Delayed Choice Lorentz Transformations on a Qubit

TL;DR

This work introduces a four-dimensional spacetime representation of continuously monitored qubits by identifying unnormalized qubit states with a four-momentum-like object and showing that the enlarged transformation group $SL(2,\,\mathbb{C})$ governs their dynamics. Unitary (Hermitian) evolution corresponds to elliptic rotations, while measurement backaction corresponds to hyperbolic boosts, with stochastic, velocity-dependent electromagnetic-like fluctuations and a novel delayed-choice (retrocausal) backaction emerging in the Gaussian readout model. The authors derive deterministic and stochastic correspondences between point-charge Lorentz dynamics and qubit evolution, derive constraint conditions on the fluctuations needed to reproduce qubit stochastic master equations, and discuss how delayed-choice measurement angles determine past backaction. The framework offers a spacetime-based visualization of monitored quantum dynamics, clarifying nonclassical features and highlighting the role of additional degrees of freedom and retrocausal interpretation in quantum trajectories. Overall, the paper connects quantum measurement dynamics to Lorentz-transform-inspired geometry, providing both conceptual insight and a practical visualization tool for analyzing monitored qubit behavior, underpinned by a rigorous SL$(2,\,\mathbb{C})$–Lorentz correspondence.

Abstract

A continuously monitored quantum bit (qubit) exhibits competition between unitary Hamiltonian dynamics and non-unitary measurement-collapse dynamics, which for diffusive measurements form an enlarged transformation group equivalent to the Lorentz group of spacetime. We leverage this equivalence to develop a four-dimensional generalization of the three-dimensional Bloch ball to visualize the state of a monitored qubit as the four-momentum of an effective classical charge affected by a stochastic electromagnetic force field. Unitary qubit dynamics generated by Hermitian Hamiltonians correspond to elliptic spatial rotations of this effective charge while non-unitary qubit dynamics generated by non-Hermitian Hamiltonians or stochastic measurement collapse correspond to hyperbolic Lorentz boosts. Notably, to faithfully emulate the stochastic qubit dynamics arising from continuous qubit measurement, the stochastic electromagnetic fields must depend on the velocity of the charge they are acting on. Moreover, continuous qubit measurements admit a dynamical delayed choice effect where a future experimental choice can appear to retroactively determine the type of past measurement backaction, so the corresponding point charge dynamics can also exhibit delayed choice Lorentz transformations in which a future experimental choice determines whether stochastic force fields are electric or magnetic in character long after they interact with the particle.

Figures (5)

• Figure 1: Visualization of the correspondence between the structure of a qubit and a relativistic point particle for (a-c) hyperbolic purification of a maximally mixed state (corresponding to a massive particle at rest) and (d-f) rotational motion of a pure state $\ket{+x}$ (corresponding to a massless particle moving in the $x$ direction). Time evolution is represented in color going from light to dark. (a) The qubit state $\hat{\rho}$ (point particle velocity $\hat{\beta}$) is purified in the $z$ direction by measurement and asymptotically approaches a pure state (the speed of light). (b) The unnormalized qubit state $\hat{s}$ (particle momentum $\hat{p}$) is boosted in the $z$ direction, causing the momentum $p_z$ to asymptotically approach the surface of the lightcone. (c) The integrated velocity traces out a relativistic worldline $\hat{x}$ (cumulative state $\hat{\phi}$) in which the particle's lightcone is tangent at each point. (d) The pure qubit state $\hat{\rho}$ (particle velocity $\hat{\beta}$) undergoes Rabi oscillations about $\hat{\sigma}_y$. (e) The unnormalized qubit state $\hat{s}$ corresponds to a particle with momentum $\hat{p}$ initially in the $x$ direction, accelerated around the $y$ axis. (f) The integrated velocity traces out a helical worldline $\hat{x}$ (cumulative state $\hat{\phi}$), which normalizes to the mean velocity $\overline{\hat{\beta}}$ (time averaged state $\overline{\hat{\rho}}$) in panel (d).
• Figure 2: Combined hyperbolic and rotational motions for a pure qubit state initially in $\ket{+y}$, corresponding to a massless particle with velocity initially moving along $+y$. (a-c) Evolution generated by $\hat{H}_\mathrm{eff} = \mu (B_z + iE_z/c)\hat{\sigma}_z$ with $E_z / c B_z = 1/8$. (a) The non-Hermitian boost in $\hat{\sigma}_z$ generates hyperbolic evolution of $\hat{\rho}$ ($\hat{\beta}$) towards the qubit $\ket 0$ state ($z$ velocity), with simultaneous rotation about $\hat{\sigma}_z$ resulting in spiraling motion that stabilizes at the fixed point $\ket 0$. (b-c) The helical momentum $\hat{p}$ (unnormalized qubit state $\hat{s}$) (b) and worldline $\hat{x}$ (c) appear increasingly timelike in the $x$-$y$ slice of spacetime as the particle is boosted in the $z$ direction, corresponding to the normalized qubit state $\hat{\phi}$ (blue, panel (a)) converging to the fixed point. (d-f) Evolution generated by $\hat{H}_\mathrm{eff} = \mu (B_x\hat{\sigma}_x + i(E_z/c)\hat{\sigma}_z)$ with $E_z / c B_x = 2/3$. (d) The non-commutativity of $\hat{\sigma}_x$ and $\hat{\sigma}_z$, with $E_z /c B_x < 1$, gives rise to Bloch sphere rotations with modulated effective frequency, evidenced by the bias of the mean velocity (blue) towards $\ket{-y}$. (f) Elliptical helix of the worldline, elongated due to the competing electric and magnetic forces.
• Figure 2: Summary of the dynamical correspondence between a relativistic point charge (left) and a qubit (right). Unnormalized quantities exist in a 4-dimensional spacetime, while normalized quantities exist in a 3-dimensional Bloch ball. The commutator and anticommutator are defined as $[\hat{A}, \hat{B}] := \hat{A} \hat{B} - \hat{B} \hat{A}^\dagger$ and $\{ \hat{A}, \hat{B}\} = \hat{A} \hat{B} + \hat{B} \hat{A}^\dagger$, respectively.
• Figure 3: A superconducting qubit is dispersively coupled to a resonator via $\hat{H}_\mathrm{int}(t - \delta t_q)$ (a). This causes the entangled qubit-resonator state to reach a steady-state superposition of $\ket{0}\ket{\alpha_+}$ and $\ket{1}\ket{\alpha_-}$, sketched in (b) as the Husimi-Q function $Q(\alpha)$ of the conditioned resonator states. The resonator is pumped in reflection via the local oscillator, which entangles the qubit-resonator states with traveling transmission line states $\ket{\sqrt{\kappa \Delta t} \, \alpha_\pm}$ discretized by the detector integration time $\Delta t$ (c). The traveling modes are amplified along a quadrature at angle $\theta(t)$ that can be freely and continuously varied throughout the experiment, which determines whether the qubit undergoes unitary ($\theta = \pi/2$) backaction, non-unitary hyperbolic ($\theta = 0$) backaction, or a combination of both. The mixer multiplies the leaked, amplified resonator field with the LO and low-pass filters the output, yielding a slowly varying homodyne signal $I_{\theta(t)}(t + \delta t_r)$ randomly distributed as a mixture of two real Gaussians due to postselection of the transmission line modes $\ket{\sqrt{\kappa\Delta t}\, \alpha_\pm}$ by the tilted quadrature eigenstate $\bra{I_{\theta(t)}}$ (d). This observation at time $t + \delta t_r$ collapses the entangled state, retroactively determining the reduced qubit state at the prior original interaction time $t-\delta t_q$ as a result of the choice of amplification quadrature at time $t$.
• Figure 4: Quantum trajectory simulation depicting the normalized $\hat{\rho}(t)$ and unnormalized $\hat{s}(t)$ state evolution of a continuously monitored qubit under dynamical delayed choice of the homodyne (squeezing) angle $\theta(t)$. Equivalently, one may interpret these dynamics as the (normalized) velocity $\hat{\beta}$ and (unnormalized) momentum $\hat{p}$ evolution of a point charge in a velocity-dependent stochastic electromagnetic field with duality phase $\theta(t)$, in the form given by Eq. \ref{['eq:advanced-stochastic-field']}. (a) The squeezing angle is varied smoothly between informational ($\theta = 0$, red) and non-informational ($\theta=\pi/2$, blue) measurement backaction. (b) The time delay $\Gamma \delta t_q = 0.1$ between qubit and amplifier leads the choice of $\theta$ at times $t + \delta t_q$ to retroactively determine the inferred qubit dynamics at times $t$, as witnessed in the Bloch coordinates of the normalized state $\hat{\rho}$. (c) On the Bloch sphere, informational measurement (red) generates hyperbolic backaction between measurement eigenstates $\ket 0$ and $\ket 1$; non-informational measurement (blue) generates rotational backaction around $\hat{\sigma}_z$; combined informational and non-informational measurement (purple) leads to a combination of both. The initial and final states are marked by a triangle and square, respectively. (d) The unnormalized state $\hat{s}$ shows backaction in $s_z$ and $s_0$ for $\theta = 0$ and in $s_x$ and $s_y$ for $\theta = \pi/2$. The absence of stochasticity in the $x,y$ components for $\theta = 0$ (red background) highlights the purely hyperbolic nature of informational backaction, which is otherwise obscured in the timeseries data of $\hat{\rho}$ by renormalization. (e) The unnormalized state lives on a four-dimensional lightcone, represented here as the $s_x$-$s_y$-$s_0$ slice. Noninformational backaction is visible as rotations in $s_x$-$s_y$, while informational backaction manifests as vertical motion of $s_0$ ($s_z$ is not visible in this slice).