We present a tomographic method which requires only 4d-3 measurement outcomes to reconstruct any pure quantum state of arbitrary dimension d. Using the proposed scheme, we have experimentally reconstructed a large number of pure states of dimension d=7, obtaining a mean fidelity of 0.94. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the a priori assumption of purity can be certified within the same set of measurements, which represents an improvement with respect to other similar methods and contributes to answering the question of how many observables are needed to uniquely determine any pure state.
We present a proof-of-principle demonstration of a method to characterize any pure spatial qudit of arbitrary dimension
d, which is based on the classic phase-shift interferometry technique. In the proposed scheme a total of only
4d measurement outcomes are needed, implying a significant reduction with respect to the standard schemes for quantum-state tomography which require on the order of
d2. By using this technique, we have experimentally reconstructed a large number of states ranging from
d=2 up to 14 with mean fidelity values higher than 0.97. For that purpose the qudits were codified in the discretized transverse-momentum position of single photons, once they are sent through an aperture with
d slits. We provide an experimental implementation of the method based in a Mach-Zehnder interferometer, which allows one to reduce the number of measurement settings to four since the
d slits can be measured simultaneously. Furthermore, it can be adapted to consider the reconstruction of the unknown state from the outcome frequencies of $4d\ensuremath{-}3$ fixed projectors independently of the encoding or the nature of the quantum system, allowing one to implement the reconstruction method in a general experiment.
We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary}
d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only}
d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of
d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from
42 to only
7.
We present a quantum state tomography method that enables the reconstruction of arbitrary d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using only d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are spatially multiplexed on the interference pattern of the projected state. Using the proposed scheme, we performed, as a proof-of-principle, an experimental reconstruction of d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 42 to only 7.
We consider the task of performing quantum state tomography on a
d-state spin qudit, using only measurements of spin projection onto different quantization axes. By an exact mapping onto the classical problem of signal recovery on the sphere, we prove that full reconstruction of arbitrary qudit states requires a minimal number of measurement axes,
rdmin, that is bounded by
2d−1≤rdmin≤d2. We conjecture that
rdmin=2d−1, which we verify numerically for all
d≤200. We then provide algorithms with
O(rd3) serial runtime, parallelizable down to
O(rd2), for (i) computing a priori upper bounds on the expected error with which spin projection measurements along
r given axes can reconstruct an unknown qudit state, and (ii) estimating a posteriori the statistical error in a reconstructed state. Our algorithms motivate a simple randomized tomography protocol, for which we find that using more measurement axes can yield substantial benefits that plateau after
r≈3d.