Optimal Pure-State Qubit Tomography via Sequential Weak Measurements
The spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continuous measurement with random controls, one can approach the optimal bound.
In the standard paradigm of quantum tomography, one is given copies of a quantum state that one seeks to estimate. When limited only by these finite quantum statistics and no other systematic experimental errors, what is the measurement that achieves the optimal average estimation fidelity? For the case of qubits, given a priori knowledge that the state is pure, this problem was solved long ago in a seminal paper by Massar and Popescu (MP) Massar and Popescu (1995). The optimal average fidelity is , and one can only reach this bound with a measurement that acts collectively on all copies. “Local” measurements acting nonadaptively on one copy at a time can only achieve at best a scaling of Jones (1994); Bagan et al. (2002); Mahler et al. (2013).
The MP bound is achieved by a measurement whose positive-operator-valued-measure (POVM) is an overcomplete basis whose elements are proportional to projectors onto spin-coherent states (SCS) of the collective spin in the symmetric subspace of qubits. The SCS-POVM is a fundamental measurement in quantum information science, with applications including metrology Holevo (1982); Appleby (2000), teleportation Braunstein et al. (2000), benchmarking Yang et al. (2014), and measurement of Husimi phase space probabilities Kofler and Brukner (2008). While the Glauber-coherent-state-POVM in infinite dimensions has a well-known implementation via heterodyne measurement D’Ariano (1997), despite various attempts Peres (2006); D’Ariano et al. (2001, 2002), there is no known implementation of POVMs over generalized-coherent-states for other Lie groups Gilmore (1972); Perelomov (1972), such as the SU(2)-coherent-states considered here (except for one qubit, , ) D’Ariano et al. (2002).
The SCS-POVM has been considered physically unattainable and previous works have constructed alternative POVMs that also attain the optimal bound for tomography of qubits and qudits Derka et al. (1998); Latorre et al. (1998); Bruß and Macchiavello (1999); Acín et al. (2000); Hayashi et al. (2005). While in principle one can use the Neumark extension to realize these POVMs consisting of a finite number of measurement outcomes, such constructions have limited applicability, particularly as grows beyond a few qubits.
In this Letter we show that the SCS-POVM is in fact physically realizable in a direct manner for the application of optimal tomography and other quantum information protocols. In particular, we show that we can realize the SCS-POVM by measuring the collective spin, , weakly and isotropically over a sufficiently long time. This sequence of weak measurements is in a similar spirit to continuous collective measurement tomography Silberfarb et al. (2005); Riofrio et al. (2011); Cook et al. (2014), which has been used for reconstructing states in a fast and robust manner Smith et al. (2006, 2013) as well as in the “retrodiction” of initial quantum states Gammelmark et al. (2013); Tan et al. (2015); Dressel et al. (2017); Hacohen-Gourgy et al. (2016); Guevara and Wiseman (2015). Here we show that the sequential isotropic protocol asymptotically saturates the MP bound in the appropriate limit.
To establish the foundation and notation, we briefly review the MP bound Massar and Popescu (1995). We consider pure qubits all prepared with the same unknown Bloch vector, . The -qubit state is , a SCS in the -dimensional symmetric subspace, where is the Dicke state along , . The SCS form a POVM according to Ref. Peres (2006)
where denote the POVM elements, proportional to SCS projectors along unit directions , denotes integration over the steradians of the sphere, and is the identity on the symmetric subspace.
If one considers a more general collective POVM, with outcomes , Banaszek and Devetek have shown Banaszek and Devetak (2001) that the state assignment which maximizes the average fidelity is , where
If is proportional to a SCS along , this result is consistent with the MP protocol, since .
We show that one can approximate the SCS-POVM to arbitrary precision through a sequence of weak collective measurements. The weak measurement of a collective spin component in the direction is described by a Kraus operator Caves and Milburn (1987),
where is a continuous variable outcome, is the measurement rate, and is the measurement duration. Given a state , the probability density for outcome is determined by the Born rule, , and is the postmeasurement state. As a result, the weak measurement backaction generally squeezes the uncertainty along the measured direction and gives the mean spin a random kick.
If the direction is fixed, then the measurement will continually squeeze the uncertainty, ultimately leading to a projective measurement onto an eigenstate of . On the other hand, if we consider a collection of the directions that are chosen isotropically, and each measurement is sufficiently weak such that , then we expect the effect of squeezing to “average out” and the state to remain close to a SCS Cook et al. (2014). Thus, the net effect of the measurement backaction will be a random walk of the mean spin on the sphere. After some time the postmeasurement state will have diffused sufficiently far from the initial state, a distance of order , and no further information about the initial state will remain. The maximum fidelity is limited, thus, by the total number of copies due to the measurement backaction.
With this physical intuition, we specify our protocol for approaching the MP bound with a physically implementable unraveling of the SCS-POVM. Consider a sequence of weak measurements along the directions, . A measurement record defines a total effect specified by the POVM element , where the total Kraus operator is , with given in Eq. (3). Operators in an indexed product are understood here as ordered from right to left.
In order to achieve a SCS-POVM, one must be able to remove the effects of squeezing due to the quadratic operators therein. This can be done by grouping together weak measurements into time intervals . For the th interval, , the resulting Kraus operator is
as follows from the Baker-Campbell-Hausdorff expansion. If the measurements are isotropic, then
Thus, for sufficiently weak measurements such that , the quadratic squeezing terms average out because is proportional to the identity.
Let us define the “operator valued” part of the total Kraus operator such that
In the limit , is the solution to the differential equation
with initial condition . The collection of these operator values enumerated by the coarse-grained measurement records define a completely positive superoperator
where we have defined the Wiener measure
Given this Gaussian form, we see that the operator values are elements in an ensemble of paths generated by an isotropic Wiener process. Since the measure is isotropic for each the resulting POVM will be rotationally invariant, as expected.
Significantly, the commutators of the generators in Eq. (7) are in the six-dimensional span of which is a representation of the Lie algebra . Therefore each at every time step is proportional to the representation of a member of the Lie group , rather than the entire , which would be generated if generators such as were present in the differential equation. Such operators can be decomposed into a restricted polar form,
where is a representation of an element of , is real, and is a unit vector. It follows that the operator-valued part of the POVM element corresponding to the sequence of weak measurements, sampled isotropically over the sphere after a time is
We will show that has a variance which increases diffusively with time, . This implies that for , the probability that decreases asymptotically in time as , and thus in the long time limit, only projectors of highest are statistically significant instances of the superoperator of Eq. (8). Thus, each POVM element converges to , proportional to a SCS projector along an asymptotically constant direction . Together with the rotation-invariant property, one can thus conclude that the sequential weak isotropic measurement protocol realizes the SCS-POVM.
To prove this, write the polar decomposition as
where . We define the generator of this unitary map as , where is a real vector that we choose to satisfy for convenience. It then follows that
For a rotation by an imaginary angle,
Comparing Eq. (7) to Eq. (Optimal Pure-State Qubit Tomography via Sequential Weak Measurements) and taking the Hermitian part,
where we define . Equating the components orthogonal and parallel to ,
Integrating Eq. (16)
By Eq. (9), the are isotropically Gaussian distributed, and thus the variables in the integrand are Gaussian distributed with the same (time-independent) variance. It follows that increases diffusively with the number of isotropic weak measurements, where .
This growth of implies that every statistically significant element of the Kraus ensemble is proportional to an operator of the form . Specifically, according to Eq. (17), as , so must and thus . This means that becomes asymptotically constant and thus . Therefore, the direction of the SCS POVM element converges to an estimate of the initial qubit direction.
Let us further define . Comparing the anti-Hermitian parts of Eq. (7) and Eq. (Optimal Pure-State Qubit Tomography via Sequential Weak Measurements), and substituting Eq. (17) into the result one finds,
As , becomes constant in magnitude and thus wanders perpetually. This implies that in any realization of a sequence of weak measurements, the postmeasurement state continues to diffuse over the sphere for all times, as expected.
Any physical realization of this measurement protocol will differ from the idealized model in a number of fundamental respects. First, each measurement will have a finite duration . Second, if we choose the directions as a random sampling of measurements over the sphere, it will be only approximately isotropic. Finally the idealized measurement will be corrupted by decoherence at a rate . Throughout we assume and ignore decoherence in the simulations below.
As an example of a physical realization, consider tomography on atomic spins via continuous measurement as studied in Refs. Silberfarb et al. (2005); Smith et al. (2006); Riofrio et al. (2011); Cook et al. (2014); Smith et al. (2013). Using the Faraday interaction and polarization spectroscopy, one can perform a collective measurement of the spins when the laser probe couples uniformly to the atomic ensemble (here is the propagation direction of the probe) Smith et al. (2004). The measurement rate is , where is the photon scattering rate and is the cooperativity per atom. The measurement will be weak when the duration of the probe pulse ; decoherence is negligible if . For example, the requisite strong atom-light interface has been demonstrated for atoms in an optical fiber cavity, with observed Haas et al. (2014). In such a geometry, one could perform a QND measurement sequence that is decoherence-free to good approximation in a time . Finally, to measure an arbitrary spin projection one can precede the measurement with a physical rotation of the atomic spin direction .
To demonstrate how one attains the optimal measurement we have performed two types of numerical simulations: (i) sequential random weak measurements; (ii) continuous weak measurements concurrent with time-dependent Hamiltonian control. In type (i), we consider a set of measurement directions randomly sampled on the sphere by the Haar measure. We simulate random measurement outcomes sampled from the probability distribution using Monte Carlo simulations. The postmeasurement state is determined by , which forms the input that determines the probability distribution for the next measurement outcome, , and the procedure is iterated for outcomes. In our simulations we choose .
For a given simulated measurement record, , the POVM element is where . We can test to see how this converges through the “coherency parameter” which satisfies the inequality
for any positive operator . The upper bound is achieved iff is a rank-1 operator, proportional to a SCS projector. Figure 1 shows , for , i.e., 5000 random directions, for copies of the qubit, and 50 different simulated measurement records of a given initial SCS. We see that quickly converges to one for all realizations. The simulation also shows the expected diffusion of the postmeasurement state over longer times, once the POVM element converges.
The time constant for the POVM to converge will depend on the number of copies qubit . As new information is gained, we gain finer resolution of the spin direction. Eventually, the resolution will be better than the spin projection uncertainty and measurement backaction will erase the initial condition. If the measurement direction is fixed, the resolution , and we expect the time at which backaction becomes nonnegligible to scale as . Here, for an isotropic measurement, we can use the coherency parameter to set a timescale for measurement backaction and convergence of the POVM. We expect from Eq. (Optimal Pure-State Qubit Tomography via Sequential Weak Measurements) that , and thus
In this case we see that the POVM converges when , which depends on the diffusive growth of , or .
We also test how well this measurement protocol achieves the MP bound by using the simulated record to estimate the initial state according to Eq. (2). Figure 2 shows the simulated infidelity averaged over 400 Haar random initial SCS as a function of . The total measurement time is taken to be in all cases. The MP bound is shown for comparison. The simulation is consistent with near optimal tomography.
In type (ii) we simulate continuous weak measurement while simultaneously subjecting the system to a time-dependent external control Silberfarb et al. (2005); Smith et al. (2006); Riofrio et al. (2011); Smith et al. (2013); Cook et al. (2014). In this case the measurements occur in infinitesimal time intervals, and random controls can be used to sample random directions on the sphere, but there are correlations between measurement directions for short times, contrary to the idealizations of our proof. The state evolves according to the stochastic Schrödinger equation , where the differential Kraus operator is
We simulate the evolution by updating the state with this differential Kraus operator for time increments such that . The control Hamiltonian is taken to be with ; is the angle of a time-dependent magnetic field in the - plane. We choose to be piecewise constant so the spins precess about a magnetic field that has a fixed amplitude but a random direction in the equator that changes every . Such a control policy is sufficient to achieve an informationally complete measurement record Cook et al. (2014).
Given a measurement record, we estimate the initial Bloch vector of a qubit in the atomic ensemble, using Eq. (2), with and . Figure 2 shows how the continuous measurement performs compared to our random sequential weak protocol and the MP bound.
In summary, we have shown that one can implement a POVM whose outcomes are specified by the overcomplete set of spin-coherent states via a sequence of weak measurements that are isotropic over the sphere. The SCS-POVM allows for optimal tomography of pure qubits, metrology, and other applications. The mathematical proof and techniques we have developed are generalizable to qudits, continuous variable systems, and other generalized-coherent-state POVMs of an arbitrary compact semisimple Lie group Jackson (to be published). Of particular interest is the possibility of a generalized weak measurement protocol to measure the initial -body correlation functions in a symmetric ensemble.
We thank Alexandre Korotkov and Hendra Nurdin for helpful discussions and insights. This work was supported by the National Science Foundation under Grants No. PHY-1606989 and No. PHY-1630114. A.K. acknowledges support from the U.S. Department of Defense. C.A.R. was supported by the Freie Universität Berlin within the Excellence Initiative of the German Research Foundation.
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