Direct characterization of quantum dynamics: General theory

The characterization of the dynamics of quantum systems is a task of both fundamental and practical importance. A general class of methods which have been developed in quantum information theory to accomplish this task is known as quantum process tomography (QPT). In an earlier paper [M. Mohseni and D. A. Lidar Phys. Rev. Lett. 97, 170501 (2006)] we presented an algorithm for direct characterization of quantum dynamics (DCQD) of two-level quantum systems. Here we provide a generalization by developing a theory for direct and complete characterization of the dynamics of arbitrary quantum systems. In contrast to other QPT schemes, DCQD relies on quantum error-detection techniques and does not require any quantum state tomography. We demonstrate that for the full characterization of the dynamics of $n$ $d$-level quantum systems (with $d$ prime), the minimal number of required experimental configurations is reduced quadratically from $d^{4n}$ in separable QPT schemes to $d^{2n}$ in DCQD.

Figure 1

(Color online) Schematic of DCQD for a single qubit, consisting of Bell-state-type preparations, application of the unknown quantum map, $\mathcal{E}$, and Bell-state measurement (BSM).

Figure 2

(Color online) A schematic diagram of quantum error detection (QED). The projective measurements corresponding to eigenvalues of stabilizer generators are represented by arrows. For a nondegenerate QEC code, after the QED, the wave function of the multiqubit system collapses into one of the orthogonal subspaces each of which is associated with a single error operator. Therefore, all errors can be unambiguously discriminated. For degenerate codes, by performing QED the code space also collapses into a set orthogonal subspaces. However, each subspace has multiple degeneracies among $k$ error operators in a subset of the operator basis, i.e., $\{E_m{\}}_{m=1}^k\subset \{E_i{\}}_{i=0}^{d^2-1}$. In this case, one cannot distinguish between different operators within a particular subset $\{E_m{\}}_{m=1}^{k_0}$.

Figure 3

(Color online) A diagram of the error-detection measurement for estimating quantum dynamical population. The arrows represent the projection operators $P_k{P}_{k^{\prime }}$ corresponding to different eigenvalues of the two stabilizer generators $S$ and $S^{\prime }$. These projective measurements result in a projection of the wave function of the two-qudit systems, after experiencing the dynamical map, into one of the orthogonal subspaces each of which is associated to a specific error operator basis. By calculating the joint probability distribution of all possible outcomes, $P_k{P}_{k^{\prime }}$, for $k,k^{\prime }=0,\dots ,d$, we obtain all $d^2$ diagonal elements of the superoperator in a single ensemble measurement.

Figure 4

(Color online) A diagram of the error-detection measurement for estimating quantum dynamical coherence: we measure the sole stabilizer generator at the output state, by applying projection operators corresponding to its different eigenvalues $P_k$. We also measure $d-1$ commuting operators that belong to the normalizer group. Finally, we calculate the probability of each stabilizer outcome, and joint probability distributions of the normalizers and the stabilizer outcomes. Optimally, we can obtain $d^2$ linearly independent equations by appropriate selection of the normalizer operators as it is shown in the next section.

Figure 5

(Color online) Procedure (a): Measuring the quantum dynamical population (diagonal elements ${\chi }_{mm}$). The arrows indicate direction of time. (1) Prepare $\mid {\phi }_0⟩=\mid 0{⟩}_A\otimes \mid 0{⟩}_B$, a pure initial state. (2) Transform it to $\mid {\phi }_C⟩=\frac{1}{\sqrt{d}}{\sum }_{k=0}^{d-1}\mid k{⟩}_A\mid k{⟩}_B$, a maximally entangled state of the two qudits. This state has the stabilizer operators $S=X^A{X}^B$ and $S^{\prime }=Z^A(Z^B{)}^{d-1}$. (3) Apply the unknown quantum dynamical map to the qudit $A$, $\mathcal{E}\left(\rho \right)$, where $\rho =\mid {\varphi }_C⟩⟨{\varphi }_C\mid$. (4) Perform a projective measurement $P_k{P}_{k^{\prime }}$, for $k,k^{\prime }=0,\dots ,d-1$, corresponding to eigenvalues of the stabilizer operators $S$ and $S^{\prime }$. Then calculate the joint probability distributions of the outcomes $k$ and $k^{\prime }$: $\mathrm{Tr}\left[P_k{P}_{k^{\prime }}\mathcal{E}\right(\rho \left)\right]={\chi }_{mm}$. The elements ${\chi }_{mm}$ represent the population of error operators that anticommute with the stabilizer generators $S$ and $S^{\prime }$ with eigenvalues ${\omega }^k$ and ${\omega }^{k^{\prime }}$, respectively. The number of ensemble measurements for procedure (a) is one.

Figure 6

(Color online) Procedure (b): Measuring the quantum dynamical coherence (off-diagonal elements ${\chi }_{mn}$). (1) Prepare $\mid {\phi }_0⟩=\mid 0{⟩}_A\otimes \mid 0{⟩}_B$, a pure initial state. (2) Transform it to $\mid {\phi }_C⟩={\sum }_{i=0}^{d-1}{\alpha }_i\mid i{⟩}_A\mid i{⟩}_B$, a nonmaximally entangled state of the two qudits. This state has a sole stabilizer operator of the form $S=E_i^A(E_i^B{)}^{d-1}$. (3) Apply the unknown quantum dynamical map to the qudit $A$, $\mathcal{E}\left(\rho \right)$, where $\rho =\mid {\varphi }_C⟩⟨{\varphi }_C\mid$. (4) Perform a projective measurement $P_k$, and calculate the probability of the outcome $k$, $\mathrm{tr}\left[P_k\mathcal{E}\right(\rho \left)\right]$. (5) Measure the expectation values of all normalizer operators $T_{rs}$ that simultaneously commute with the stabilizer generator $S$. There are only $d-1$ such operators $T_{rs}$ that are independent of each other, within a multiplication by a stabilizer generator; and they belong to an Abelian subgroup of the normalizer group. (6) Repeat the steps (1)–(5) $d+1$ times, by preparing the eigenkets of the other stabilizer operator $E_i^A(E_i^B{)}^{d-1}$ for all $i∊\{1,2,\dots ,d+1\}$, such that states $\mid i{⟩}_A$ in step (2) belong to a mutually unbiased basis [32]. (7) Repeat the step (6) up to $d-1$ times, each time choosing normalizer elements $T_{rs}$ from a different Abelian subgroup of the normalizer, such that these measurements become maximally noncommuting, i.e., their eigenstates form a set of mutually unbiased bases. The number of ensemble measurements for Procedure (b) is $(d+1)(d-1)$.