The Reeh–Schlieder Property

The Reeh–Schlieder property says that acting on the vacuum with observables from any suitable nonempty local region produces a dense set of vectors in the global Hilbert space. For bounded regions the vacuum is also separating: no nonzero local observable annihilates it. These are exact statements about an algebra–state pair under spectrum, additivity, causality, or analytic-microlocal hypotheses. They neither say that every state has the property nor permit a controllable superluminal signal.

Required background. Quantum Information supplies local operations, postselection, and the distinction between selective and trace-preserving processes.

Helpful background. Flat-Spacetime Quantum Field Theory provides the vacuum spectrum condition; Partial Differential Equations and Green Functions defines field equations and unique continuation; Causality and Global Structure establishes causal complements; Quantum Fields in Curved Spacetime provides local covariance and Hadamard or analytic-microlocal state conditions.

For orientation, a vector $\Omega\in\mathcal H$ is cyclic for an algebra $\mathcal M$ when $\overline{\mathcal M\Omega}=\mathcal H$, and separating when $A\Omega=0$ with $A\in\mathcal M$ implies $A=0$. These are independent definitions. Cyclicity for $\mathcal M'$ implies separatingness for $\mathcal M$, a fact used below.

The flat-spacetime theorem

Consider a vacuum representation of a local net in Minkowski spacetime. Assume positive energy, translation covariance, weak additivity, and Einstein causality. If $O$ is a bounded causally convex open region, then

$$\overline{\{A\Omega:A\in\mathcal A(O)\}}=\mathcal H, \qquad A\Omega=0\Longrightarrow A=0.$$

The first assertion is cyclicity; the second is separatingness. Fewster and Rejzner state both conclusions, with weak additivity and the spectrum condition explicit, in §5.2, Theorem 26. Boundedness matters to the second step because one needs a nonempty causally disjoint region whose algebra commutes with $\mathcal A(O)$.

Density is a norm statement about vectors, not a promise that a desired vector can be prepared with a bounded operator of modest norm or with appreciable probability. For each target $\Psi$ and tolerance $\epsilon>0$, cyclicity gives some $A\in\mathcal A(O)$ with $\lVert A\Omega-\Psi\rVert<\epsilon$. The theorem supplies no uniform bound on $\lVert A\rVert$, no efficient construction, and no lower bound on the success probability when $A$ is implemented as one outcome of a physical instrument.

Why analyticity creates density

The proof architecture explains the surprising reach of a small region. Suppose $\Psi$ is orthogonal to every $A\Omega$ with $A$ localized in a slightly smaller region $O_1\Subset O$. Translate finitely many local operators through small vectors $x_j$ while remaining inside $O$. Matrix elements of the form

$$F(x_1,\ldots,x_n) =\langle\Psi,U(x_1)A_1U(x_2)A_2\cdots U(x_n)A_n\Omega\rangle$$

vanish in an open set. Positive energy extends $F$ holomorphically into a forward tube. Edge-of-the-wedge uniqueness then forces the boundary value to vanish for all translations. Weak additivity makes translated local products weakly dense in the global algebra, while the GNS vector is globally cyclic; hence $\Psi=0$. The orthogonal complement is trivial, proving local cyclicity Fewster and Rejzner 2020, §5.2, Theorem 26.

Separatingness has a shorter causal proof. If $A\in\mathcal A(O)$ and $A\Omega=0$, choose $O_2$ causally disjoint from $O$. For every $B\in\mathcal A(O_2)$,

$$A B\Omega=B A\Omega=0.$$

Cyclicity of $\Omega$ for $\mathcal A(O_2)$ makes the vectors $B\Omega$ dense, so the bounded operator $A$ vanishes. This derivation displays the two distinct inputs: analyticity or spectral support establishes cyclicity, while locality transfers cyclicity of the complement into separatingness.

Curved-spacetime versions and their hypotheses

A generic curved spacetime has no translation group and no global energy spectrum, so the flat proof cannot simply be copied. On real analytic spacetimes, the analytic microlocal spectrum condition replaces forward-tube analyticity. For free fields, analytic wave-front control and unique continuation can yield the Reeh–Schlieder property; quasifree analytic ground and KMS states in suitable stationary analytic settings provide important examples Strohmaier, Verch, and Wollenberg 2002, complete article, pp. 5514–5530. This is stronger than the smooth Hadamard condition, so “Hadamard” alone must not be silently substituted for “analytic microlocal.”

There is also a deformation route based on local covariance and the time-slice axiom. If two globally hyperbolic spacetimes have diffeomorphic Cauchy surfaces, a chain of Cauchy morphisms can transport algebraic state information between them. Sanders proves existence of states that are Reeh–Schlieder for specified bounded causally convex regions under the locally covariant and time-slice hypotheses; with causality, the vector is cyclic and separating for the corresponding local von Neumann algebra Sanders 2009, §4, Theorem 4.1 and Corollary 4.2.

The stronger assertion that one state is Reeh–Schlieder for all regions requires additional assumptions, including noncompact diffeomorphic Cauchy surfaces, local quasi-equivalence, a maximal state space, and an input Reeh–Schlieder state. Its physical regularity is a separate question; Sanders explicitly notes that most Reeh–Schlieder states produced abstractly need not be Hadamard Sanders 2009, §4, Theorem 4.3 and Corollary 4.4. Thus neither local deformation nor analytic continuation proves the property for every curved-spacetime state.

Why no signaling follows

Let $O_A$ and $O_B$ be spacelike separated, $B\in\mathcal A(O_B)$, and let a nonselective operation in $O_A$ have Kraus operators $C_i\in\mathcal A(O_A)$ with $\sum_i C_i^*C_i=\mathbf1$. Local commutativity gives

$$\sum_i\langle\Omega,C_i^* B C_i\Omega\rangle =\langle\Omega,B\sum_i C_i^*C_i\Omega\rangle =\langle\Omega,B\Omega\rangle.$$

Therefore the statistics observable in $O_B$ do not change. A selected outcome can change the conditional state assigned at $O_B$, just as in ordinary entanglement, but the distant observer cannot know which subensemble occurred until an ordinary causal message arrives. Reeh–Schlieder does not permit or enable a superluminal signal.

The success-probability issue can be made quantitative. Suppose $A\Omega$ approximates a normalized target. For $c\ge\max(1,\lVert A\rVert)$, $C=A/c$ is a valid contraction, with successful branch probability

$$p_C=\lVert C\Omega\rVert^2 =\frac{\lVert A\Omega\rVert^2}{c^2}.$$

The other Kraus operator $D=(\mathbf1-C^*C)^{1/2}$ belongs to the same local von Neumann algebra and completes a trace-preserving instrument. If the approximation requires large $\lVert A\rVert$, its successful branch can be very unlikely. Keeping both outcomes gives precisely the unchanged remote statistics derived above. The theorem promises neither a favorable probability nor control over which outcome occurs.

Countermodel and scope check

Take a finite bipartite system $\mathcal H_A\otimes\mathcal H_B$, local algebra $\mathcal B(\mathcal H_A)\otimes\mathbf1$, and product vector $\Omega=|0_A\rangle\otimes|0_B\rangle$. Its local orbit is

$$\{(A\otimes\mathbf1)\Omega\} =\mathcal H_A\otimes\operatorname{span}\{|0_B\rangle\},$$

which is not dense if $\dim\mathcal H_B>1$. A nonzero projection in $A$ that annihilates $|0_A\rangle$ also annihilates $\Omega$, so the vector is not separating. This is not a counterexample to the QFT theorem: the product state lacks the vacuum’s spectral–analytic entanglement and the finite subsystem net lacks weak additivity across translations. It shows why the hypotheses, rather than “infinitely many degrees of freedom” alone, carry the conclusion.

Conversely, a full-Schmidt-rank vector in equal finite factors can be cyclic and separating. Reeh–Schlieder-like algebraic properties are therefore not by themselves signatures of relativistic locality. What is special in QFT is their occurrence for every suitable small region in a physically distinguished state under causal and spectral or microlocal conditions.

Solved diagnostic

A vacuum vector is cyclic and separating for $\mathcal A(O)$. An experimenter in $O$ claims: “For any remote target vector I can choose a deterministic local channel that prepares it to arbitrary accuracy, so I can signal.” Locate both errors.

Solution

Cyclicity provides algebraic vectors $A\Omega$ dense in Hilbert space; it does not make $A$ a deterministic trace-preserving channel or bound its norm and success probability. Normalizing $A$ generally describes a selected outcome. When all outcomes of a local instrument are retained, commutativity and completeness therefore leave every spacelike observable’s statistics unchanged. A remote observer can sort the conditionally altered ensemble only after learning the local outcome through a causal communication channel. Separatingness adds that no nonzero local observable annihilates the vacuum; it does not upgrade postselection into control.

References

• Fewster, Christopher J., and Katarzyna Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI.
• Sanders, Ko. “On the Reeh–Schlieder Property in Curved Spacetime.” Communications in Mathematical Physics 288 (2009): 271–285. DOI; Open PDF.
• Strohmaier, Alexander, Rainer Verch, and Manfred Wollenberg. “Microlocal Analysis of Quantum Fields on Curved Spacetimes: Analytic Wave Front Sets and Reeh–Schlieder Theorems.” Journal of Mathematical Physics 43, no. 11 (2002): 5514–5530. DOI; Open PDF.