the first system (which you would with reasonably high probability 1/4) this would not imply the second system is in the state \vert s=99 \rangle_2. In fact, it would still just be in the state \vert s=2 \rangle_2.b   on \vert \eta \rangle with the operator A^{(\chi)} \otimes I where A^{(\chi)} = \sum_{s,s'} 2^{s'/2} c_{s,s'} \vert s\rangle_1 \langle s' \vert.

Our objective is to extend this to a state with more than two parts, so lets start with three: \mathcal{X}, \mathcal{Y}, and \mathcal{Z}. If the Hilbert space of each part — henceforth “site” — were single dimensional, then the Reeh–Schlieder property is enjoyed trivially by the (only) state \vert 1,1,1\rangle. But if each site has at least a 2-dimensional subspace, then each pair of sites like \mathcal{X} \otimes \mathcal{Y} has at least a 4-dimensional subspace, and we’d want our “vacuum” state \vert \Omega \rangle to contain components like

(1)   \begin{align*} \vert 1,1,3 \rangle, \qquad \vert 1,2,4 \rangle, \qquad \vert 2,1,5 \rangle, \qquad \vert 2,2,6 \rangle. \end{align*}

so that we can pick them out with projectors on \mathcal{Z}. To do this, we have expanded the dimensionality of \mathcal{Z} to 6 in order to ensure there’s a state we can project on to get any state of the 4-dimensional subspace of \mathcal{X} \otimes \mathcal{Y}.

We’d want this to be symmetric with respect to permuting the three systems, so we’d like to also include the states

(2)   \begin{align*} \vert 1,3,1 \rangle, \qquad \vert 1,4,2 \rangle, \qquad \vert 2,5,1 \rangle, \qquad \vert 2,6,2 \rangle,\\ \vert 3,1,1 \rangle, \qquad \vert 4,1,2 \rangle, \qquad \vert 5,2,1 \rangle, \qquad \vert 6,2,2 \rangle. \end{align*}

where we have expanded the dimensionality of \mathcal{X} and \mathcal{Y} similarly. If our vacuum was proportional to a sum of these 12 components, Eq. (12), then it would be possible to project on any one of the three sites and get (up to normalization) any of the 8 states state where each site is either \vert 1 \rangle or \vert 2 \rangle. For instance, to get state \vert 1,2,1 \rangle by acting only on \mathcal{X}, we would apply the operator \vert 1 \rangle_{\mathcal{X}} \langle 5 \vert \otimes I_{\mathcal{Y}\mathcal{Z}} to the vacuum \vert \Omega \rangle, which picks out the state \vert 5,2,1 \rangle and flips the “5” to a “1” at \mathcal{X}.

But of course, now that each site has a 6-dimensional Hilbert space, we need to include another level of states

(3)   \begin{align*} \vert 1,1,\phantom{1}7 \rangle,\quad \vert 1,2,\phantom{1}8 \rangle,\quad \vert 1,3,\phantom{1}9 \rangle,\quad & \vert 1,4,10 \rangle,\quad \vert 1,5,11 \rangle,\quad \vert 1,6,12 \rangle, \\ \vert 2,1,13 \rangle,\quad \vert 2,2,14 \rangle,\quad \vert 2,3,15 \rangle,\quad& \vert 2,4,16 \rangle,\quad \vert 2,5,17 \rangle,\quad \vert 2,6,18 \rangle, \\ \vdots \quad &\\ \vert 6,1,37 \rangle,\quad \vert 6,2,38 \rangle,\quad \vert 6,3,39 \rangle,\quad& \vert 6,4,40 \rangle,\quad \vert 6,5,41 \rangle,\quad \vert 6,6,42 \rangle. \end{align*}

to allow us to select, by acting only on \mathcal{Z}, any of the 6^2 = 36 states formed from the two 6-dimensional subspaces of \mathcal{X} and \mathcal{Y}. This would be complemented by the other two sets obtained by swapping \mathcal{X} or \mathcal{Y} for \mathcal{Z}, making each site 42-dimensional because 42=2+6^2=2+(2+2^2)^2.

Continuing this process ad infinitum, and generalizing from 3 sites to an arbitrary number N, we would constructFor compactness, we’ve switched to the (CS) convention where the indices start from 0 rather than 1.c   our vacuum state as a sum of the above collections of states, appropriately weighted:

(4)   \begin{align*} \vert \Omega \rangle =  \sum_{\ell = 1}^{\infty} g_\ell\sum_{n=0}^{N-1} \sum_{s_1 = 0}^{d_{\ell}-1} \cdots \sum_{s_{N-1} = 0}^{d_{\ell}-1} \left\vert R_n \left(s_1, \ldots, s_{N-1}, d_{\ell}+\sum_{m=1}^{N-1} d_\ell^{m-1}s_m\right)\right\rangle  \end{align*}

where d_{\ell+1} = d_{\ell} + d_{\ell}^{N-1}, d_1 = 2 is the (recursively defined) dimensionality of each “level”Our explicit construction above got to level \ell = 2.d  , g_\ell = 1/\sqrt{N2^\ell d_\ell^{N-1}} are normalization coefficients, and R_n(b_0,\ldots,b_{N-1}) = (b_n,\ldots,b_{N-1},b_1,\ldots,b_{n-1}) is the function that cyclically shifts an N-tuple n places to the left. The state \vert \Omega \rangle is normalized and manifestly invariant under permutation of its N subsystems

The careful reader will notice that if \vert \Omega \rangle contains the entire tower of levels labeled by \ell, then we can no longer act on one site to cleanly project out, from level \ell, an arbitrary joint state of the other sites because we will necessarily pick up some contributions from higher levels (i.e., larger \ell). However, the coefficients g_\ell necessarily fall off exponentially. Therefore, for any desired degree of accuracy, we can always project onto a component from some level large enough that the higher levels have negligible relative norm. A little thought also shows we can use the same trick employed with the Bell-ish state \vert \eta \rangle to act on \vert \Omega \rangle with a single-site operator to get any state in the full Hilbert space (up to arbitrarily smaller error). In this sense we have constructed a state that has the Reeh–Schlieder property on a finite number of degrees of freedom.

Of course, to create a normalized state on the other sites from a projected component of \vert \Omega \rangle, we’d need to multiply by roughly g_\ell^{-1}. That means the norm of our local operator A generically needs to be exponentially large to get \vert \chi \rangle = A\vert \Omega \rangle. Physically, this would correspond to performing a local measurement and obtaining, with exponentially small probability, an outcome that assures you that an arbitrary state has been prepared elsewhere. (See previous post.)

Note that a state satisfying the Reeh–Schlieder property has all possible forms of multi-partite entanglement “in its belly”, e.g., EPR pairs, GHZ states, W states, etc. That is, it’s possible to distill any sort of entanglement with an exponential number of copies of the state. Presumably, it’s also straightforward to modify such a state to exhibit the so-called “split structure”.

Edit: Thank you to Zoltan Zimboras for pointing me to Clifton et al., which includes a very similar construction of an N-partite state with the Reeh-Schlieder property. (The arXiv version calls it “hyperentanglement”, but the PRA calls it “superentanglement”; I guess the editors made them tone it down :). I think my construction is a lot easier to read!

[I thank Peter Morgan for discussion.]

Footnotes

(↵ returns to text)

  1. Most likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code…
  2. Note that this sort of “acting” on \vert \eta \rangle is possible mathematically but not physically. That is, it’s tempting to think that a local agent could “act” on the first system \mathcal{H}_1 with a local operator like A^{(s)} \otimes I = {\sqrt{2^{s}} \vert s \rangle_1 \langle s \vert \otimes I to ensure the creation of any (normalized) state \vert s \rangle_2 on the second system \mathcal{H}_2. But this operator simply does not correspond to a physically realizable action that can be taken by a local agent! Indeed, such an ability would allow superluminal signaling. Rather, the agent on the first system can only perform unitaries or make (POVM/PVM) measurements. For instance, the agent could applying a local unitary that evolves the state \vert s=2 \rangle_1 to \vert s=99 \rangle_1, but now if you got outcome 99 from a measurement of s in the first system (which you would with reasonably high probability 1/4) this would not imply the second system is in the state \vert s=99 \rangle_2. In fact, it would still just be in the state \vert s=2 \rangle_2.
  3. For compactness, we’ve switched to the (CS) convention where the indices start from 0 rather than 1.
  4. Our explicit construction above got to level \ell = 2.
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