We show that for any positive integer
n, the maps
x∈Cn↦{∣⟨x,zi⟩∣2}i=14n∈R4n, where
zi are the columns of four
n×n unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of
CPn−1 into
R4n−4. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for
n≥6. To complement this information-theoretic study, we establish in a companion paper that the PhaseLift algorithm yields efficient phase retrieval from quadratic measurements with
O(1) unitary matrices, with high probability, where the unitaries are iid according to Haar measure.
A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms.
We analyze the following strategy: given
n copies of an unknown state
ρ, convert them into copies of a purification
∣ρ⟩; run a pure state tomography algorithm to produce an estimate of
∣ρ⟩; and output the resulting estimate of
ρ. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-
r d-dimensional state, it uses
n=O((rd+log(1/δ))/ε) samples to output an estimate which is
ε-close in fidelity with probability at least
1−δ. This algorithm also uses poly
(n) gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension
d alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.
We consider a variant of the phase retrieval problem, where vectors are replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U, and the measurements consist of squared inner products |tr(C†U)|2with unitary matrices C that are chosen by the observer. This problem has applications to quantum process tomography, when the unknown process is a unitary operation. We show that PhaseLift, a convex programming algorithm for phase retrieval, can be adapted to this matrix setting, using measurements that are sampled from unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show that PhaseLift can reconstruct all unitary matrices, using a nearoptimal number of measurements. This extends previous work on PhaseLift using spherical 4-designs. In the case of unitary 2-design measurements, we show that PhaseLift still works pretty well on average: it recovers almost all signals, up to a constant additive error, using a near-optimal number of measurements. These 2-design measurements are convenient for quantum process tomography, as they can be implemented via randomized benchmarking techniques. This is the first positive result on PhaseLift using 2-designs.
We present an efficient algorithm that solves the quantum state tomography problem from an arbitrary number of projective measurements in any finite dimension
d. The algorithm is flexible enough to allow us to impose any desired rank
r to the state to be reconstructed, ranging from pure (
r=1) to full rank
(r=d) quantum states. The method exhibits successful and fast convergence under the presence of realistic errors in both state preparation and measurement stages, and also when considering overcomplete sets of observables. We demonstrate that the method outperforms semidefinite programming quantum state tomography for some sets of physically relevant quantum measurements in every finite dimension.