Bounds on quantum communication via Newtonian gravity

Consider our hypothetical situation: gravitational interactions are observed to follow a $1/r$ behavior at low energies but are not allowed to transmit quantum information. Within this context, we would like a consistent model for understanding the low-energy (Newtonian) limit for gravity. For simplicity, we consider a length cutoff a below which our low-energy theory breaks down. Given this setup, we can consider a range of possible classical communication only theories. Fortunately [14] allows us to consolidate all such theories by only considering a measure and feedback approach. We regard this type of theory as an 'impostor' theory of gravity, with no particularly good physical basis. However, the fact that it saturates the minimal noise bound necessary for a classical communication-only theory [14] indicates that it is as good as any other such theory, and thus we use it for its conceptual and calculational simplicity.

Starting with single particles in one dimension, our length cutoff can be made exact by replacing the position basis $\left| x \right\rangle$ with a discrete variable representation corresponding to the projector $\hat{\Pi }={{\int }_{|p|\lt {{p}_{c}}}}{\rm d}p|p\rangle \langle p|$. This is analogous to moving to a tight-binding-type model⁶ . Defining ${{p}_{c}}=\frac{\pi \hbar }{a}$ as a momentum cutoff in the single particle picture, this yields an integer-indexed set of states $\mid {{x}_{j}}\rangle$ with a position basis representation, in one dimension

$$\begin{eqnarray}&&{{f}_{j}}(x)\equiv \langle x|{{x}_{j}}\rangle =\sqrt{\frac{\hbar }{\pi {{p}_{c}}}}\frac{{\rm sin} [{{p}_{c}}(x-{{x}_{j}})/\hbar ]}{x-{{x}_{j}}}\end{eqnarray} \tag{ 1 }$$

with ${{x}_{j}}=aj$, as shown pictorially in figure 1(a). A similar picture was suggested in [17].

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Looking in the second quantized picture, we work with a scalar field with mass m and a field operator $\hat{\psi }(x)$. This operator can be approximated to our length scale a by transforming x to k-space, then truncating, then transforming back:

$$\begin{eqnarray}\begin{array}{rcl} {{{\hat{a}}}_{j}} & = & \sqrt{\frac{\pi \hbar }{{{p}_{c}}}}\int \frac{{\rm d}q}{\sqrt{2\pi }}{{{\rm e}}^{-{\rm i}q{{x}_{j}}}}\theta ({{p}_{c}}/\hbar -|q|){{{\hat{\psi }}}_{q}} \\ {} & = & \sqrt{\frac{\pi \hbar }{{{p}_{c}}}}\int \frac{{\rm d}q}{\sqrt{2\pi }}{{{\rm e}}^{-{\rm i}q{{x}_{j}}}}\theta ({{p}_{c}}/\hbar -|q|)\int \frac{{\rm d}x^{\prime} }{\sqrt{2\pi }}{{{\rm e}}^{{\rm i}qx^{\prime} }}\hat{\psi }(x^{\prime} ) \\ {} & = & \int {{f}_{j}}(x^{\prime} )\hat{\psi }(x^{\prime} ){\rm d}x^{\prime} \\ \end{array}\end{eqnarray} \tag{ 2 }$$

with the inversion relation for a Fourier transform on a finite domain

$$\begin{eqnarray*}&&{{\hat{\psi }}_{q}}=\sqrt{\frac{\hbar }{2{{p}_{c}}}}\mathop{\sum }\limits_{j}{{\hat{a}}_{j}}{{{\rm e}}^{{\rm i}q{{x}_{j}}}}\ {\rm and}\ \Pi \hat{\psi }(x)\Pi =\mathop{\sum }\limits_{j}{{f}_{j}}(x){{\hat{a}}_{j}}.\end{eqnarray*}$$

Regarding the commutation relations, we have for a bosonic field

$$\begin{eqnarray*}&&[{{\hat{a}}_{j}},\hat{a}_{l}^{\dagger }]=\int {\rm d}xf(x-{{x}_{j}})f(x-{{x}_{l}})={{\delta }_{jl}}.\end{eqnarray*}$$

More generally, these position-based field operators satisfy the same commutation relations as the continuum field, as expected. Higher dimensions are straightforward to add, and we assume three spatial dimensions in the remainder of this discussion. The free field Hamiltonian is

$$\begin{eqnarray*}&&{{\hat{H}}_{0}}=\frac{{{\hbar }^{2}}}{2m}\int {{{\rm d}}^{3}}q\theta ({{p}_{c}}/\hbar -|\bar{q}|)|\bar{q}{{|}^{2}}\hat{\psi }_{{\bar{q}}}^{\dagger }{{\hat{\psi }}_{{\bar{q}}}}.\end{eqnarray*}$$

We can now re-express our expected gravitational interaction using these operators. Specifically, starting with

$$\begin{eqnarray*}&&\hat{V}=-\frac{G{{m}^{2}}}{2}\int {{{\rm d}}^{3}}x{{{\rm d}}^{3}}\;y\frac{{{{\hat{\psi }}}^{\dagger }}(\bar{x}){{{\hat{\psi }}}^{\dagger }}(\bar{y})\hat{\psi }(\bar{y})\hat{\psi }(\bar{x})}{|\bar{x}-\bar{y}|},\end{eqnarray*}$$

where m is the mass of our scalar field, we project with Π. This produces a sum $-\frac{G{{m}^{2}}}{2}{{\sum }_{ijkl}}{{I}_{ijkl}}\hat{a}_{i}^{\dagger }\hat{a}_{j}^{\dagger }{{\hat{a}}_{k}}{{\hat{a}}_{l}}$ with the integral ${{I}_{ijkl}}=\int {{{\rm d}}^{3}}x{{{\rm d}}^{3}}\;y\frac{{{f}_{i}}(x){{f}_{j}}(y){{f}_{k}}(y){{f}_{l}}(x)}{|x-y|}$. Looking at I_ijkl, we have three terms that contribute after projection with Π: a direct, Coulomb-like term (I_ijji, $i\ne j$) which contributes at long range, exchange terms (I_iijj, I_ijij, and I_ijkl with at most one index repeated) which decays over a length scale $\gtrsim a$ due to the reducing overlap of the sinc functions at long distance, and a self-interaction from the finite-size wave packet I_iiii which we neglect as a background (constant) contribution for any particle conserving theory.

With these approximations, we have a softened Newtonian interaction

$$\begin{eqnarray}&&\hat{\Pi }\hat{V}\hat{\Pi }\approx {{\hat{V}}_{a}}\equiv -\frac{G{{m}^{2}}}{2}\mathop{\sum }\limits_{i\ne j}\hat{a}_{i}^{\dagger }\hat{a}_{j}^{\dagger }{{\hat{a}}_{i}}{{\hat{a}}_{j}}\frac{1}{|{{{\bar{x}}}_{i}}-{{{\bar{x}}}_{j}}|+a}.\end{eqnarray} \tag{ 3 }$$

At this point, a Hamiltonian that contains both the kinetic terms $\hat{H}$₀ and interaction ${{\hat{V}}_{a}}$ provides a toy model for the quantum mechanics of interacting, gravitating particles. We now consider what happens when we add the crucial constraint of this paper: that interactions through ${{\hat{V}}_{a}}$ do not convey quantum information. More specifically, defining ${{\hat{n}}_{i}}=\hat{a}_{i}^{\dagger }{{\hat{a}}_{i}}$ as a number operator measuring the local particle number, we can emulate a set of pairwise interactions of the form ${{\hat{V}}_{ij}}=-\frac{G{{m}^{2}}}{2(|{{{\bar{x}}}_{i}}-{{{\bar{x}}}_{j}}|+a)}{{\hat{n}}_{i}}{{\hat{n}}_{j}}$ while disallowing quantum information transfer by using a measure-and-feedback approach with a (hypothetical) ancillary field. In particular, the ancillary field stores a weak measurement result of the mass $m{{\hat{n}}_{i}}$ at a given location, which is then used to create a feedback force at distant locations.

Note that if we have multiple fields, we might write ${{\hat{n}}_{i}}$ as a weighted sum over occupation numbers of fields of different masses. More general, relativistic scenarios for parameter estimation of a local Hamiltonian are also possible [18], and a simple analysis suggests that the T₀₀ component of the stress energy tensor is the more natural variable to observe in that regime. Here we are working strictly in the non-relativistic regime, and use ${{\hat{n}}_{i}}$ in what follows.

From the measure-and-feedback perspective, we expect for a protocol that mimics the effects of ${{\hat{V}}_{a}}$ necessarily acts via weak measurement of the set $\{{{\hat{n}}_{j}}\}$. Consequently, we anticipate an unavoidable noise in observables that do not commute with elements of this set. We can understand this unavoidable noise as arising due to local estimation of ${{\hat{n}}_{j}}$, i.e., a local measurement of the mass in a region of size a³.

We now describe the measurement and feedback process illustrated in figure 1(b). We recall that the interaction of interest is

$$\begin{eqnarray}&&{{\hat{V}}_{a}}=\hbar \mathop{\sum }\limits_{j}\mathop{\sum }\limits_{k\ne j}{{\chi }_{jl}}{{\hat{n}}_{j}}{{\hat{n}}_{k}}\end{eqnarray} \tag{ 4 }$$

with ${{\chi }_{jk}}=-\frac{1}{2\hbar }\frac{G{{m}^{2}}}{|{{{\bar{x}}}_{j}}-{{{\bar{x}}}_{k}}|+a}$. For each position j and at each time step $[t,t+\tau )$ we define an auxiliary harmonic oscillator with (dimensionless) canonical variables satisfying $[{{\hat{X}}_{j}},{{\hat{P}}_{j}}]=i$. Weak measurement with a strength ξ corresponds to the unitary ${{\hat{U}}_{1}}={\rm exp} (-{\rm i}\sqrt{\xi \tau }{{\hat{n}}_{j}}{{\hat{P}}_{j}})$, where ξ has units [s⁻¹]. This measurement is stored in ${{\hat{X}}_{j}}$, which can then induce a force via ${{\hat{U}}_{2}}={\rm exp} (-{\rm i}\sqrt{2\tau /\xi }{{\hat{X}}_{j}}{{\sum }_{k\ne j}}{{\chi }_{jk}}{{\hat{n}}_{k}})$. Using ${\rm exp} (rA){\rm exp} (rB)={\rm exp} (rA+rB+\frac{{{r}^{2}}}{2}[A,B]+O({{r}^{3}}))$, we have that

$$\begin{eqnarray*}&&{{\hat{U}}_{2}}{{\hat{U}}_{1}}={\rm exp} (-{\rm i}\sqrt{2\tau }(\sqrt{\xi }{{\hat{n}}_{j}}{{\hat{P}}_{j}}+\sqrt{1/\xi }{{\hat{O}}_{j}}{{\hat{X}}_{j}})-{\rm i}\tau {{\hat{n}}_{j}}{{\hat{O}}_{j}}+O({{\tau }^{3/2}})),\end{eqnarray*}$$

where we have defined the feed-back operator

$$\begin{eqnarray*}&&{{\hat{O}}_{j}}=\mathop{\sum }\limits_{k\ne j}{{\chi }_{jk}}{{\hat{n}}_{k}}.\end{eqnarray*}$$

The product ${{\hat{U}}_{2}}$\${{\hat{U}}_{1}}$ thus produces the desired interaction $-{\rm i}\tau {{\hat{n}}_{j}}{{\sum }_{k}}{{\chi }_{jk}}{{\hat{n}}_{k}}$, as well as terms representing the strength of the measurement and feedback⁷ . Importantly, after each time step τ, the auxiliary harmonic oscillator is discarded and replaced with another in the ground state, $\left| {\rm vac} \right\rangle$.

The effective time evolution emerges in the limit $\tau \to 0$, so the master equation ${{\partial }_{\tau }}\hat{\rho }=\mathcal{L}(\hat{\rho })$ is obtained by expanding ${{\hat{U}}_{2}}{{\hat{U}}_{1}}\hat{\rho }\otimes |{\rm vac}\rangle \langle {\rm vac}|\hat{U}_{1}^{\dagger }\hat{U}_{2}^{\dagger }$ to order τ then tracing out the ancillary oscillator (this process is repeated at each site). Substituting from the above equation we compute

$$\begin{eqnarray*}\begin{array}{lll} {} & {} & \quad (\hat{\rho }-{\rm i}\tau [{{{\hat{n}}}_{j}}{{{\hat{O}}}_{j}},\hat{\rho }])\otimes |{\rm vac}\rangle \langle {\rm vac}| \\ {} & {} & -{\rm i}\sqrt{2\tau }[\sqrt{\xi }{{{\hat{n}}}_{j}}{{{\hat{P}}}_{j}}+\sqrt{1/\xi }{{{\hat{O}}}_{j}}{{{\hat{X}}}_{j}},\hat{\rho }\otimes |{\rm vac}\rangle \langle {\rm vac}|] \\ {} & {} & -\tau [\sqrt{\xi }{{{\hat{n}}}_{j}}{{{\hat{P}}}_{j}}+\sqrt{1/\xi }{{{\hat{O}}}_{j}}{{{\hat{X}}}_{j}},[\sqrt{\xi }{{{\hat{n}}}_{j}}{{{\hat{P}}}_{j}}+\sqrt{1/\xi }{{{\hat{O}}}_{j}}{{{\hat{X}}}_{j}},\hat{\rho }\otimes |{\rm vac}\rangle \langle {\rm vac}|]]. \\ \end{array}\end{eqnarray*}$$

Here we have dropped all terms of order τ^3/2 and higher, as after taking the derivative they vanish in the τ→0 limit. Taking the trace over the ancilla, the first line gives the expected interaction term $\hat{\rho }-{\rm i}\tau [{{\hat{n}}_{j}}{{\hat{O}}_{j}},\hat{\rho }]$, while the second line vanishes as $\left\langle {\rm vac} \right|{{\hat{X}}_{j}}\left| {\rm vac} \right\rangle =\left\langle {\rm vac} \right|{{\hat{P}}_{j}}\left| {\rm vac} \right\rangle =0$. The final line is associated with the back-action and feedback noise introduced into the system. Using $\langle \hat{P}_{j}^{2}\rangle =\langle \hat{X}_{j}^{2}\rangle =1/2$, $\langle {{\hat{X}}_{j}}{{\hat{P}}_{j}}+{{\hat{P}}_{j}}{{\hat{X}}_{j}}\rangle =0$, and the cyclic property of the trace, one can show it is equal to

$$\begin{eqnarray}&&\mathcal{L}_{{\rm noise}}^{j}(\hat{\rho })=-\frac{\xi }{2}[{{\hat{n}}_{j}},[{{\hat{n}}_{j}},\hat{\rho }]]-\frac{1}{2\xi }[{{\hat{O}}_{j}},[{{\hat{O}}_{j}},\hat{\rho }]].\end{eqnarray} \tag{ 5 }$$

The complete master equation is obtained by summing over effects from all sites j, as well as including local time evolution due to $\hat{H}$₀ for time τ.

Equation (5) represents the central identity of this paper. Choice of the measurement strength parameter ξ may be optimized, as we describe below, to bound various potential non-quantum theories. Curiously, this equation is reminiscent of the similar master equation deduced by Diosi [15], which was seen as necessary to maintain reversibility in the Newton–Schrodinger equation evolution. Here we have explicitly constructed the theory to be compatible with standard quantum mechanics, and find that the intuitive picture elucidated by Diosi appears to be comparable to the measure-and-feedback approach.

Using this formalism, we now calculate the effects of the measurement and feedback noise super-operator, ${{\mathcal{L}}_{{\rm noise}}}={{\sum }_{j}}\mathcal{L}_{{\rm noise}}^{j}$. To do so we change to the Heisenberg picture, for which the relevant super-operator is its Hermitian adjoint, defined explicitly through the Hilbert–Schmidt inner product, ${\rm Tr}\left[ \mathcal{L}_{{\rm noise}}^{\dagger }({{{\hat{A}}}^{\dagger }})\hat{B} \right]\equiv {\rm Tr}\left[ {{{\hat{A}}}^{\dagger }}{{\mathcal{L}}_{{\rm noise}}}(\hat{B}) \right]$. By the cyclic property of the trace, we have that in this case the noise super-operator is self-adjoint, i.e. $\mathcal{L}_{{\rm noise}}^{\dagger }={{\mathcal{L}}_{{\rm noise}}}$.

Before proceeding to calculating the noise and dephasing, we first must address a technical point. Strictly speaking, the action of $\mathcal{L}_{{\rm noise}}^{\dagger }$ is not well defined on all operators, as the sum implicit in ${{\hat{O}}_{j}}$ in equation (5) may diverge. In our analysis we therefore only consider operators which conserve the total particle number, since (as we show below) the action of $\mathcal{L}_{{\rm noise}}^{\dagger }$ is bounded on these. In essence, we are restricting the class of physical operations and observables to those satisfying a total particle number super-selection rule (which the Hamiltonian and ${{\mathcal{L}}_{{\rm noise}}}$ already satisfy). More specifically, we consider the dynamics of operators of the form

$$\begin{eqnarray}&&\hat{C}=\hat{a}_{{{j}_{1}}}^{\dagger }\hat{a}_{{{j}_{2}}}^{\dagger }\;\ldots \;\hat{a}_{{{j}_{N}}}^{\dagger }{{\hat{a}}_{{{i}_{1}}}}{{\hat{a}}_{{{i}_{2}}}}\;\ldots \;{{\hat{a}}_{{{i}_{N}}}}\end{eqnarray} \tag{ 6 }$$

for some positive integer N. We posit that any (possibly unbounded) operator which conserves total particle number $({{\sum }_{k}}{{\hat{n}}_{k}})$ and can be written in normal order can necessarily be written as sums of terms looking like $\hat{C}$. To consider bounded operators which conserve total particle number, we may equivalently assume $\hat{C}$ is of the form $\hat{a}_{{{j}_{1}}}^{\dagger }\hat{a}_{{{j}_{2}}}^{\dagger }\;\ldots \;\hat{a}_{{{j}_{N}}}^{\dagger }|{\rm vac}\rangle \langle {\rm vac}|{{\hat{a}}_{{{i}_{1}}}}{{\hat{a}}_{{{i}_{2}}}}\;\ldots \;{{\hat{a}}_{{{i}_{N}}}}$, where $\left| {\rm vac} \right\rangle$ is the many-body vacuum state corresponding to zero mass. Since $[{{\hat{n}}_{i}},|{\rm vac}\rangle \langle {\rm vac}|]=0$ for all i, the following arguments are identical in either case.

To begin, we note the following identity

$$\begin{eqnarray*}&&[\hat{A},{{\hat{B}}_{1}}{{\hat{B}}_{2}}\;\ldots \;{{\hat{B}}_{N}}]=[\hat{A},{{\hat{B}}_{1}}]({{\hat{B}}_{2}}\;\ldots \;{{\hat{B}}_{N}})+{{\hat{B}}_{1}}[\hat{A},{{\hat{B}}_{2}}]({{\hat{B}}_{3}}\;\ldots \;{{\hat{B}}_{N}})+\;\cdots \;+({{\hat{B}}_{1}}\;\ldots \;{{\hat{B}}_{k-1}})[\hat{A},{{\hat{B}}_{N}}].\end{eqnarray*}$$

Given $[{{\hat{n}}_{k}},{{\hat{a}}_{i}}]=-{{\delta }_{ki}}{{\hat{a}}_{i}}$ and $[{{\hat{n}}_{k}},\hat{a}_{j}^{\dagger }]={{\delta }_{kj}}\hat{a}_{j}^{\dagger }$, we then quickly compute

$$\begin{eqnarray*}&&[{{\hat{n}}_{k}},[{{\hat{n}}_{k^{\prime} }},\hat{C}]]=(\mathop{\sum }\limits_{n=1}^{N}({{\delta }_{k{{j}_{n}}}}-{{\delta }_{k{{i}_{n}}}}))(\mathop{\sum }\limits_{m=1}^{N}({{\delta }_{k^{\prime} {{j}_{m}}}}-{{\delta }_{k^{\prime} {{i}_{m}}}}))\hat{C}\;.\end{eqnarray*}$$

Hence the action of $\mathcal{L}_{{\rm noise}}^{\dagger }$ on $\hat{C}$ is

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{L}_{{\rm noise}}^{\dagger }(\hat{C}) & = & -\frac{1}{2}\mathop{\sum }\limits_{l}\left[ \xi {{(\mathop{\sum }\limits_{n=1}^{N}({{\delta }_{l{{j}_{n}}}}-{{\delta }_{l{{i}_{n}}}}))}^{2}} \right. \\ {} & {} & +\left. \frac{1}{\xi }\mathop{\sum }\limits_{kk^{\prime} }{{\chi }_{lk}}{{\chi }_{lk^{\prime} }}(\mathop{\sum }\limits_{n=1}^{N}({{\delta }_{k{{j}_{n}}}}-{{\delta }_{k{{i}_{n}}}}))(\mathop{\sum }\limits_{m=1}^{N}({{\delta }_{k^{\prime} {{j}_{m}}}}-{{\delta }_{k^{\prime} {{i}_{m}}}})) \right]\hat{C}. \\ \end{array}\end{eqnarray} \tag{ 7 }$$

Considering the term proportional to ξ, observe that the inner sum is just $\#\{{{j}_{n}}=l\}-\#\{{{i}_{n}}=l\}$, i.e. the number of j_n indices equal to l minus the same for i_n. Hence the sum over l is simply a (finite) combinatorial function of the indices

$$\begin{eqnarray}&&-\xi \frac{1}{2}\mathop{\sum }\limits_{l}{{(\#\{{{j}_{n}}=l\}-\#\{{{i}_{n}}=l\})}^{2}}.\end{eqnarray} \tag{ 8 }$$

Likewise, the term proportional to $\frac{1}{\xi }$ is

$$\begin{eqnarray}&&\begin{array}{l} -\frac{1}{2\xi }\mathop{\sum }\limits_{l}(\mathop{\sum }\limits_{kk^{\prime} }{{\chi }_{lk}}{{\chi }_{lk^{\prime} }}(\mathop{\sum }\limits_{n=1}^{N}({{\delta }_{k{{j}_{n}}}}-{{\delta }_{k{{i}_{n}}}}))(\mathop{\sum }\limits_{m=1}^{N}({{\delta }_{k^{\prime} {{j}_{m}}}}-{{\delta }_{k^{\prime} {{i}_{m}}}}))) \\ =-\frac{1}{2\xi }\mathop{\sum }\limits_{l}(\mathop{\sum }\limits_{n=1}^{N}({{\chi }_{l{{j}_{n}}}}-{{\chi }_{l{{i}_{n}}}}))(\mathop{\sum }\limits_{m=1}^{N}({{\chi }_{l{{j}_{m}}}}-{{\chi }_{l{{i}_{m}}}})) \\ =-\frac{1}{2\xi }\mathop{\sum }\limits_{n=1}^{N}\mathop{\sum }\limits_{m=1}^{N}\mathop{\sum }\limits_{l}({{\chi }_{l{{j}_{n}}}}-{{\chi }_{l{{i}_{n}}}})({{\chi }_{l{{j}_{m}}}}-{{\chi }_{l{{i}_{m}}}}), \\ \end{array}\end{eqnarray} \tag{ 9 }$$

where we have only interchanged orders of summation involving finite sums. Using the definition of ${{\chi }_{ij}}$, this sum can be rewritten as

$$\begin{eqnarray}&&{{\left( \frac{G{{m}^{2}}}{2\hbar } \right)}^{2}}\mathop{\sum }\limits_{l}\frac{({{d}_{{{j}_{n}}l}}-{{d}_{{{i}_{n}}l}})({{d}_{{{j}_{m}}l}}-{{d}_{{{i}_{m}}l}})}{({{d}_{{{j}_{n}}l}}+a)({{d}_{{{j}_{m}}l}}+a)({{d}_{{{i}_{n}}l}}+a)({{d}_{{{i}_{m}}l}}+a)},\end{eqnarray} \tag{ 10 }$$

where ${{d}_{ij}}=|{{\bar{x}}_{i}}-{{\bar{x}}_{j}}|$.

We note that, as in the first term of equation (7), the above sum is finite: from the triangle inequality, $|{{d}_{ij}}-{{d}_{kj}}|\leqslant {{d}_{ik}}$, we can bound the absolute value of the numerator by ${{d}_{{{j}_{n}}{{i}_{n}}}}{{d}_{{{j}_{m}}{{i}_{m}}}}$, which is independent of ${{\bar{x}}_{l}}$. Hence the sum above is absolutely convergent, and is bounded in magnitude by the sum

$$\begin{eqnarray*}&&{{d}_{{{j}_{n}}{{i}_{n}}}}{{d}_{{{j}_{m}}{{i}_{m}}}}{{\left( \frac{G{{m}^{2}}}{2\hbar } \right)}^{2}}\mathop{\sum }\limits_{l}\frac{1}{({{d}_{{{j}_{n}}l}}+a)({{d}_{{{j}_{m}}l}}+a)({{d}_{{{i}_{n}}l}}+a)({{d}_{{{i}_{m}}l}}+a)}\end{eqnarray*}$$

which is finite in three dimensions.

We now calculate the spatial dephasing rate of a single particle evolving under these dynamics, and find ξ that minimizes such dephasing. In the first quantized picture, its density matrix ${{\hat{\rho }}_{s}}$ would be written as

$$\begin{eqnarray}&&{{\hat{\rho }}_{s}}=\mathop{\sum }\limits_{ij}{{\rho }_{ij}}|i\rangle \langle j|\end{eqnarray} \tag{ 11 }$$

with $\left| j \right\rangle$ being the single particle state at site j. This maps into the second quantized picture as

$$\begin{eqnarray}&&\hat{\rho }=\mathop{\sum }\limits_{ij}{{\rho }_{ij}}a_{i}^{\dagger }|{\rm vac}\rangle \langle {\rm vac}|{{a}_{j}},\end{eqnarray} \tag{ 12 }$$

where $\left| {\rm vac} \right\rangle$ is the vacuum state. Defining the operators

$$\begin{eqnarray}&&{{\hat{C}}_{ji}}\equiv \hat{a}_{j}^{\dagger }|{\rm vac}\rangle \langle {\rm vac}|{{\hat{a}}_{i}}\end{eqnarray} \tag{ 13 }$$

and noticing that ${{\rho }_{ij}}={\rm Tr}\left[ {{{\hat{C}}}_{ji}}\hat{\rho } \right]$, we have that for dynamics solely determined by the operators in equation (7),

$$\begin{eqnarray}&&{{\dot{\rho }}_{ij}}={\rm Tr}\left[ {{{\hat{C}}}_{ji}}{{\mathcal{L}}_{{\rm noise}}}(\hat{\rho }) \right]={\rm Tr}\left[ \mathcal{L}_{{\rm noise}}^{\dagger }({{{\hat{C}}}_{ji}})\hat{\rho } \right].\end{eqnarray} \tag{ 14 }$$

Finally, from equations (8) and (10), for $j\ne i$ we have

$$\begin{eqnarray}&&\mathcal{L}_{{\rm noise}}^{\dagger }({{\hat{C}}_{ji}})=-(\xi +\frac{1}{2\xi }\kappa _{ji}^{2}){{\hat{C}}_{ji}},\end{eqnarray} \tag{ 15 }$$

where the latter frequency is

$$\begin{eqnarray}&&\kappa _{ji}^{2}={{\left( \frac{G{{m}^{2}}}{2a\hbar } \right)}^{2}}\mathop{\sum }\limits_{l}\frac{{{({{d}_{il}}/a-{{d}_{jl}}/a)}^{2}}}{{{({{d}_{jl}}/a+1)}^{2}}{{({{d}_{il}}/a+1)}^{2}}}.\end{eqnarray} \tag{ 16 }$$

The dephasing rate is minimized at ${{\xi }^{2}}=\frac{1}{2}\kappa _{ji}^{2}$, giving a minimal decay rate of $\sqrt{2}{{\kappa }_{ji}}$. We now proceed to bound this rate from below.

Assuming that $|\mathcal{D}|=\frac{1}{a}{{d}_{ji}}\gg 1$, the sum of equation (16) is approximated by the dimensionless integral expression

$$\begin{eqnarray}&&I=\int {{{\rm d}}^{3}}\;u\frac{{{(|u-{{{\bar{r}}}_{j}}|-|u-{{{\bar{r}}}_{i}}|)}^{2}}}{{{(1+|u-{{{\bar{r}}}_{j}}|)}^{2}}{{(1+|u-{{{\bar{r}}}_{i}}|)}^{2}}},\end{eqnarray} \tag{ 17 }$$

where ${{\bar{r}}_{j}}={{\bar{x}}_{j}}/a$. By translating $u\to u+\frac{{{{\bar{r}}}_{i}}+{{{\bar{r}}}_{j}}}{2}$ and then rotating so that the vector $\bar{\mathcal{D}}={{\bar{r}}_{j}}-{{\bar{r}}_{i}}$ is on the z axis, this expression can be written in cylindrical coordinates as

$$\begin{eqnarray*}&&2\pi \int _{0}^{\infty }{\rm d}r\int _{-\infty }^{\infty }{\rm d}z\frac{r{{(\sqrt{{{r}^{2}}+{{(z-|\mathcal{D}|/2)}^{2}}}-\sqrt{{{r}^{2}}+{{(z+|\mathcal{D}|/2)}^{2}}})}^{2}}}{{{(1+\sqrt{{{r}^{2}}+{{(z-|\mathcal{D}|/2)}^{2}}})}^{2}}{{(1+\sqrt{{{r}^{2}}+{{(z+|\mathcal{D}|/2)}^{2}}})}^{2}}}.\end{eqnarray*}$$

It is then straightforward to show that the numerator is larger than $r{{(z|\mathcal{D}|)}^{2}}/({{z}^{2}}+{{r}^{2}}+|\mathcal{D}{{|}^{2}}/4)$, and the denominator is smaller than ${{(1+\sqrt{{{r}^{2}}+{{z}^{2}}+|\mathcal{D}{{|}^{2}}/4})}^{4}}$ . The integral I is therefore larger than

$$\begin{eqnarray*}&&I\gt \int _{0}^{\infty }{\rm d}r\int _{-\infty }^{\infty }{\rm d}z\frac{2\pi {{(z|\mathcal{D}|)}^{2}}r}{{{\left( 1+\sqrt{{{r}^{2}}+{{z}^{2}}+|\mathcal{D}{{|}^{2}}/4} \right)}^{6}}}=\frac{{{\pi }^{2}}}{2}|\mathcal{D}|(1+O\left( |\mathcal{D}{{|}^{-1}} \right)).\end{eqnarray*}$$

Hence for a particle of mass m in a spatial superposition of distance ${{d}_{ij}}=|\mathcal{D}|a$, we should observe a spatial dephasing rate on the order of at least ${{\kappa }_{ij}}\gtrsim \frac{G{{m}^{2}}}{2\;a\hbar }\sqrt{\frac{{{d}_{ij}}}{a}}$. We remark that this result is similar, at short distances, to the result of [4], where the decoherence rate for a particle separated by one distance of 'measurement' uncertainty goes as $\frac{G{{M}^{2}}}{2a\hbar }$.

We can already constrain such dephasing by considering interference of large molecules as described in [19, 20]. To do so, we treat the center of mass of each molecule as the effective position coordinate of the particles in our theory. These effective particles do not occupy the full mass distribution of the molecules, but rather are spread over a length on the order of their de Broglie wave length ($\hbar /p\sim {{10}^{-12}}\;{\rm m}$). This treatment is valid if, over the timescale of the experiment, this coordinate does not strongly interact with other (internal or environmental) degrees of freedom. Our assumption is supported, a fortiori, from the fact that interference is observed in the experiment. For a mass of $m\sim 5000$ amu, a spatial superposition $\Delta \sim 0.5\ \mu {\rm m}$ at the second diffraction grating (set by the standing wave length scale), and an observed coherence time $\sim 1$ ms, we find that $a\gt {{10}^{-19}}$ m. This is 16 orders of magnitude larger than the Planck length, or about 100 GeV, a substantially lower energy scale than expected for gravitational physics to emerge.

Having established that restricting potential communication via gravity to being classical would cause a mass superposition to dephase, we next consider what happens in the case of a large object composed of many particles. Although a detailed description requires the addition of other forces and particle species, for simplicity we focus on the consequences of including ${{\mathcal{L}}_{{\rm noise}}}$ of equation (5) in the general equation of motion. Towards this end we will consider the evolution of the particles' momentum, which requires understanding how 'hopping' terms in the discrete variable representation evolve under the super-operator ${{\mathcal{L}}_{{\rm noise}}}$.

To start, we look at the evolution of $\hat{a}_{j}^{\dagger }{{\hat{a}}_{i}}$, for $j\ne i$:

$$\begin{eqnarray}&&{{\mathcal{L}}_{{\rm noise}}}(\hat{a}_{j}^{\dagger }{{\hat{a}}_{i}})=-(\xi +\frac{1}{2\xi }\kappa _{ji}^{2})\hat{a}_{j}^{\dagger }{{\hat{a}}_{i}},\end{eqnarray} \tag{ 18 }$$

where the latter constant is defined as in equation (16). From our previous bound for ${{\kappa }_{ij}}$ at large d_ij, we have that for any $i\ne j$, the evolution of $\hat{a}_{j}^{\dagger }{{\hat{a}}_{i}}$ is damped at a rate on the order of at least $\frac{G{{m}^{2}}}{\hbar a}(1+O(\sqrt{\frac{|{{d}_{ij}}|}{a}}))$. We now consider a finite region of space S, having a total mass operator

$$\begin{eqnarray*}&&{{\hat{M}}_{S}}=m\mathop{\sum }\limits_{j\in S}\hat{a}_{j}^{\dagger }{{\hat{a}}_{j}}.\end{eqnarray*}$$

Letting ${{\hat{\Pi }}_{S}}$ denote the projector onto the indices $j\in S$, the net momentum in this space is

$$\begin{eqnarray}&&{{\bar{p}}_{S}}={{\hat{\Pi }}_{S}}({{a}^{3}}\hbar \int {{{\rm d}}^{3}}k\;\bar{k}\;\tilde{a}_{{\bar{k}}}^{\dagger }{{\tilde{a}}_{{\bar{k}}}}){{\hat{\Pi }}_{S}},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}&&=\;{{a}^{3}}\hbar \mathop{\sum }\limits_{j,j^{\prime} \in S}\;\hat{a}_{j^{\prime} }^{\dagger }{{\hat{a}}_{j}}\int {{{\rm d}}^{3}}k\;\bar{k}{{{\rm e}}^{-{\rm i}({{{\bar{x}}}_{j}}-{{{\bar{x}}}_{j^{\prime} }})\cdot \bar{k}}},\end{eqnarray} \tag{ 20 }$$

where the integral is over $\bar{k}\in {{[-\pi /a,\pi /a]}^{\otimes 3}}$, and we used the defintion

$$\begin{eqnarray*}&&{{\tilde{a}}_{{\bar{k}}}}=\frac{1}{\sqrt{{{(2\pi )}^{3}}}}\mathop{\sum }\limits_{j}{{{\rm e}}^{-{\rm i}{{{\bar{x}}}_{j}}\cdot \bar{k}}}{{\hat{a}}_{j}}.\end{eqnarray*}$$

From the above analysis, we see that ${{\mathcal{L}}_{{\rm noise}}}$ causes $\hat{a}_{j^{\prime} }^{\dagger }{{\hat{a}}_{j}}$ to decays exponentially to 0 at a rate at least $\sim \frac{G{{m}^{2}}}{\hbar a}$, hence, ${{\bar{p}}_{S}}$ tends to 0 at this rateas well.

Likewise, at a similar rate $\sim \frac{G{{m}^{2}}}{\hbar a}$ the variance of the net momentum approaches

$$\begin{eqnarray}\begin{array}{rcl} \hat{p}_{S}^{2} & = & {{{\hat{\Pi }}}_{S}}({{\hbar }^{2}}{{a}^{3}}\int {{{\rm d}}^{3}}k\;|\bar{k}{{|}^{2}}\;\tilde{a}_{{\bar{k}}}^{\dagger }{{\tilde{a}}_{{\bar{k}}}}){{{\hat{\Pi }}}_{S}} \\ {} & = & {{\hbar }^{2}}{{a}^{3}}\mathop{\sum }\limits_{j,j^{\prime} \in S}(\int {{{\rm d}}^{3}}\;k{{{\rm e}}^{-{\rm i}({{{\bar{x}}}_{j}}-\bar{x}_{j}^{{\prime} })\cdot \bar{k}}}\;|\bar{k}{{|}^{2}})\hat{a}_{j^{\prime} }^{\dagger }{{{\hat{a}}}_{j}} \\ {} & \to & {{\hbar }^{2}}\frac{{{a}^{3}}}{{{(2\pi )}^{3}}}\int {{{\rm d}}^{3}}k\;|\bar{k}{{|}^{2}}\mathop{\sum }\limits_{j\in S}\hat{a}_{j}^{\dagger }{{{\hat{a}}}_{j}} \\ {} & = & \frac{1}{m}{{\left( \frac{2\pi \hbar }{a} \right)}^{2}}{{{\hat{M}}}_{S}}. \\ \end{array}\end{eqnarray} \tag{ 21 }$$

Since this asymptotic value is arbitrarily large as $a\to 0$, we can use it to approximate a heating rate of the space S as

$$\begin{eqnarray}&&{{\mathcal{L}}_{{\rm noise}}}(\frac{\hat{p}_{S}^{2}}{2m})\sim (\frac{G{{m}^{2}}}{\hbar a})\frac{1}{2m}(\frac{1}{m}{{\left( \frac{2\pi \hbar }{a} \right)}^{2}}{{\hat{M}}_{S}})\gtrsim \frac{G\hbar }{{{a}^{3}}}{{\hat{M}}_{S}}.\end{eqnarray} \tag{ 22 }$$

For an arbitrary region of space S, our theory therefore predicts a generic heating rate proportional to the the total mass contained in that space.

We can use this heating rate, a consequence of restricting the gravitational potential to only convey classical information, to get experimental bounds for the length scale a of our theory. Namely, a critical temperature $\sim 500$ nK and typical lifetime of 5 s for Rubidium-87 BECʼs yields [21]a per-atom heating rate of ${{\kappa }_{{\rm BEC}}}\sim {{10}^{-30}}$ J s⁻¹ . Comparing this to the above bound with $\langle {{\hat{M}}_{S}}\rangle ={{M}_{{\rm Rb}}}\sim {{10}^{-25}}$ kg yields $a\gg {{\left( G{{M}_{{\rm Rb}}}\hbar /{{\kappa }_{{\rm Rb}}} \right)}^{1/3}}\sim {{10}^{-13}}\;{\rm m}$. At a much different mass scale, we can also compare the predicted heating rate of the Earth to its total absorbed radiative energy from the Sun, on the order of $\sim 1\times {{10}^{17}}$ J s⁻¹. Using $\langle {{\hat{M}}_{S}}\rangle ={{M}_{E}}$ gives a slightly tighter bound $a\gg {{10}^{-12}}$ m.

We have shown that combining observational constraints of interactions indicative of a $1/{{r}^{2}}$ force law with the concept that such forces cannot communicate quantum information leads to a simple model for 'classical communication only' theories of gravity. Our particular approach, using measurement and feedback as calculational tools, saturates the minimum noise and dephasing rates anticipated for any such model. Examination of two particular cases—matter wave interference and heating of massive objects—indicate that most likely any 'classical communication only' model of gravity breaks down at length scales larger than expected for new gravitational physics to emerge. However, stringent tests of softening of Newtonian interactions at short length scales remain before these bounds become complete. Further extensions of these concepts to relativistic domains are in principle straightforward, and may provide a framework within which one can understand a wide variety of different heuristic models of gravitational decoherence.

The authors would like to thank J Preskill, C Caves, B-L Hu, and S Schlamminger for helpful commentary and discussions on these topics, and the hospitality of the Kavli Institute for Theoretical Physics, where these ideas were first considered. Support is provided by the NSF-funded physical frontier center at the JQI.