{"id":218466,"date":"2025-10-22T18:28:09","date_gmt":"2025-10-22T18:28:09","guid":{"rendered":"https:\/\/www.newsbeep.com\/uk\/218466\/"},"modified":"2025-10-22T18:28:09","modified_gmt":"2025-10-22T18:28:09","slug":"classical-theories-of-gravity-produce-entanglement","status":"publish","type":"post","link":"https:\/\/www.newsbeep.com\/uk\/218466\/","title":{"rendered":"Classical theories of gravity produce entanglement"},"content":{"rendered":"

To illustrate this further, we now demonstrate how equation (4<\/a>) leads to entanglement in a version of Feynman\u2019s experiment. Two spherical mass distributions, each with total mass M and radius R, are prepared in a quantum superposition of two locations. This could be achieved by, for example, implementing matter-wave beam splitters6<\/a>, manipulating potentials32<\/a> or exploiting internal degrees of freedom, such as quantum spins in Stern\u2013Gerlach experiments5<\/a> (Fig. 3<\/a>). Gravity is assumed to be the only interaction between the particles, and in the non-relativistic limit and describing matter within first quantization, it just acts as a quantum phase \u03c6ij\u2009\u2254\u2009GM2t\/(\u0127\u2009dij) on each superposition branch5<\/a>,6<\/a>, where dij is the distance between the matter distributions in the branch labelled by i,\u00a0j\u2009\u2208\u2009{L, R}, and GM2\/dij is the Newtonian potential energy. With the superposition size \u0394x much greater than the smallest distance dRL, only the quantum phase \u03c6\u2009\u2254\u2009\u03c6RL is significant, so that the systems are clearly entangled, with the entanglement depending solely on \u03c6 (refs. 5<\/a>,6<\/a>). By contrast, when \u0394x\u2009\u226a\u2009dRL, the relevant parameter for entanglement becomes essentially \\(\\overline{\\varphi }:= \\varphi \\,\\Delta {x}^{2}\/{d}_{{\\rm{RL}}}^{2}\\) (ref. 33<\/a>). To measure the entanglement, the superposed paths could be recombined and correlations sought between the interferometer outputs6<\/a> or internal degrees of freedom5<\/a>.<\/p>\n

Fig. 3: Visualization of a version of Feynman\u2019s experiment.\"figure<\/a><\/p>\n

Two spherical mass distributions (1 and 2) of radius R are placed in quantum superpositions at two locations as N00N states, with blue and red denoting the components separated by \u0394x. After gravitationally interacting for a short time, the paths are recombined and entanglement is sought5<\/a>,6<\/a>. Although Stern\u2013Gerlach interferometry with internal spins is illustrated5<\/a>, alternative set-ups, such as parallel Mach\u2013Zehnders, are also possible6<\/a>. Here, \u0394x is depicted larger than the minimum separation dRL, but a general configuration can be implemented, including \u0394x\u2009\u226a\u2009dRL.<\/p>\n

Quantum gravity<\/p>\n

We now analyse this experiment using perturbative quantum gravity with a QFT description of matter. With electromagnetic interactions ignored, we take the initial state of the objects immediately after being placed in a quantum superposition as a product of N00N states:<\/p>\n

$$\\begin{array}{l}|\\varPsi \\rangle =\\frac{1}{2}({|N\\rangle }_{{\\rm{1L}}}{|0\\rangle }_{{\\rm{1R}}}{|\\uparrow \\rangle }_{1}+{|0\\rangle }_{{\\rm{1L}}}{|N\\rangle }_{{\\rm{1R}}}{|\\downarrow \\rangle }_{1})\\\\ \\qquad \\otimes ({|N\\rangle }_{{\\rm{2L}}}{|0\\rangle }_{{\\rm{2R}}}{|\\uparrow \\rangle }_{2}+{|0\\rangle }_{{\\rm{2L}}}{|N\\rangle }_{{\\rm{2R}}}{|\\downarrow \\rangle }_{2}),\\end{array}$$<\/p>\n

\n (5)\n <\/p>\n

where |N\u27e9\u03bai is a product of N independent position states of matter particles obeying a complex scalar field34<\/a>, with \u03ba\u2009\u2208\u2009{1, 2} and i\u2009\u2208\u2009{L, R} labelling the position of the spherical objects (matching Fig. 3<\/a>). N is the number of particles in the objects, such that M\u2009=\u2009mN, with m the mass of the particles. We also include possible internal spin states {|\u2191\u27e9, |\u2193\u27e9}, which could be used to generate the quantum superpositions.<\/p>\n

After a time t, the state of the matter system in the Schr\u00f6dinger picture is<\/p>\n

$$| \\varPsi (t)\\rangle ={{\\rm{e}}}^{{\\rm{i}}{\\widehat{H}}_{0}t\/\\hbar }\\widehat{T}{{\\rm{e}}}^{-({\\rm{i}}\/\\hbar ){\\int }_{0}^{t}{\\rm{d}}\\tau {\\widehat{H}}_{{\\rm{I}}}(\\tau )}| \\varPsi \\rangle ,$$<\/p>\n

\n (6)\n <\/p>\n

where \\(\\widehat{T}\\) is the time-ordering operator, \u03c4 is a dummy time variable, \\({\\widehat{H}}_{0}\\) is the Hamiltonian of perturbative quantum gravity in the absence of matter\u2013gravity interactions and \\({\\widehat{H}}_{{\\rm{I}}}:= \\exp ({\\rm{i}}{\\widehat{H}}_{0}t\/\\hbar ){\\widehat{H}}_{{\\rm{int}}}\\,\\exp (-{\\rm{i}}{\\widehat{H}}_{0}t\/\\hbar )\\). Just before the superposition paths are bought back together, for example through reverse Stern\u2013Gerlachs1<\/a>,5<\/a>, we can write<\/p>\n

$$\\begin{array}{l}|\\,\\varPsi (t)\\rangle \\propto {\\alpha }_{{\\rm{LL}}}{|N\\rangle }_{{\\rm{1L}}}{|N\\rangle }_{{\\rm{2L}}}+{\\alpha }_{{\\rm{LR}}}{|N\\rangle }_{{\\rm{1L}}}{|N\\rangle }_{{\\rm{2R}}}\\\\ \\qquad \\,+\\,{\\alpha }_{{\\rm{RL}}}{|N\\rangle }_{{\\rm{1R}}}{|N\\rangle }_{{\\rm{2L}}}+{\\alpha }_{{\\rm{RR}}}{|N\\rangle }_{{\\rm{1R}}}{|N\\rangle }_{{\\rm{2R}}},\\end{array}$$<\/p>\n

\n (7)\n <\/p>\n

where \\({\\alpha }_{ij}\\in {\\mathbb{C}}\\). We have now neglected any internal spin states and the vacuum states for simplicity. We can calculate the amplitudes \u03b1ij by taking the inner product of equation (7<\/a>) (and also equation (6<\/a>)), with the basis states |N\u27e91i\u2009|N\u27e92j and expanding the time-ordered unitary operation in equation (6<\/a>) as the Dyson series26<\/a>,35<\/a>. The amplitudes can then be written as a perturbative series with each term corresponding to that of the Dyson series: \\({\\alpha }_{ij}={\\alpha }_{ij}^{(0)}+{\\alpha }_{ij}^{(1)}+{\\alpha }_{ij}^{(2)}+\\cdots \\,\\). The first process where a virtual graviton is exchanged between the matter objects occurs at second order in the series and corresponds to the Feynman diagram in Fig. 1<\/a> (top left). The amplitude for this Feynman diagram, within the very good approximation ct\u2009\u226b\u2009dij (Methods<\/a>), is<\/p>\n

$$\\frac{G{M}^{2}t}{\\hbar {V}^{2}}\\iint {{\\rm{d}}}^{3}{\\bf{x}}\\,{{\\rm{d}}}^{3}{\\bf{y}}\\frac{{\\theta }_{1i}({\\bf{x}}){\\theta }_{2j}({\\bf{y}})}{| {\\bf{x}}-{\\bf{y}}| }\\equiv {\\varphi }_{ij},$$<\/p>\n

\n (8)\n <\/p>\n

where \u03b8\u03bai(x)\u2009\u2254\u2009\u03b8(R\u2009\u2212\u2009\u2223x\u2009\u2212\u2009X\u03bai\u2223) is the unit-step function defining the spherical shape of the matter distribution \u03ba in branch i, X\u03bai is the coordinate for the centre of mass for the distributions and V\u2009\u2254\u20094\u03c0R3\/3. The above amplitude directly contributes to \\({\\alpha }_{ij}^{(2)}\\) such that, when \u0394x\u2009\u226b\u2009dRL, the \\({\\alpha }_{{\\rm{RL}}}^{(2)}\\) amplitude dominates over all others and equals i\u03c6. Then, given that \\({\\alpha }_{ij}^{(0)}=1\\) from the Dyson series and that \\({\\alpha }_{ij}^{(1)}\\) contains no contractions for the gravitational field, we are left with \u03b1ij\u2009\u2248\u20091 except for \u03b1RL\u2009\u2248\u20091\u2009+\u2009i\u03c6, which matches the first-quantized result to first order in the quantum phase \\(\\exp ({\\rm{i}}\\varphi )\\). The full non-perturbative result is straightforwardly obtained by considering the form of the corresponding higher-order Feynman diagrams and extrapolating the result26<\/a>.<\/p>\n

Classical gravity<\/p>\n

We now consider the above experiment within the context of classical gravity. The calculation follows the above but with the interaction Hamiltonian of equation (4<\/a>) rather than equation (2<\/a>). At second order in the Dyson series there are no non-vanishing Wick contractions corresponding to Feynman diagrams that contain quantum communication between the matter objects, and the diagram responsible for entanglement in quantum gravity, Fig. 1<\/a> (top left), becomes Fig. 2<\/a> (top middle). This diagram represents the two matter objects sitting in their combined classical gravitational field, with the amplitude just contributing a local relative quantum phase between the branches of each matter object, which does not lead to entanglement13<\/a>. However, at fourth order in the series, a diagram appears where the matter distributions are connected quantum mechanically through virtual matter particles (Fig. 2<\/a>, top right). Within the same approximations as in the quantum gravity case, the amplitude for the Feynman diagram is (Methods<\/a>)<\/p>\n

$${{\\vartheta }}_{ij}:= \\frac{{m}^{6}{t}^{2}{N}^{2}}{4{{\\rm{\\pi }}}^{2}{\\hbar }^{6}{V}^{2}}{\\left(i\\int {{\\rm{d}}}^{3}{\\bf{x}}{{\\rm{d}}}^{3}{\\bf{y}}\\frac{\\varPhi ({\\bf{x}})\\varPhi ({\\bf{y}}){\\theta }_{1i}({\\bf{x}}){\\theta }_{2j}({\\bf{y}})}{| {\\bf{x}}-{\\bf{y}}| }\\right)}^{2},$$<\/p>\n

\n (9)\n <\/p>\n

where \u03a6(x)\u2009\u2254\u2009\u2212c2h00(x)\/2 is the total gravitational potential of the matter objects, and \u03d1ij directly contributes to \\({\\alpha }_{ij}^{(4)}\\) in equation (7<\/a>). As we have a classical theory of gravity, \u03a6(x) is the same in each superposition branch, otherwise the Newtonian force would be in a quantum superposition. Certain works have considered gravity to be classical but still allow the field or Newtonian force to be in a quantum superposition36<\/a>,37<\/a>,38<\/a>,39<\/a>. Here, we keep to the notion that quantum superposition is a purely quantum-mechanical phenomena such that \u03a6(x) is not superposed in equation (9<\/a>) and is not a quantum operator. However, despite this, equation (9<\/a>) generically results in entanglement, as \u03b81i(x) and \u03b82i(x) are different for the different superposition paths and are connected through \u2223x\u2009\u2212\u2009y\u2223, such that \\({\\alpha }_{ij}^{(4)}\\) is different for each superposition path, except for \\({\\alpha }_{{\\rm{LL}}}^{(4)}={\\alpha }_{{\\rm{RR}}}^{(4)}\\) from symmetry. We can understand this from the Feynman diagram in Fig. 2<\/a> (top right), where, unlike the gravitational potential, the virtual matter particles enter into a quantum superposition with the different mass states and the distance the virtual matter particles propagate is different in each branch, resulting in the spatial integrals in equation (9<\/a>) being connected through \u2223x\u2009\u2212\u2009y\u2223.<\/p>\n

As \u03a6(x) comes from a superposition of matter in equation (9<\/a>), we must consider exactly how gravity is sourced by quantum matter in a fundamental theory of classical gravity. In the approach most considered40<\/a>,41<\/a>, \u03a6(x) in equation (9<\/a>) is the sum of the average potentials of the superposition states of the two objects. In this case, \u03d1ij is inversely proportional to dij such that, if \u0394x\u2009\u226b\u2009dRL, the RL amplitude dominates over all others and is \u2223\u03d1RL\u2223\u2009\u2248\u2009\u03d1, where (Methods<\/a>)<\/p>\n

$$\\sqrt{{\\vartheta }}=\\frac{6}{25}\\frac{{G}^{2}{m}^{2}{M}^{3}Rt}{{\\hbar }^{3}{d}_{{\\rm{RL}}}}.$$<\/p>\n

\n (10)\n <\/p>\n

The state in equation (7<\/a>) is then clearly entangled, because, just as in quantum gravity, \u03b1RL contains a contribution \u03d1 that is not in any of the other amplitudes \u03b1ij. Furthermore, like quantum gravity33<\/a>, the inverse scaling of \u03d1ij with dij allows \\(\\overline{{\\vartheta }}:= {\\vartheta }\\,\\Delta {x}^{2}\/{d}_{{\\rm{RL}}}^{2}\\) to be identified as the relevant parameter for entanglement when \u0394x\u2009\u226a\u2009dRL.<\/p>\n","protected":false},"excerpt":{"rendered":"To illustrate this further, we now demonstrate how equation (4) leads to entanglement in a version of Feynman\u2019s…\n","protected":false},"author":2,"featured_media":218467,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[93502,4230,4231,2302,4835,90,56,54,55],"class_list":{"0":"post-218466","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-general-relativity-and-gravity","9":"tag-humanities-and-social-sciences","10":"tag-multidisciplinary","11":"tag-physics","12":"tag-quantum-information","13":"tag-science","14":"tag-uk","15":"tag-united-kingdom","16":"tag-unitedkingdom"},"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/218466","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/comments?post=218466"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/posts\/218466\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media\/218467"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/media?parent=218466"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/categories?post=218466"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/uk\/wp-json\/wp\/v2\/tags?post=218466"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}