Quantum Uncertainty Principles for Measurements with Interventions

Heisenberg’s uncertainty principle implies fundamental constraints on what properties of a quantum system we can simultaneously learn. However, it typically assumes that we probe these properties via measurements at a single point in time. In contrast, inferring causal dependencies in complex processes often requires interactive experimentation—multiple rounds of interventions where we adaptively probe the process with different inputs to observe how they affect outputs. Here, we demonstrate universal uncertainty principles for general interactive measurements involving arbitrary rounds of interventions. As a case study, we show that they imply an uncertainty trade-off between measurements compatible with different causal dependencies.

Figure 1

Interactive measurements. Our uncertainty relations apply to all interactive measurements, including (a) passive measurements (framed by standard uncertainty relations) and (b) two-time measurements, where a quantum system first passes a quantum instrument that incorporates both a measurement outcome and the output state and later gets measured, as described by the framework of a pseudo-density matrix [18, 19]. Our results also pertain to (c) non-Markovian interactive measurements that involve coherently interacting the system with a quantum register $R$ and doing some joint measurement at a subsequent time step and, most generally, (d) any interactive measurement ${\mathcal{T}}_x$ with interventions at $a-1$ different time steps.

Figure 2

Quantum roulette is a game that aids in interpreting lower bounds for the combined uncertainty of two general interactive measurements $\{{\mathcal{T}}_b{\}}_{b=1,2}$. ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$ are shown in (b). Now introduce a quantum “roulette table” with $2\times m$ grid of cells (c), labeled $(b,x)$ with $x=1,\dots ,m$. In the $k$th-order game, Bob begins with $k$ chips, of which he can allocate to $k$ of these cells. Bob then supplies Alice with a dynamical process $\mathrm{\Phi }$ (a). Alice selects a $b$ at random and measures $\mathrm{\Phi }$ with ${\mathcal{T}}_b$ to obtain outcome $x$. Bob wins if he has a chip on the cell $(b,x)$. Lemma 1 and Theorem 1 then relate Bob’s winning probabilities with the incompatibility between ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$.

Figure 3

Quantum description of causal structures. There are three possible causal structures for two events $X$ and $Y$ (a), all of which can be expressed by a quantum dynamic process ${\mathrm{\Phi }}_{B\to AC}$ (b). (i) Direct cause, ${\mathrm{\Phi }}_{B\to AC}$ involves preparing a state $A$ to be observed at $X$, whose output is sent directly to $Y$ via quantum channel from $B$ to $C$. (ii) Common cause, correlations between $X$ and $Y$ can be attributed to measurements on some preprepared correlated state ${\rho }_{AC}$ (event $Z$). Most generally (iii), ${\mathrm{\Phi }}_{B\to AC}$ consists of a state-preparation process ${\mathrm{\Psi }}_{\mathbb{C}\to AE}^{\mathrm{Pre}}$ and a postprocessing quantum channel ${\mathrm{\Psi }}_{BE\to C}^{\text{Post}}$ [(b-iii); $E$ is an ancillary system]. This then corresponds to a (possibly coherent) mixture of direct and common cause.

Figure 4

Maximal common cause (CC) and direct cause (DC) indicators. We introduce (a) ${\mathcal{M}}_{\mathrm{CC}}=\{{\mathcal{T}}_{\mathrm{CC}}(U_1,U_2)\}$ and (b) ${\mathcal{M}}_{\mathrm{DC}}=\{{\mathcal{T}}_{\mathrm{DC}}(U_3,U_4)\}$ as two respective families of interactive measurements with a single intervention. Here, systems $A$, $B$, and $C$ are $d$-level quantum systems (qudits), and each $U_k$, $k=1$, 2, 3, 4 is some single-qudit unitary, and $|{\mathrm{\Phi }}_1⟩≔{\sum }_{k=0}^{d-1}|kk⟩/\sqrt{d}$. Measurements are done with respect to a maximally entangling basis $\{{\mathrm{\Phi }}_i{\}}_i$ with $d^2$ possible outcomes. The two measurement families are incompatible and satisfy the causal uncertainty relation in Eq. (3).

Figure 5

Causal uncertainty relations on non-Markovian dynamics. Consider a single qubit—bottom rail of the circuit in (a)—undergoing non-Markovian evolution. Here the systems are initialized in a maximally entangled state, $Z(\theta )$ and $X(\theta )$ represent single-qubit rotation gates in $X$ and $Z$ axis. (b) Illustrates the combined uncertainty $H({\mathcal{T}}_1)+H({\mathcal{T}}_2)$, where ${\mathcal{T}}_1\in M_{\mathrm{CC}}$ and ${\mathcal{T}}_2\in M_{\mathrm{DC}}$ are, respectively, common- and direct-cause indicators in Fig. 4 with all $U_k$ set to the identity. Observe that this never goes below the fundamental lower bound of 2 (gray plane). (c) Illustrates $H({\mathcal{T}}_1)$ (green dashed), $H({\mathcal{T}}_2)$ (red dashed), and their sum (blue solid) for $\alpha =-\pi /4$ and various values of $\beta$, corresponding to various coherent superpositions of common- and direct-cause circuits.