1 Introduction↩︎

Quantum Field Theory (QFT) has proven very successful in modeling particle scattering experiments at high energies. However, as noted by Sorkin [1], the use of seemingly innocuous quantum operations within the QFT formalism would allow three or more separate parties to violate Einstein’s causality. The accepted resolution of Sorkin’s paradox is to acknowledge that the set of quantum operations which can be conducted within a finite region of space-time is smaller than previously envisaged. This raises the question of how to model local operations within QFT.

This problem has been greatly advanced in recent years. In [2], Jubb studies which quantum channels (or update rules) do not lead to Sorkin-type violations of causality in QFT, a property he calls strong causality. Among other things, he concludes that weak “Gaussian” measurements of locally smeared fields are strongly causal. Oeckl, who calls this property causal transparency, reaches the same conclusion in [3].

Independently, Fewster and Verch proposed in [4], [5] a general framework to model measurements within QFT. In a nutshell, the FV framework consists in making the ‘target’ (or ‘system’) field theory interact in a compact region with another ‘probe’ field theory. Measurement outcomes are obtained by measuring the probe in an arbitrary bounded region outside the causal past of the interaction region. The probe is then discarded, i.e., one is not allowed to measure it a second time. Remarkably, all quantum operations achievable within the FV framework are strongly causal [6]. Moreover, they suffice to carry out full tomography when the target QFT is a scalar field [7]. However, as we argue later on, this is different from modeling actual measurements. In this regard, the FV framework presents at least two unresolved issues:

• First, given the specification of a quantum measurement, or even of the corresponding quantum instrument, it is not straightforward to determine whether the FV framework allows implementing one or the other—even approximately. Doing so involves specifying a probe QFT, its initial state, the form of the interaction, and the measurement on the probe, or proving that no such elements exist.

• Second, if we regard the probe QFT as a physical entity and not as part of a mere mathematical construction to obtain a set of strongly causal operations, then it is unclear how come one is allowed to measure the probe arbitrarily; a standing assumption of the framework. Consistency therefore demands that the probe measurement itself be modeled within the FV framework. That would involve introducing another probe, whose measurement should be realizable within the FV framework, and so on. In other words, any physical measurement should be expressible as an arbitrarily long chain of consecutive interactions with independent probes, in such a way that we predict the same measurement statistics irrespective of where in the chain we invoke Born’s rule.

  In non-relativistic quantum mechanics, the specific probe within a measurement chain where one applies Born’s rule is known as “Heisenberg cut". That the position of the cut can be chosen at will was already observed by von Neumann [8]. What we have argued above is that, if the FV framework is fundamental and its probe field is meant to be physical, then feasible QFT measurements should have a similarly movable”FV-Heisenberg cut". This leads us to ask: Are there non-trivial FV-realizable measurements with a movable FV-Heisenberg cut?

In this paper, we address the first problem by proving that Gaussian-modulated measurements of a locally smeared quantum field can be modeled within the FV framework. As it turns out, the probe measurement that induces a Gaussian-modulated field measurement can itself be chosen a Gaussian-modulated field measurement. This solves the second problem: Gaussian field measurements are consistent with the FV framework, even if we accept the physicality of the probe field. As a byproduct, we establish that projective field measurements can also be modeled within the FV framework.

The structure of this paper is as follows. In 2 we specify the linear QFT setting to which our results apply. In 3 we introduce how to model general as well as Gaussian-modulated measurements in QFT, distinguishing between positive-operator-valued measures (POVMs) and quantum instruments, and illustrate how this differs from state tomography. In 4 we summarize the FV framework and characterize the POVMs and instruments that are realizable within the framework in order to state our main results (1). Finally, in 5 we prove our results by presenting a concrete scheme to induce Gaussian-modulated measurements, projective measurements, and more—even iteratively. We conclude in 6. In Appendix 7 we review the construction of the Weyl CCR algebra of a linear scalar field. In Appendix 8, we establish that state transformations induced by finite-rank perturbations of the classical phase space underlying a linear scalar field preserve the Hadamard property. In Appendix 9 we collect some proofs on (dephased) Gaussian instruments.

2 Technical setting for quantum field theory↩︎

We will be dealing with the standard framework of AQFT: namely, starting from a fixed globally hyperbolic spacetime manifold \({\mathscr M}\), we associate to any region \({\mathscr R}\subset {\mathscr M}\) a unital *-algebra \({\cal A}({\mathscr R})\). These local algebras, which define our QFT, satisfy the usual axioms: isotony, microcausality, time-slice and Haag’s property; confer [5] and references therein for details. States of this QFT are linear, positive functionals \(\omega:{\cal A}\to\mathbb{C}\) on the algebra of observables \({\cal A}:={\cal A}({\mathscr M})\) with \(\omega(1)=1\). An element \(A \in {\cal A}\) is said to be localizable in region \({\mathscr R}\subset {\mathscr M}\) if \({A}\in{\cal A}({\mathscr R})\). To express limits, we assume \({\cal A}\) and all \({\cal A}({\mathscr R})\) are equipped and completed with respect to a suitable topology. Specifically, we work with von-Neumann algebras; for some fixed Hilbert space \(\mathcal{H}\) these are *-subalgebras of \(B(\mathcal{H})\) closed under the strong-operator topology¹. Limits, infinite sums, and integrals pertaining to \({\cal A}\) will be taken with respect to this topology, with integrals understood as Bochner integrals in this topology. Note that in this setting, any density matrix, i.e., \(\rho \in B(\mathcal{H})\) such that \(\rho \geq 0\) and \(\operatorname{tr} \rho = 1\), gives rise to a state \(\omega_\rho\) on \({\cal A}\) through \(\omega_\rho({A}) :=\operatorname{tr} (\rho {A})\).

Our main study will be conducted in the setting of linear scalar QFT. In this setting the smeared field \(\Phi(f)\) corresponds to the quantization of \[\phi(f) = \int_{\mathscr M}\phi(x) f(x) \mathrm{vol}(x) =: \braket{\phi,f},\] a classical point field \(\phi\) satisfying a classical linear differential equation \(T\phi(x)= 0\) and smeared over spacetime through a test function \(f\) using \(\mathrm{vol}\), the volume form² on \({\mathscr M}\). We assume that \(T\) is symmetric with respect to \(\braket{,}\) and normally hyperbolic, which implies it has well-defined retarded/advanced Green operators \(E^\pm\) and associated causal propagator \(E= E^--E^+\) [9]. This holds, e.g., for the Klein-Gordon operator \(T = \square_{\mathscr M}- m^2\), \(m > 0\). We denote the space of real-valued smooth compactly supported functions on \({\mathscr M}\) by \({\mathcal{D}}({\mathscr M})\) and by \({\mathcal{D}}({\mathscr R})\) if restricting to functions having support in \({\mathscr R}\subset {\mathscr M}\). With \(\phi\) having spatially compact support, it is natural to identify \(Tf \sim 0\) for all \(f \in {\mathcal{D}}({\mathscr M})\) since \(\phi(Tf) = \braket{T\phi,f} = 0\) by partial integration. We denote the resulting quotient space by \(\mathcal{C}_T :={\mathcal{D}}({\mathscr M})/T{\mathcal{D}}({\mathscr M})\) and say that \(f \in {\mathcal{D}}({\mathscr M})\) is localizable in \({\mathscr R}\subset {\mathscr M}\) iff there exists \(g \in {\mathcal{D}}({\mathscr R})\) such that \(f \sim g\). In this case we denote \([f] \in \mathcal{C}_T({\mathscr R}) \subset \mathcal{C}_T\). The classical timeslice property [10] entails that any \(f\in {\mathcal{D}}({\mathscr M})\) is localizable in any globally hyperbolic neighbourhood \(N\) of any Cauchy surface of \(M\). We define the classical phase space as the symplectic space \((\mathcal{C}_T,E)\), where \(E\) denotes also the symplectic form \(E([f],[g]) :=\braket{f, Eg}\), \(f,g \in {\mathcal{D}}({\mathscr M})\) , associated with the causal propagator. This concludes the classical part of the construction.

Quantization may then be performed in a fully algebraic setting which we allude to in Appendix 7. For the main text, however, we are content with a fixed Hilbert space representation and make these assumptions: The smeared field \(\Phi\) is a map from \(\mathcal{C}_T\) to self-adjoint operators on a Hilbert space \(\mathcal{H}\) such that \[\begin{align} \tag{1}&\boldsymbol{[Continuity]} \, && e^{it\Phi(f)} \text{ is strongly operator continuous in }t,\\ \tag{2}&\boldsymbol{[Weyl relation]} \, &&e^{i\Phi(f)} e^{i\Phi(g)} = e^{-\frac{i}{2} \braket{f,Eg}} e^{i\Phi(f+g)}, \end{align}\] for all \(f,g \in {\mathcal{D}}({\mathscr M})\), where we agree to \(\Phi(f):=\Phi([f])\) by abuse of notation. Note here that the Weyl relation is a strong and covariant form of the canonical commutation relations. Finally, we obtain a net of von-Neumann algebras \(\{ {\cal A}({\mathscr R})\}_{\mathscr R}\subset B(\mathcal{H})\) labeled by open causally convex regions \({\mathscr R}\subset {\mathscr M}\) by defining \({\cal A}({\mathscr R})\) to be the algebra generated by the Weyl operators \(\{ e^{i\Phi(f)}: \, f \in {\mathcal{D}}({\mathscr R})\}\), equipped and completed with respect to the strong operator topology of \(B(\mathcal{H})\). These local algebras are known to satisfy the usual axioms as listed above. We will call this a linear scalar QFT (induced by \(T\)) and denote it by \({\cal A}_T\).

We conclude this section by specifying suitable states and state transformations on \({\cal A}_T\). For QFTs (at least for linear ones) there is a broad class of physically reasonable states given by the quasi-free Hadamard states. There are many such states on any globally hyperbolic spacetime [11] and for stationary spacetimes this includes ground and thermal states [12]. There is also a broad class of state transformations: Any symplectic transformation \(F\) of \((\mathcal{C}_T,E)\) induces an automorphism \(\alpha_F\) of \({\cal A}_T\) by imposing \(\alpha_F(e^{i\Phi(v)}) :=e^{i\Phi(Fv)}\), \(v \in \mathcal{C}_T\) and a new \(F\)-transformed state \(\omega_F\) via pullback, i.e., \[\label{eq:transformedstate} \omega_F(\bullet) :=\omega(\alpha_F(\bullet)).\tag{3}\] To deem a transformation of the form \[\label{eq:statetrafo} \omega \mapsto \omega_F\tag{4}\] as ‘physical’, e.g., arising as a state-update for a localized measurement, one would at least require that it preserves the class of physically reasonable states.

It is well known (confer, e.g., [13]) that the class of quasi-free states is preserved by all transformations which have the form 4 . In Appendix 8 we prove that these preserve also the Hadamard class if \(F\) transforms only a finite-dimensional subspace of \(\mathcal{C}_T\). Although possibly known to experts, this result, to the best of our knowledge, has not been addressed in the existing literature and might be of independent interest.

Since our main study does not require to work with Hadamard states, we will only define quasi-free states in more detail: A state \(\omega\) on \({\cal A}_T\) is referred to as quasi-free (sometimes also as “Gaussian”) iff there exists a positive bilinear symmetric form \(\Gamma:\mathcal{C}_T^{\times 2}\to \mathbb{R}\) such that \[\label{eq:qfree} \omega(e^{i\Phi(v)})=e^{-\frac{\Gamma(v,v)}{2}}, \quad v \in \mathcal{C}_T.\tag{5}\] Conversely, 5 defines a state on \({\cal A}_T\) iff \[\label{eq:uncertaintyrel} \Gamma(v,v) \Gamma(w,w) \geq \tfrac{1}{4} E(v,w)^2, \quad v,w \in \mathcal{C}_T.\tag{6}\] We refer to such \(\Gamma\) as a covariance. Given any symplectic transformation \(F\) of \((\mathcal{C}_T,E)\), it is straightforward to check that the \(F\)-transformed state \(\omega_F\) is also quasi-free with covariance \[\Gamma_F(v,w) :=\Gamma(Fv,Fw), \quad v,w \in \mathcal{C}_T.\] A quasi-free state is called pure iff 6 can be saturated, i.e., for any given nonzero \(v \in \mathcal{C}_T\), \[\label{eq:optuncertainty} \underset{0\neq w \in \mathcal{C}_T}{\sup}\, \frac{E(v,w)^2}{4 \Gamma(v,v)\Gamma(w,w)} = 1.\tag{7}\] Note that also pure quasi-free Hadamard states exist in arbitrary globally hyperbolic spacetimes. In stationary spacetimes, these exist since, at least for a free scalar field with a positive stationary potential, i.e., where \(T= \square_{\mathscr M}+ V\) with stationary \(0 < V \in C^\infty({\mathscr M})\), there is a unique (pure) ground state [12]. More generally, existence follows by a deformation argument³: For any globally hyperbolic spacetime \({\mathscr M}\), there is a deformation in the form of a certain time-orientation preserving diffeomorphism such that the deformed spacetime, say \({\mathscr M}'\), is static in the causal past of some Cauchy surface [14]. A quasi-free Hadamard state on \({\mathscr M}\) may then be constructed by taking suitable extensions and restrictions of a pure quasi-free Hadamard state on the static patch of \({\mathscr M}'\). That this state is also pure follows using the defining relation 7 and the classical timeslice property.

3 Quantum field measurements and instruments↩︎

We wish to model local operations within QFT, specifically measurements and associated instruments. In particular, we will use the technical setting from the previous section and fix some QFT with algebra of observables \({\cal A}\). Later on we illustrate these notions defining Gaussian operations in a linear scalar QFT.

Generally, a (quantum) measurement comes with a set of (classical) measurement outcomes \(\mathfrak{X}\) and associates with it a (classical) probability distribution which depends on the state of the (quantum) system—in our case a QFT—before the measurement. If we add a description of the state of the system after the measurement, we speak of a (quantum) instrument. To avoid technicalities, we treat the two paradigmatic cases of a finite- and a continuous-outcome measurement and refer to [15] for generalizations. Moreover, as it is convenient, we define our instruments in the Heisenberg picture acting directly on \({\cal A}\)—confer e.g. [16], [17]—and provide the associated state changes in form of state update rules.

To begin with, let \(\mathfrak{X}\) be finite. In this case, an instrument is defined by a family of linear completely positive maps \(I:=\{ I_j \}_{j\in \mathfrak{X}}\), where \(I_j :{\cal A}\to{\cal A}\) is such that \[\label{eq:sum1} \bar{I}({A}) :=\sum_j I_j ({A}) \quad\text{satisfies}\quad \bar{I}(1) = 1.\tag{8}\] The map \(\bar{I}: {\cal A}\to {\cal A}\) is called the measurement channel. Intuitively, if our QFT is initially in state \(\omega\), the instrument \(I\) describes a measurement that returns an outcome \(j\) with probability \[\label{prob:1} p(j |\omega)=\omega(I_j (1)),\tag{9}\] in which case the new state of the QFT is updated to \[\bar{\omega}_j :=\frac{1}{\omega(I_j (1))}\omega\circ I_j .\] If we ignore or discard the measurement outcome \(j\), then the new QFT state will be \[\label{eq:sum2} \bar{\omega} :=\sum_j p(j |\omega)\bar{\omega}_j =\omega\circ\bar{I}.\tag{10}\]

If we are just interested in computing the probabilities of the different outcomes, it suffices to work with the algebra elements \(M :=\{ M_j :=I_j(1) \}_j\) so that \(p(j |\omega)=\omega(M_j )\). These elements satisfy \[\label{eq:sum3} \sum_j M_j =1 \quad \text{and}\quad \forall j : \, M_j \geq 0.\tag{11}\] Any such family \(M\) is called a positive operator-valued measure (POVM).

The formalism above can also be used to model continuous-outcome measurements. In this case we suppose the set of outcomes \(\mathfrak{X}\subset \mathbb{R}\) to be a Borel subset of the real line, e.g., an interval or the full real line, we label the outcomes with indices \(s\) instead of \(j\) for distinction⁴and we replace \(\sum_{j\in\mathfrak{X}}\) by \(\int_{\mathfrak{X}} \bullet \,ds\) whenever it appears—in particular within 8 , 10 and 11 . Further, we require that \(I_s({A})\) is integrable in \(s\) for all \({A}\in {\cal A}\). Here, "integrable" refers to Bochner integrability with respect to the strong-operator topology (confer Appendix 9 for a definition). Note that this assumes more than strictly necessary to keep the statistical interpretation from above, Eqs. 9 –10 , which relies only on the existence of limits within states; for example, the weak-operator topology would be sufficient in this regard. However, our choice of topology implies the existence of the measurement channel, \(\overline{I}({A}) :=\int_{\mathfrak{X}} I_s({A}) ds\) as a completely positive map within \({\cal A}\) and we will establish this property for all given examples.

3.1 Gaussian-modulated measurements↩︎

Now, we are ready to discuss concrete examples of instruments and POVMs. In order to do this, we take \({\cal A}\) to be a linear scalar QFT as specified in the preceding section (dropping the index \(T\)) and denote the associated quantized field by \(\Phi\). We remark that expressions of the form \(\mu(\Phi(f))\) for any bounded Borel function \(\mu:\mathbb{R}\to \mathbb{C}\), by means of the spectral theorem, define elements of \({\cal A}\subset B(\mathcal{H})\) in terms of (Borel) functional calculus. From this perspective, most statements made in the remainder of this section are direct consequences of the properties of the functions \(\mu\), and we defer proofs to Appendix 9.

To start with, let \(f\in {\mathcal{D}}({\mathscr M})\) and \(\epsilon>0\), and consider the following continuous-outcome POVM on \({\cal A}\), \[M^{f,\epsilon} :=\{ M_s^{f,\epsilon}\}_{s\in\mathbb{R}}, \quad M_s^{f,\epsilon} :=\frac{1}{\sqrt{2\pi}\epsilon} e^{-\frac{(\Phi(f)-s)^2}{2\epsilon^2}}. \label{gaussian95POVM}\tag{12}\] We will call this POVM a Gaussian-modulated measurement of the field \(\Phi(f)\), or Gaussian measurement, for short. It is a special instance of a weak measurement [18]. Moreover, it approximates a projective measurement of \(\Phi(f)\) in the limit \(\epsilon \to 0\), i.e., for any Borel set \(B \subset \mathbb{R}\) and with \(\Pi_B(A)\) denoting the spectral projection of a self-adjoint operator \(A\) associated with the spectral subset \(B\), \[\label{eq:gaussPOVMlimit} \Pi_B(\Phi(f))=\underset{\epsilon\to 0}{\lim} \, \int_B M_s^{f,\epsilon} ds.\tag{13}\]

Note that there exist many different instruments implementing the Gaussian POVM 12 . One of them is: \[I^{f,\epsilon}_s(\bullet) :=\frac{1}{\sqrt{2\pi}\epsilon} e^{-\frac{(\Phi(f)-s)^2}{4\epsilon^2}} \bullet e^{-\frac{(\Phi(f)-s)^2}{4\epsilon^2}}. \label{gaussian95instrument95basic}\tag{14}\] This instrument appears in the literature of quantum optics and quantum foundations [19], [20]. Its causality properties have been recently studied in QFT, see [2] and [3].

Now, for any \(\delta>0\), define the dephasing map \[\label{eq:dephasing} D^{f,\delta}(\bullet) :=\frac{1}{\sqrt{2\pi}\delta}\int e^{-\frac{\nu^2}{2\delta^2}}e^{-i\nu\Phi(f)}\bullet e^{i\nu\Phi(f)} d\nu.\tag{15}\] This map is completely positive and unit preserving. Intuitively, it introduces decoherence between the generalized eigenstates of \(\Phi(f)\).

It is easy to verify that the instrument \[I^{f,\epsilon,\delta} :=\{ I^{f,\epsilon,\delta}_s\}_{s\in\mathbb{R}}, \quad I^{f,\epsilon,\delta}_s:=I_s^{f,\epsilon} \circ D^{f,\delta} =D^{f,\delta}\circ I_s^{f,\epsilon} \label{eq:dephasedgauss95instr}\tag{16}\] also induces the POVM \(M^{f,\epsilon}\) defined in (12 ) for any \(\delta > 0\). This example will have a prominent appearance in our measurement scheme and illustrates that different instruments might induce the same POVM.

The instrument \(I^{f,\epsilon,\delta}\) is fully characterized by its action on Weyl operators. In this regard, by a straightforward computation it follows that \[I^{f,\epsilon,\delta}_s(e^{i\Phi(h)})=\frac{1}{\sqrt{2\pi}\epsilon} e^{-\frac{\langle f,Eh\rangle^2}{8\underline{\epsilon}(\epsilon, \delta)^2}}e^{i\Phi(h)}e^{-\frac{(\Phi(f)-s-\frac{\langle f,Eh \rangle}{2})^2}{2\epsilon^2}}, \quad \underline{\epsilon}(\epsilon, \delta)^2 :=\frac{1}{4\delta^2+\frac{1}{\epsilon^2}} \label{action95gauss95instr}\tag{17}\] for arbitrary \(h \in {\mathcal{D}}({\mathscr M})\). Integrating over \(s\), we arrive at an expression for the measurement channel of instrument 16 , namely: \[\bar{I}^{f,\epsilon,\delta}(e^{i\Phi(h)})=e^{-\frac{\langle f,Eh\rangle^2}{8\underline{\epsilon}(\epsilon, \delta)^2}} e^{i\Phi(h)}.\] Hence, the instruments \(I^{f,\epsilon,\delta}\) and \(I^{f,\underline{\epsilon}(\epsilon,\delta),0}\), despite giving rise to different POVMs, share the same measurement channel; a measurement channel which is causal, as has been verified in [2] and [3].

3.2 State tomography↩︎

We conclude this section with some words on tomography. Any family of POVMs \(\{M^l\}_l\) with elements spanning the Hermitian part of \({\cal A}\) allows one to completely fix the underlying state \(\omega\). That is, for any other state \(\omega'\) on \({\cal A}\), the condition \[\forall s,l: \quad \omega(M^l_s)=\omega'(M^l_s)\] implies that \(\omega=\omega'\). In that case, we say that the family \(\{M^l\}_l\) is tomographically complete.

Being able to implement a tomographically complete family of POVMs does not necessarily mean being able to implement, even approximately, any possible POVM. For instance, suppose that, for fixed \(\epsilon>0\), we can measure all POVMs of the form \(\{M^{f,\epsilon} :\, f\in {\mathcal{D}}({\mathscr M}),\,\langle Ef,\alpha Ef\rangle=1\}\), for some positive \(\alpha\in L^1({\mathscr M})\). Now, take \(f\in {\mathcal{D}}({\mathscr M})\), with \(\langle Ef,\alpha Ef\rangle=1\). Suppose for an ensemble of independent preparations of a quasi-free state \(\omega\) that we measure \(M^{f,\epsilon}\) and, given the result \(s\), compute \(e^{i\lambda s}\), \(\lambda \in \mathbb{R}\). Then, the resulting random variable \(\beta(f,\lambda)\) has the expectation value \[\langle \beta(f,\lambda)\rangle =\frac{1}{\sqrt{2\pi}\epsilon}\int \omega\left( e^{-\frac{(\Phi(f)-s)^2}{2\epsilon^2}}\right)e^{i\lambda s} ds=e^{-\frac{\epsilon^2\lambda^2}{2}}\omega\left( e^{i\lambda \Phi(f)}\right).\] Since the linear span of all Weyl operators yields a dense subset of \({\cal A}\), it follows that the given family of POVMs is tomographically complete. Now, consider the spectral projectors \[\label{eq:proj1} \Pi_0= \Pi_{[-1,1]}(\Phi(f)),\quad \Pi_1=1-\Pi_0,\tag{18}\] which form the POVM \(\Pi:=\{\Pi_0,\Pi_1\}\). Then we have that \[\frac{1}{\pi} \lim_{\Lambda \to \infty} \, \int_{-\Lambda}^{\Lambda} sinc\left(\lambda\right) e^{\frac{\epsilon^2\lambda^2}{2}}\langle \beta(f,\lambda)\rangle d\lambda=\omega\left(\Pi_0\right).\] That is, given the statistics obtained by measuring independent preparations of \(\omega\) with \(M^{f,\epsilon}\), one can estimate the probability with which one would have observed outcome \(0\) (or \(1\)) had we measured \(\Pi\).

This is, however, not the same as measuring the POVM \(\Pi\). In fact, by measuring \(\Pi\), one can distinguish with certainty if the system was prepared in state \(\omega=\omega_\rho\) with \(\rho=\ket{\psi_0}\bra{\psi_0}\) or \(\rho=\ket{\psi_1}\bra{\psi_1}\) for any two vectors \(\ket{\psi_0},\ket{\psi_1}\) with \(\Pi_j\ket{\psi_j}=\ket{\psi_j}\). In contrast, by measuring \(M^{f, \epsilon}\), one cannot tell with certainty which of the two states was prepared. Depending on the value of \(\epsilon\) one would need to measure many independent preparations of \(\omega\) with \(M^{f, \epsilon}\) to reach a statistically significant conclusion.

4 The FV framework and its “Heisenberg cut”↩︎

It is tempting to think that any instrument of the form \(I_j (\bullet) :=\sum_i {A}_{i,j }\bullet {A}^*_{i,j }\) for some (say finite) family \(\{{A}_{i,j }\}_{i,j }\subset{\cal A}({\mathscr R})\) with \(\sum_{i,j } {A}_{i,j} {A}^*_{i,j }=1\) can be physically implemented by acting only on region \({\mathscr R}\) or, at most, its causal hull. However, as shown by Sorkin [1], doing so leads to contradictions with Einstein’s causality in scenarios with three or more separate experimenters. For instance \(I_j(\bullet) :=\Pi_j \bullet \Pi_j\) with POVM \(\Pi_j\) as defined in 18 would be acausal. Since Sorkin-like scenarios occur also more widely—recently an example for classical field theory has been given in [21]—the issue is not to be attributed to quantum theory or QFT per sé but rather to an insufficient characterization of local instruments which do not clash with causality.

Recently, Fewster and Verch proposed a set of quantum instruments to model measurements [4], [5] which was subsequently found to be strongly causal [6], avoiding any Sorkin-like paradoxes. Their main idea was to model the measurement of a ‘target’ QFT observable⁵ by making it interact with a ‘probe’ QFT in a region \({\mathscr K}\). The probe field is measured using a POVM with elements localized in some processing region \({\mathscr L}\) strictly separated from the past of \({\mathscr K}\). After discarding (‘tracing out’) the probe field, the resulting quantum instrument induced in the target theory is shown to be localizable in any causally complete connected region strictly containing \({\mathscr K}\). More formally, call \(\mathcal{T}\) the target QFT and \(\mathcal{P}\) the probe QFT, which we assume to be in state \(\sigma\). The coupling between system and probe is specified in terms of an automorphism \(\Theta\) acting on the uncoupled theory \(\mathcal{T}\otimes \mathcal{P}\) implementing the interaction—referred to as the scattering morphism. A key assumption is that the coupling region \({\mathscr K}\subset {\mathscr M}\) associated with \(\Theta\) is compact, both in space and time, which allows the identification of the coupled theory with the uncoupled one, \(\mathcal{T}\otimes \mathcal{P}\), in the so-called “in” and “out” regions \({\mathscr K}^\pm :={\mathscr M}\setminus {\mathscr J}^{\mp}({\mathscr K})\), where \({\mathscr J}^\pm({\mathscr R})\) denotes the causal future/past of a region \({\mathscr R}\subset {\mathscr M}\). The processing region \({\mathscr L}\) is assumed to be precompact such that \(\overline{{\mathscr L}}\subset {\mathscr K}^+\).

If we now implement a POVM \(M \subset \mathcal{P}({\mathscr L})\) on the probe and then discard it, this induces an instrument \(I\) and a POVM \(I(1)\) on the target theory \(\mathcal{T}\). Defining \(\eta_\sigma : \mathcal{T}\otimes \mathcal{P} \to \mathcal{T}\) by linear and continuous extension of \(A \otimes B \mapsto \sigma(B) A\) to “trace out” the probe degrees of freedom, the instrument \(I\) is given by linear maps \(I_\alpha :\mathcal{T}\to \mathcal{T}\), \[I_\alpha ({A}) :=\eta_{\sigma}(\Theta({A}\otimes M_\alpha )), \quad {A}\in \mathcal{T},\quad \alpha \in \mathfrak{X} \label{instr95FV}\tag{19}\] where it is easy to verify that these are completely positive and \(\bar{I}(1)=1\). Moreover, the induced POVM \(I(1)\) can be localized in any causally complete connected region containing \({\mathscr K}\).

A problem with the FV framework is that the set of allowed quantum operations is defined implicitly: Namely, any instrument of the form 19 is FV-realizable. However, given an instrument \(I\) on the target theory, the problem of deciding whether \(I\) admits an FV representation is far from trivial: It amounts to determining if there exist a probe theory \(\mathcal{P}\), a scattering morphism \(\Theta\), a probe state \(\sigma\) and a POVM \(M\) such that 19 holds—this we call a measurement scheme for \(I\). The problem of determining if a given POVM \(M\) is realizable within the FV framework is similarly challenging. In [7], Fewster, Jubb and Ruep prove that, for the case of a free scalar field, the set of FV-realizable instruments is rich enough to carry out state tomography: namely, it allows one to estimate all parameters defining the QFT state of the target theory. As illustrated at the end of the previous section, this is not the same as proving that any POVM can be realized within the FV framework. This begs the question: Can we characterize a class of relevant quantum instruments that are FV-realizable?

This question extends even further, if we regard the probe in the FV framework, not as a mere mathematical artifact to arrive at a well-behaved set of local QFT operations but as a (perhaps, effective) QFT which is at the root of any measurement we currently conduct in the lab. In that second case, the measurement of the probe should likewise be realizable within the FV framework. That would require introducing another probe, which in turn should be measured with another probe, and so on.

Now, let us call \(\mathbb{M}_1\) the set of POVMs that can be realized within the FV framework (i.e., of the form \(I(1)\) as given in 19 for some choice of a measurement scheme) and define the sets of POVMs \(\{\mathbb{M}_n\}_n\) by induction: A POVM is in \(\mathbb{M}_{n}\) if it admits an FV realization such that the probe is subject to a measurement in \(\mathbb{M}_{n-1}\). Analogously, we define \(\mathbb{I}_{n}\) as the set of instruments that admit an FV realization such that the probe is subject to a measurement in \(\mathbb{M}_{n-1}\).

Following the recursion, we find that an instrument \(I\) is in \(\mathbb{I}_n\), resp., \(I(1) \in \mathbb{M}_n\) iff there is a sequence of measurement schemes \(((\mathcal{P}_j,\Theta_j,\sigma_j,M^j))_{j=1}^n\) such that the \(j^\text{th}\) scheme induces \(M^{j-1}\) for \(j > 1\) and the first scheme induces \(I\), resp., \(I(1)\). This we call a measurement chain. By definition, the coupling regions \({\mathscr K}_j\) associated with \(\Theta_j\) obey \({\mathscr K}_{j} \subset {\mathscr K}_{j-1}^+\) so that we may interpret the measurement chain also as a single measurement scheme \((\mathcal{P},\Theta,\sigma,M)\) consisting of a “super probe” theory \(\mathcal{P}= \mathcal{P}_1 \otimes \ldots \otimes \mathcal{P}_n\) in the probe state \(\sigma\), the product state formed from \(\sigma_1\),..,\(\sigma_n\), a probe POVM \(M = \mathbb{1}_{\mathcal{P}_1\otimes ..\otimes \mathcal{P}_{n-1}} \otimes M^n\) and a scattering morphism \(\Theta = \hat{\Theta}_1 \circ \ldots \circ \hat{\Theta}_n\), where \(\hat{\Theta}_j\) denotes the scattering morphism which acts as \(\Theta_j\) on \(\mathcal{P}_{j-1}\otimes \mathcal{P}_j\) and trivially on all other factors. The scheme \((\mathcal{P},\Theta,\sigma,M)\) also induces \(I\), resp., \(I(1)\), and using \(\iota_j :=\eta_{\sigma_j} \circ \Theta_j\) we obtain⁶ the identity \[\begin{align} \label{eq:chain} I_\alpha (A) &= \eta_{\sigma}(\Theta(A \otimes M_\alpha)) \\ &= \iota_1(A \otimes \iota_2(\mathbb{1}\otimes \ldots \iota_n(\mathbb{1}\otimes M^n_\alpha))), \quad A \in \mathcal{T}. \end{align}\tag{20}\]

One wonders if there exist non-trivial POVMs \(M\) or instruments \(I\)—namely, with informative measurement outcomes—such that \(M\in \mathbb{M}_n\) or \(I\in \mathbb{I}_n\) for all \(n\).

Since some quantum operations seem only reachable as limits of other FV-realizable quantum operations, it will be convenient to work with an asymptotic notion of the sets \(\{\mathbb{M}_n\}_n\) and \(\{\mathbb{I}_n\}_n\). A POVM \(M\) on a QFT \({\cal A}\) belongs to \(\overline{\mathbb{M}}_n\) if there exists a sequence of POVMs \((M^k)_k \subset \mathbb{M}_n\) such that \(\lim_{k\to\infty} M_\alpha ^k =M_\alpha\) for all \(\alpha \in \mathfrak{X}\). Analogously, an instrument \(I\) belongs to \(\overline{\mathbb{I}}_n\) if there exists a sequence of instruments \((I^k)_k \subset \mathbb{I}_n\) such that \(\lim_{k\to\infty} I^k_\alpha ({A}) =I_\alpha ({A})\) for all \({A}\in {\cal A}\) and \(\alpha \in \mathfrak{X}\).

In the next section, for linear scalar QFT, we will prove that Gaussian-modulated local field measurements and their associated Gaussian instruments are asymptotically FV-realizable, i.e., they are elements of \(\overline{\mathbb{M}}_1\) and \(\overline{\mathbb{I}}_1\), respectively. Remarkably, the required probe measurement is also a Gaussian field measurement. Iterating our scheme, we then establish that Gaussian field measurements admit a movable FV-Heisenberg cut, i.e., they are elements of \(\overline{\mathbb{M}}_n\) and \(\overline{\mathbb{I}}_n\), respectively, for all \(n \in \mathbb{N}\). Our precise results are as follows.

In words, all Gaussian-modulated measurements and all dephased Gaussian-modulated instruments of linear scalar fields are asymptotically FV-realizable and admit a movable FV-Heisenberg cut.

5 Implementation of Gaussian measurements within the FV framework↩︎

In this section, we prove the results listed in 1. We consider two linear scalar QFTs, a target \(\mathcal{T}\) and a probe \(\mathcal{P}\), induced by, for simplicity, the same classical equation of motion \(T\phi(x) = 0\), respectively, \(T\psi(x)=0\), as described in 3. We denote the associated quantized fields by \(\Phi\) for the target and \(\Psi\) for the probe. To implement the POVM \(M^{f,\epsilon}\) and the instrument \(I^{f,\epsilon,\delta}\) in \(\mathcal{T}\), corresponding to a Gaussian measurement of the target field mode \(f\), we will use weak linear interactions and carefully chosen squeezed probe states. Our principal aim here is to define a measurement scheme \((\Theta_\lambda,\sigma_\lambda,M^\lambda)\) consisting of an interaction, a probe state, and a probe POVM, respectively, which induces \(M^{f,\epsilon}\) and \(I^{f,\epsilon,\delta}\) in the limit \(\lambda \to 0\).

5.0.0.1 Interaction:

We first use ideas from the tomographic asymptotic scheme presented in [7], which solves the “classical part” of the scheme. There, it is shown that: For all \(f \in {\mathcal{D}}({\mathscr R})\) localizable in a precompact region \({\mathscr R}\subset {\mathscr M}\) and all precompact regions \({\mathscr L}\subset \overline{{\mathscr R}}^+ = {\mathscr M}\setminus {\mathscr J}^-(\overline{{\mathscr R}})\) with \({\mathscr R}\subset D^-({\mathscr L})\), the past domain of dependence of \({\mathscr L}\), we find a region \({\mathscr K}\subset {\mathscr R}\), \(g\in {\mathcal{D}}({\mathscr L})\) and \(\beta \in {\mathcal{D}}({\mathscr K})\) such that \[\label{eq:classscheme} f=-\beta E^-g.\tag{21}\] We fix \({\mathscr R}\) (localization region), \({\mathscr K}\) (coupling region), \({\mathscr L}\) (processing region) with a schematic setup given in 1, as well as \(f\) (target field mode), \(g\) (probe field mode) and \(\beta\) as given above. For the following, we exclude the trivial case \([f] = 0\), which excludes also \([g]=0\): If we had \([g] = 0\) then \(g = Tg'\) for some \(g' \in {\mathcal{D}}({\mathscr M})\). However, by the previous assumptions, \(f = -\beta E^- g\) as well as \(\mathop{\mathrm{supp}}\beta \subset {\mathscr K}\) and \(\mathop{\mathrm{supp}}g \subset {\mathscr L}\) so that \(f = -\beta (E+E^+) T g' = 0\) since \(E T {\mathcal{D}}({\mathscr M}) = 0\) and \(\mathop{\mathrm{supp}}\beta E^+ T g' \subset \mathop{\mathrm{supp}}\beta \cap {\mathscr J}^+(\mathop{\mathrm{supp}}Tg') = {\mathscr K}\cap {\mathscr J}^+({\mathscr L}) = \emptyset\).

Following Fewster and Verch [4], we then define a linear and local interaction of strength \(\lambda \beta\) between the probe and the target field where \(\lambda\) is a real parameter. More precisely, we consider the interaction term \[\lambda\int_{\mathscr K}\mathrm{vol}(x) \, \beta(x) \phi(x) \psi(x).\] This gives rise to a coupled equation of motion for the combined theory, \[T_\lambda \left( \begin{matrix}\phi \\ \psi \end{matrix}\right) = 0, \quad T_\lambda :=\left( \begin{matrix}T & \lambda \beta \\ \lambda\beta & T\end{matrix}\right),\] which has well-defined Green operators for arbitrary real \(\lambda\).

At the QFT level, this interaction generates a scattering morphism of the form \[\Theta_\lambda\left(e^{i\Phi(u)} \otimes e^{i\Psi(v)}\right)= e^{i\Phi(u_\lambda)}\otimes e^{i\Psi(v_\lambda)}, \quad u,v \in {\mathcal{D}}({\mathscr M}), \label{def95theta95lambda}\tag{22}\] and, if we suppose that \(u,v\in {\mathcal{D}}({\mathscr K}^+)\), we have the explicit formula \[\label{eq:thetaexpl} \left(\begin{array}{c}u_\lambda\\v_\lambda\end{array}\right)=\theta_\lambda \left(\begin{array}{c}u\\v\end{array}\right), \quad \theta_\lambda :=\left(\mathbb{1}-\lambda \left(\begin{array}{cc}0&\beta\\\beta&0\end{array}\right)E^-_{\lambda} \right),\tag{23}\] where \(E^-_{\lambda}\) is the retarded Green operator associated with \(T_\lambda\). For all such \(u,v\), one finds from perturbation theory that \[\left(\begin{array}{c}u_\lambda\\v_\lambda\end{array}\right)=\left(\begin{array}{c}u-\lambda\beta E^-v+O(\lambda^2)\\v-\lambda\beta E^-u+O(\lambda^2)\end{array}\right), \label{pert95theory}\tag{24}\] where \(E^-\) is the retarded Green operator associated with \(T\) and the approximation holds in the standard test function topology.

For later use, considering arbitrary \(h \in {\mathcal{D}}({\mathscr L})\), we define the functions \(f_\lambda,g_\lambda, h_\lambda,p_\lambda\) through the identities: \[\begin{align} &\left(\begin{array}{c}\lambda f_\lambda\\ g_\lambda\end{array}\right)=\theta_\lambda\left(\begin{array}{c}0\\g\end{array}\right),\quad \left(\begin{array}{c} h_\lambda\\ \lambda p_\lambda\end{array}\right)=\theta_\lambda \left(\begin{array}{c}h\\0\end{array}\right). \label{def95funcs} \end{align}\tag{25}\] By 24 we have that \(\lim_{\lambda\to 0} g_\lambda=g\), \(\lim_{\lambda\to 0} h_\lambda=h\) and \(\lim_{\lambda\to 0} f_\lambda=-\beta E^- g = f\). Thus, by analogy we define \[p:=\lim_{\lambda\to 0}p_\lambda=-\beta E^-h. \label{def95p}\tag{26}\] We remark that \[\label{eq:symplcons} \braket{p,Eg} = \braket{f, Eh},\tag{27}\] since \[\begin{align} &\langle p, E g\rangle=\langle p, (E^--E^+) g\rangle=\langle p, E^- g\rangle\nonumber\\ &=-\langle \beta E^-h, E^- g\rangle=-\langle E^-h, \beta E^- g\rangle=\langle E^-h, f\rangle=\langle Eh, f\rangle=\langle f, E h\rangle, \end{align}\] where we used in the first line that \(\mathop{\mathrm{supp}}p \subset {\mathscr K}\) while \(\mathop{\mathrm{supp}}E^+ g \subset {\mathscr J}^+(\mathop{\mathrm{supp}}g) = {\mathscr J}^+({\mathscr L})\) such that \(\braket{p,E^+ g} = 0\) and in the second line that \(\braket{f,E^+ h} = 0\) by an analogous argument.

5.0.0.2 Probe state:

Next, let us choose a family of initial states for the probe. They all arise starting from a fixed but arbitrary quasi-free state on \(\mathcal{P}\) and applying a \(\lambda\)-dependent ‘squeezing’ transformation. We define the involved transformations first abstractly:

Associated with a fixed mode pair \((q,\bar{q}) \in \mathcal{C}_T^2\) which is canonically conjugate, i.e., \(E(q,\bar{q}) = 1\), we define the symplectic flip \(R_{q,\bar{q}}\) and, for nonzero \(\mu \in \mathbb{R}\), the squeezing transformation \(F_{q,\bar{q},\mu}\) by \[\label{eq:symplflip} R_{q,\bar{q}} v :=E(v,\bar{q})\bar{q} + E(v,q)q + v^{\perp},\tag{28}\] and \[\label{eq:symplsqueeze} F_{q,\bar{q},\mu} v :=\mu E(v,\bar{q}) q - \mu^{-1} E(v,q) \bar{q} + v^{\perp},\tag{29}\] where \[\label{eq:orth} v^{\perp} :=v - E(v,\bar{q}) q + E(v,q) \bar{q}.\tag{30}\] These are evidently symplectic transformations characterized on the linear hull of \(\{ q,\bar{q}\}\) by mapping \(q \mapsto \bar{q}\), \(\bar{q} \mapsto -q\), respectively, \(q \mapsto \mu q\), \(\bar{q} \mapsto \mu^{-1}\bar{q}\) and by acting trivially on the symplectic complement. Thus, given a state \(\sigma\) and a canonically conjugate mode pair, we obtain a flipped state \(\tilde{\sigma}\) and a family \(\sigma_\mu\) of \(\mu\)-squeezed states by applying \(R_{q,\bar{q}}\), respectively, \(F_{q,\bar{q},\mu}\) to \(\sigma\); confer 3 . If \(\sigma\) is quasi-free, then so will be \(\tilde{\sigma},\sigma_\mu\), in this case their covariances \(\tilde{\Gamma}\) and \(\Gamma_\mu\) will satisfy \[\tilde{\Gamma}(q,q) = \Gamma(\bar{q},\bar{q}), \quad \tilde{\Gamma}(q,q) = \Gamma(\bar{q},\bar{q}), \quad \tilde{\Gamma}(q,\bar{q}) = - \Gamma(q,\bar{q})\] and \[\Gamma_\mu(q,q) = \mu^2\Gamma(q,q), \quad \Gamma_\mu(\bar{q},\bar{q}) = \mu^{-2}\Gamma(\bar{q},\bar{q}), \quad \Gamma_\mu(q,\bar{q}) = \Gamma(q,\bar{q}),\] while \[\tilde{\Gamma}(r,s)=\Gamma_\mu(r,s) = \Gamma(r,s),\] for all \(r, s\) with \[E(r,q)=E(s,q)=E(r,\bar{q})=E(s,\bar{q})=0.\]

Now, let \(g \in {\mathcal{D}}({\mathscr L})\) be as fixed earlier and \(\bar{g} \in {\mathcal{D}}({\mathscr L})\) be a canonical conjugate to \(g\)⁷, so we have \(E([g],[\bar{g}])=\braket{g,E\bar{g}}=1\). For our protocol, we will need a quasi-free state \(\sigma\) with covariance \(\Gamma\) satisfying the conditions: \[\label{tecreq} \Gamma([g],[\bar{g}])=0, \quad \Gamma([g],[g])=\Gamma([\bar{g}],[\bar{g}])=:c^2,\tag{31}\] This is easily achieved: let \(\sigma_0\) be an arbitrary quasi-free state on \(\mathcal{P}\) with covariance \(\Gamma_0\) and denote its flipped state (with respect to \([g]\) and \([\bar{g}]\)) by \(\tilde{\sigma}_0\) with covariance \(\tilde{\Gamma}_0\). Then \(\sigma\) can be defined as the quasi-free state with covariance \(\Gamma :=\frac{1}{2} (\Gamma_0 + \tilde{\Gamma}_0)\), which is well-defined and satisfies the requirement 31 by inspection. In other words, we define \(\sigma\) as the liberation⁸ of the uniform mixture \(\frac{1}{2}(\sigma_0+\tilde{\sigma}_0)\).

Next, let \(g_\lambda\) be as defined in 25 , then for each \(\lambda \in [0,\lambda_0]\) for some \(\lambda_0 > 0\), we choose a canonical conjugate \(\bar{g}_\lambda \in {\mathcal{D}}({\mathscr L})\) to \(g_\lambda\) such that \[\label{eq:cconj} \underset{\lambda \to 0}{\lim} \bar{g}_\lambda = \bar{g}.\tag{32}\] This is always possible, e.g., by setting \(\bar{g}_\lambda :=(\braket{g_\lambda, E \bar{g}})^{-1} \bar{g}\), which is well-defined for sufficiently small \(\lambda_0\) since \(\braket{g_\lambda, E\bar{g}}\) is real and converges to 1 in the limit \(\lambda \to 0\). Finally, we define \(\sigma_\lambda\) as the \(\mu\lambda-\)squeezed state obtained from \(\sigma\) with respect to the mode pair \(([g_\lambda],[\bar{g}_\lambda])\), \[\label{def95sigma95lambda} \sigma_\lambda :=\sigma_{F_{[g_\lambda],[\bar{g}_\lambda],\mu \lambda }},\tag{33}\] where we take \(\lambda \in [0, \lambda_0]\) as a standing assumption and reserve a constant \(\mu > 0\) for later optimization.

Finally, note that if \(\sigma\) is chosen to be Hadamard, then \(\sigma_\lambda\) is Hadamard. This follows, since the Hadamard property is preserved under finite-rank perturbations (4). Likewise, if \(\sigma_0\) is chosen to be Hadamard, then also \(\sigma\) is Hadamard, since, as can easily be verified, the Hadamard property is preserved under liberation and taking convex mixtures. Thus, our probe state preparation preserves the class of physically reasonable states: Starting with an arbitrary quasi-free Hadamard state, all probe states involved will have this property.

5.0.0.3 Probe measurement:

We have provided an interaction and a probe state. The last element we need to specify in the FV measurement scheme is the POVM to be implemented on the probe. We choose it to be Gaussian and dependent on \(\lambda\). Namely, for some \(\varepsilon > 0\), \[M_s^{\lambda} :=M_s^{\lambda^{-1}g,\varepsilon} =\frac{1}{\sqrt{2\pi}\varepsilon} e^{-\frac{\left(\frac{\Psi(g)}{\lambda}-s\right)^2}{2\varepsilon^2}}. \label{probe95POVM}\tag{34}\]

5.0.0.4 Implementing the measurement scheme:

We are ready to formulate our first result which establishes that Gaussian measurements admit an asymptotic FV-realization.

5.0.0.5 Moving the “FV Heisenberg cut”:

After establishing that Gaussian measurements are asymptotically FV-realizable in 2, there are essentially two ways to extend our scheme; cf. [5]. One option is to treat multiple probes all coupled to the target theory, which allows for modeling successive Gaussian measurements on the target theory. The other is to treat a measurement chain as discussed in 4. In this paragraph, we will show that for a suitable iteration of our scheme we obtain arbitrarily long measurement chains inducing Gaussian measurements asymptotically in the limit \(\lambda \to 0\). Thus, we prove that Gaussian measurements admit a movable FV Heisenberg cut and complete the proof of our main theorem (1). Our precise result is

Proof. The proof is easy, but cumbersome, so we omit some of the details. To begin with, we fix arbitrary \(\varepsilon > 0\) and \(n \in \mathbb{N}\) and suppose \(j\in \{1,..,n\}\) throughout the proof. Also, we fix \(f \in \mathcal{C}_T({\mathscr R})\) for some precompact \({\mathscr R}\subset {\mathscr M}\) and assume without loss of generality¹⁰ that \([f] \neq 0\) within \(\mathcal{C}_T\). We consider a target QFT \(\mathcal{T}\) whose field we denote by \(\Phi\) and \(n\) probe QFTs \(\{\mathcal{P}_j\}\) whose fields we denote by \(\{ \Psi_j\}\), all induced by the same equation of motion operator \(T\).

Next, we select localization regions \(\{´\mathscr{R}_j\}\), coupling regions \(\{ {\mathscr K}_j \}\) and processing regions \(\{{\mathscr L}_j\}\) within \({\mathscr M}\) as well as probe modes \(\{ g^j\}\) and interaction strengths \(\{\beta_j\}\) within \({\mathcal{D}}({\mathscr M})\) suitable to induce the target mode \(f\) classically. In particular, we choose \(\{ \mathscr{R}_j\}\) and \(\{ {\mathscr L}_j\}\) to be arbitrary precompact regions within \(\overline{{\mathscr R}}^+\) such that \({\mathscr L}_j \subset \overline{{\mathscr L}_{j-1}}^+\) and \(\mathscr{R}_j \subset {\mathscr L}_{j-1} \cap D^{-}({\mathscr L}_{j})\) for all \(j\); here \({\mathscr L}_0 = {\mathscr R}\). The involved spacetime regions are illustrated in 2. Then we invoke the classical part of the scheme from above iteratively: Starting with \(j=1\) and \(g^0=f\), we find regions \({\mathscr K}_j \subset \mathscr{R}_j\), \(g^j \in {\mathcal{D}}({\mathscr L}_j)\) and \(\beta_j \in {\mathcal{D}}({\mathscr K}_j)\) such that \[g^{j-1} = -\beta_{j} E^- g^{j};\] which implies also \([g^j] \neq 0\) for all \(j\) since \([f] \neq 0\).

Given this, we can define the interaction. Fix \(\lambda\in\mathbb{R}^+\). For each \(j\), we consider the interaction term \[\lambda\int_{{\mathscr K}_j} \mathrm{vol}(x) \, \beta_j(x)\psi_{j-1}(x)\psi_{j}(x)\] with \(\psi_0 = \phi\). We define the corresponding scattering morphism \(\Theta^j_{\lambda}\) on \(\mathcal{P}_{j-1} \otimes \mathcal{P}_{j}\) with \(\mathcal{P}_0 = \mathcal{T}\) in analogy with 22 . That is, \[\label{eq:itint} \Theta^j_{\lambda}(e^{i\Psi_{j-1}(u^{j-1}_\lambda)} \otimes e^{i\Psi_{j}(u^j_\lambda)})=e^{i\Psi_{j-1}(u^{j-1})} \otimes e^{i\Psi_{j}(u^j)},\tag{40}\] with \[\left(\begin{array}{c}u^{j-1}_\lambda\\ u^j_\lambda\end{array}\right) = \theta^j_{\lambda}\left(\begin{array}{c}u^{j-1}\\ u^j\end{array}\right)\] and \(\theta^j_\lambda\) as in 23 for \(\beta = \beta_j\).

Starting from \(g^n_{-}(\lambda):=g^n\), we inductively define the \(\lambda\)-dependent functions \(\{ g^j_{\pm}(\lambda) \}_j\) through the relation \[\left(\begin{array}{c}\lambda g^{j-1}_{+}(\lambda) \\ g^j_{-}(\lambda)\end{array}\right)=\theta^j_{\lambda}\left(\begin{array}{c}0\\g^j_{-}(\lambda) \end{array}\right). \label{def95g95pm}\tag{41}\] Note that \(\lim_{\lambda\to 0} g^j_{\pm}(\lambda) =g^j\). We thus choose \(\lambda\) small enough so that \([g^j_{\pm}(\lambda)]\not=0\) for all \(j\). Hence, for every \(g^j_{+}(\lambda)\), there exists a canonical conjugate \(\bar{g}^j_{+}(\lambda)\).

Let \(\sigma\) be a quasi-free probe state satisfying the technical condition 31 , with \(c = c_1\) and \(g=g^1\). For each \(j\) and arbitrary \(\mu_j>0\), on \(\mathcal{P}_j\), we define the probe state \[\label{eq:itstate} \sigma^j_{\lambda}:=\sigma_{F_{[\bar{g}^j_{+}(\lambda)], [g^j_{+}(\lambda)],\mu_j\lambda^j}}.\tag{42}\]

Starting from \(M^n(\lambda):=M^{g^n_{-}(\lambda)\lambda^{-n},\varepsilon}\), we define the POVMs \((M^j(\lambda))_{j=1}^{n-1}\) through the recursive relation: \[M_s^j(\lambda)=\eta_{\sigma^\lambda_j}\circ\Theta_\lambda^{j+1}(\mathbb{1}\otimes M^{j+1}_s(\lambda)). \label{recur95POVMs}\tag{43}\] By construction, \(M^j(\lambda)\in \mathbb{M}^{n+1-j}\), for \(j=1,...,n\).

To arrive at an explicit expression for the \(j^{th}\) POVM, we note that, for \(\omega\) quasi-free, \(g\in {\mathcal{D}}({\mathscr M})\) and \(\varepsilon\in\mathbb{R}^+\), it holds that \[\eta_{\omega}\circ\Theta_\lambda(\mathbb{1}\otimes M^{g,\varepsilon}_s)=M^{f_\lambda,\varepsilon'}, \label{explicit95recur95POVMs}\tag{44}\] with \[\varepsilon'=\sqrt{\varepsilon^2+\lambda^{-2}\Gamma_\omega(g_\lambda,g_\lambda)},\] where \(g_\lambda, f_\lambda\) are defined from \(g\) by the first identity in eq. (25 ). From eqs. (41 ), (43 ), (44 ), it then follows that \[\label{eq:itpovm} M^j(\lambda)=M^{g^{j}_{-}(\lambda)\lambda^{-j},\epsilon_j},\tag{45}\] with \[\epsilon_j:=\sqrt{\varepsilon^2+\sum_{k=j+1}^n \mu_j^2 \Gamma(g_{+}^j(\lambda),g_{+}^j(\lambda))}.\]

Next, we consider the effect of measurement \(M^1(\lambda)\) on the target system. To do so, we note that, for \(\omega\), quasi-free, \(\varepsilon\in \mathbb{R}^+\), it holds that \[\eta_{\omega}\circ \Theta_\lambda(e^{i\Phi(h)}\otimes M^{g,\varepsilon}_s)=e^{i\Phi(h_\lambda)}\int \frac{dz}{2\pi} e^{-\frac{\varepsilon^2z^2}{2}}e^{i(\phi(f_\lambda)-s-\frac{1}{2}\langle f_\lambda, E h_\lambda\rangle)z}\omega\left(e^{i\Psi\left(\frac{g_\lambda}{\lambda}z+\lambda p_\lambda^\perp \right)}\right),\] with \(f_\lambda,g_\lambda,p_\lambda\) defined from \(g,h\) through Eq. (25 ) and \(p^\perp_\lambda\) as in 30 , the symplectic complement of \(p(\lambda)\) with respect to \(span\{g^1(\lambda),\bar{g}^1(\lambda)\}\)

Substituting \(\omega\) by \(\sigma_1\); \(g\), by \(\lambda^{-1} g^1_-(\lambda)\), and \(\varepsilon\), by \(\epsilon_1\), the instrument induced on the target system is: \[I_s^\lambda=\int \frac{dz}{2\pi} e^{\frac{-\epsilon_1^2z^2}{2}} e^{i(\Phi(g^0_{-}(\lambda))-s-\frac{1}{2}\langle g^0_{-}, E h_\lambda\rangle)z}\sigma\left(e^{i\Psi(\mu g^1_{+}(\lambda)(z-\lambda^2\langle \bar{g}^1_{+}(\lambda),E p(\lambda)\rangle +\lambda p^\perp(\lambda)}\right), \label{final95instrument}\tag{46}\] where \(h(\lambda),p(\lambda)\) are defined by the relation: \[\left(\begin{array}{c}h(\lambda)\\ \lambda p(\lambda) \end{array}\right)=\theta^1_{\lambda}\left(\begin{array}{c}h\\ 0\end{array}\right).\] Note that \(\lim_{\lambda\to 0}h(\lambda)=h\), \(\lim_{\lambda\to 0}p(\lambda)=-\beta_1E^-h\).

By construction, \(I^\lambda\in \mathbb{I}_n\). From Eq. (46 ), it is obvious that \[\lim_{\lambda\to 0} I_s^\lambda=I^{f,\epsilon,\delta},\] with \(\epsilon, \delta\) as defined in ?? and \(c_j^2 :=\Gamma(g^j,g^j)\). ◻

5.0.0.6 Projective measurements and recovery of the “projection postulate”:

The FV scheme outlined in the proof of 2 allows one to realize many POVMs other than Gaussian-modulated field measurements. Indeed, consider a Gaussian measurement \(\{ M_s^{f,\epsilon}\}_s\) and let \(\{p_{\tilde{s}}:\mathbb{R}\to\mathbb{R}^+\}_{\tilde{s}\in\tilde{\mathfrak{X}}}\) be any set of Borel-integrable functions with normalized parameter-dependence \(\int_{\tilde{\mathfrak{X}}}p_{\tilde{s}}(s)d\tilde{s}=1\), for all \(s\in\mathbb{R}\). Then the POVM \(\{ \tilde{M}_{\tilde{s}}^{f,\epsilon} \}_{\tilde{s}}\) that results when we measure \(\{ M_s^{f,\epsilon}\}_s\) and, upon obtaining the result \(s\), sample \(\tilde{s}\) from the probability distribution \(\{p_{\tilde{s}}(s)\}_{\tilde{s}}\) will have the form \[\tilde{M}_{\tilde{s}}^{f,\epsilon}=\left(p_{\tilde{s}} \ast g_\epsilon \right)\left(\Phi(f)\right), \quad g_\epsilon(x) :=\tfrac{1}{\sqrt{2\pi}\epsilon}e^{-\frac{x^2}{2\epsilon^2}}, \label{meas95with95noise}\tag{47}\] using \(\ast\) to denote convolution of functions. Since \(\widehat{p_{\tilde{s}} \ast g_\epsilon}(z) = e^{-\frac{\epsilon^2 z^2}{2}} \hat{p}_{\tilde{s}}(z)\) is clearly integrable in \(z\) for all \(\epsilon > 0\), the proof of 2 with initial probe measurement \(\tilde{M}_s^{\lambda^{-1}g,\epsilon}\) goes through and we obtain \(\tilde{M}^{f,\epsilon} \in \overline{\mathbb{M}}_1\). Taking the limit \(\epsilon\to 0\), we have that \(\tilde{M}_{\tilde{s}}^{f,\epsilon}\) tends to \(p_{\tilde{s}}(\Phi(f))\) which is thus also within \(\overline{\mathbb{M}}_1\).

Combined with 3, this observation implies that any POVM of the form \(\{ p_{\tilde{s}}(\Phi(f))\}_{\tilde{s}}\) also belongs to \(\overline{\mathbb{M}}_n\), for all \(n\in \mathbb{N}\). Clearly, also discrete-outcome variants of these POVMs can be modeled; in this case normalization of \(\{ p_j \}_j\) changes to \(\sum_j p_j(s) = 1\). This includes the option \(p_j=\chi_{B^j}\), where \(\{B^j\subset \mathbb{R}\}_j\) is a discrete family of pairwise disjoint Borel-measurable sets that satisfy \(\bigcup_j B^j=\mathbb{R}\). As a result, also the projective measurement \(\{ \Pi_j^f\}_j\) with \(\Pi^f_j :=\Pi_{B^j}(\Phi(f))\) is asymptotically FV-realizable.

However, note that, if we follow the construction of 2, then the measurement channel of the instrument associated to the POVM 47 is always \(I^{f,\epsilon,\delta}\), regardless of how we choose to process the “real" outcome \(s\). So, even in the limit \(\epsilon\to 0\) and transforming to discretely many outcomes \(j\), the measurement channel will not resemble anything like \(\sum_{j} \Pi_j^f \bullet \Pi_j^f\) which would run again into Sorkin’s causal paradox [1]. The FV formalism neatly avoids them, despite the fact that it allows measuring the projectors \(\Pi^f_j\) up to arbitrary accuracy. The”projection postulate" is restored only for observers in the causal complement region to the field mode \(f\): For an observer who knows the classical outcome \(s\) to be contained in some Borel set \(B \subset \mathbb{R}\) but has access only to field modes \(h\) which are supported in the causal complement to the support of \(f\), we obtain Born’s rule from 17 in the (strong-operator) limit \[\lim_{\epsilon \to 0} \, \int_B I^{f,\epsilon,\delta}_s(e^{i\Phi(h)}) ds= \Pi_B(\Phi(f)) e^{i\Phi(h)} \Pi_B(\Phi(f)),\] while the overall measurement channel is trivial, \[\lim_{\epsilon \to 0} \, \overline{I}^{f,\epsilon,\delta} (e^{i\Phi(h)}) = e^{i\Phi(h)}.\] We conclude that if the observer cannot resolve the causal structure of the apparatus which led to the measurement of \(f\), then the projection postulate might be a valid idealization.

5.0.0.7 Movable “Heisenberg cut” in the purely *-algebraic framework:

We conclude this section by noting that, even when \({\cal A}\) is just a *-algebra and not completed with respect to any topology (as done in the original formulation of the FV framework [4]), it still allows to describe a large class of non-trivial, FV-realizable measurements with a movable FV-Heisenberg cut. Let \({\cal A}^f\) denote the *-algebra spanned by all abstract Weyl operators of the form¹¹ \(\{e^{iz\Phi(f)}:\, z\in\mathbb{R}\}\) and consider a finite-outcome POVM \(\{M_j\}_j \subset {\cal A}^f\) such that, for some finite family \(\{{A}_j^k: \, j,k\}\subset {\cal A}^f\) and every outcome \(j\), \[\label{eq:abspovm} M_j=\sum_k {A}_j^k({A}_j^k)^*.\tag{48}\] Given the (unambiguous) expression of \(M_j\) as a finite linear combination of imaginary exponentials of \(\Phi(f)\), define \(\hat{M}_j(s)\) by replacing every instance of \(\Phi(f)\) by \(s\). Then, for any \(\lambda\in\mathbb{R}\), \(\{\hat{M}_j(\Psi(g_\lambda/\lambda))\}_j\) defines a POVM in the probe field. Moreover, the induced POVM element \[\hat{M}^\lambda_j:= \eta_{\sigma_\lambda}\circ\Theta_\lambda\left(\hat{M}_j\left(\Psi\left(g_\lambda/\lambda\right)\right)\right)\] is also a positive semidefinite element of \({\cal A}\). One can then verify that \[\lim_{\lambda\to 0}\omega(\hat{M}^\lambda_j)=\omega(M_j) \label{eq:explimit}\tag{49}\] for any regular state \(\omega\) on \({\cal A}\), i.e., a state such that \(\omega(e^{it\Phi(f)})\) is continuous in \(t\) (see also Appendix 7 for details). Therefore, the FV framework—also in the purely algebraic setting—includes certain “Weyl type" POVMs linearly generated from Weyl operators of a single field mode.

6 Conclusion↩︎

We have argued that the tomographic results of [7] are insufficient to infer that the FV scheme is rich enough to model field measurements. Rather, it is proven there that averages of powers of the field operator can be estimated asymptotically. As explained in the text, this is not the same as being able to implement a (Gaussian-modulated) field measurement: in fact, their asymptotic tomographic scheme in the limit \(\lambda\to 0\) induces the identity map as a measurement channel on the target QFT.

Instead, we have shown that Gaussian-modulated measurements of a (local) smeared field operator can be asymptotically implemented within the FV scheme. The degree \(\epsilon\) of the modulation can be chosen arbitrarily low, which allows one to approximate a projective measurement of the field with arbitrary precision. For non-zero \(\epsilon\), there is a simple map to update the QFT state conditioned on the measurement outcome: namely, the composition of the Gaussian instrument proposed by Jubb [2] and further studied by [3] with a single-mode dephasing channel.

Notably, we find that the probe measurement required to induce a Gaussian measurement in a QFT is itself a Gaussian measurement. That is, we can model the measurement carried out on this first probe by making it interact with a second probe, which is subsequently measured. This measurement, in turn, can be also conducted by making the second probe QFT interact with a third, and so on ad infinitum.

It would be interesting to know if the property of having a movable FV-Heisenberg cut is applicable to all quantum measurements admitting an FV representation. The answer would be a resounding “yes!" if all POVMs with elements localizable within a local algebra happened to be FV realizable.

The authors thank Henning Bostelmann, Chris Fewster, Markus Fröb, Claudio Iuliano, Robert Oeckl, Maximilian Heinz Ruep, Leonardo Sangaletti, Rainer Verch for valuable discussions on the subject as well as remarks and questions which helped improving this text.

J. Mandrysch was funded by the quantA core project “Local operations on quantum fields".

References↩︎