Phase transitions in sequential weak measurements

Quantum measurement can be strong or weak, the former being important in readout and initialization of quantum objects and the latter being useful for monitoring and maneuvering quantum evolution. However, the boundary between weak and strong measurement is unclear. Here we show that a phase transition occurs in sequential quantum measurement, which unambiguously separates the weak and strong measurement by a critical value of measurement strength or duration. We formulate the probability distribution of the output of a sequence of quantum measurements as the Boltzmann distribution of an interacting spin model. The measurement results present phase transitions similar to those in the spin model. In particular the sequential commuting positive operator-valued measurement is mapped to a long-range Ising model, and a projective measurement emerges from sequential weak measurements when the strength or the number of measurements becomes above certain critical values, corresponding to the ferromagnetic phase transition of the spin model. This finding sheds insights on sequential quantum measurement, and also provides the theoretical foundation for constructing projective measurements from sequential weak measurements, which have applications in steering the quantum evolution and initializing quantum systems where strong measurement in a single shot is often not possible.

Figure 1

Free energy of the spin model corresponding to a sequential POVM. (a) Free energy as a function of measurement polarization $X$ for different measurement strengths $\lambda$ with the measurement time $m=1000$. (b) Free energy as a function of $X$ for different $m$ with $\lambda =0.0004$. Here we use the initial state of the TLS with $r_0^z=0$.

Figure 2

The histogram of the number of samples with respect to the measurement polarization $X$ in the Monte Carlo simulation of sequential POVMs in case II for different measurement times: (a) $m=50$, (b) $m=100$, (c) $m=200$, and (d) $m=500$. The red solid lines represent the exact probability distribution in Eq. (10). The measurement strength is $\lambda =0.01$. The simulation contains ${10}^4$ samples of sequential POVMs from the same initial state ($r_0^z=0$).

Figure 3

Phase transitions of sequential weak measurement. (a) Critical measurement time $m_C$ as a function of the measurement strength $\lambda$. (b), (d) The measurement polarization $X$ and $\partial X/\partial \sqrt{\lambda }$ as functions of $\sqrt{\lambda }$ for different measurement times $m$. (c), (e) $X$ and $\partial X/\partial \sqrt{\lambda }$ as functions of $m$ for different $\lambda$. In (b)–(d), the lines without (with) crosses represent the results from the exact model (the long-range Ising model). Here we use the initial state with $r_0^z=0$.

Figure 4

Probability distribution of a sequential POVM. (a) Probability distribution $P\left(X\right)$ of measurement polarization $X$ for different measurement strengths $\lambda$ with the measurement time fixed at $m=1000$. (b) Modified probability distribution $\sqrt{m}P\left(X\right)$ of measurement polarization $X$ for different measurement times $m$ with the measurement strength fixed at $\lambda =0.09$. In (a) and (b), the initial state of the TLS is in the equatorial plane of the Bloch sphere with $r_0^z=0$ and $r_0=1$. (c) and (d) are similar to (a) and (b) but with $r_0^z=0.2$ and $r_0=1$.

Figure 5

Determining the minima of free energy of the spin model corresponding to a sequential POVM. (a) The intersections between the two curves $f_1\left(X\right)=ln\left[\left(1+2X\right)/\left(1-2X\right)\right]$ and $f_2\left(X\right)=ln\left(\eta \right)tanh\left[ln\left(\eta \right)mX\right]$, where $\eta =ln\left[\left(1+\sqrt{\lambda }\right)/\left(1-\sqrt{\lambda }\right)\right]$, determine the free-energy minima. (b) Free energy as a function of the measurement polarization $X$ for different measurement strengths $\lambda$. The parameters are such that $m=1000$, $r_0^z=0$, and $r_0=1$.

Figure 6

Effects of initial states on the phase transitions of sequential weak measurement. Different initial states (given by the initial polarization $r_0^z$) are represented by different line colors. (a) Free energy as a function of the measurement polarization $X$. The measurement strength and measurement time are $m=1000$ and $\lambda =0.01$, respectively. (b), (d) The measurement polarization $X$ and the final Bloch vector polarization $r_m$ as functions of the square root of the measurement strength with measurement times fixed at $m=100$. (c), (e) The measurement polarization $X$ and the final Bloch vector polarization $r_m$ as functions of the measurement time with the measurement strength fixed at $\lambda =0.01$. In (b) and (c), the lines without (with) crosses represent the results from the exact model (the approximate long-range Ising model). In (d) and (e), the lines without (with) plus signs represent the $z$ ($x$) component of the final Bloch vector. Initially the TLS is in a pure state with $r_0=1$ and $r_0^y=0$.