*Huge* thanks to my amazing collaborators Lennart Bittel, @jenseisert, and @QuAntonioMele 🥳.9/.

There remain many fundamental and exciting open questions in quantum learning theory for CV systems. Solving these would unveil new structural properties of CV systems (e.g. as done previously with trace distance bounds, symplectic rank, "derivative" of Gaussian states, . ) .9/.

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Finally, as a by-product of our analysis, we also provide state-of-the-art estimates on the sample complexity of heterodyne tomography, which is routinely performed in quantum optics laboratories. 8/.

As a bonus result, we also show that this mild log(log(energy))-dependence can be removed entirely if access to the *transposed state* is available. 7/.

In other words, estimating a Gaussian state in trace distance can be much more efficient than estimating its covariance matrix e.g. in operator norm. 6/.

Note: our algorithm is *not* the intuitive one of "estimate first moment and covariance matrix + propagate the error". 5/.

Notably, the tomography algorithm that achieves such energy-independent performance is also experimentally feasible. It exploits only: single-copy passive Gaussian unitaries, homodyne measurements, and auxiliary squeezed states. 4/.

In today’s paper, we prove a rather surprising fact: the time complexity of tomography of Gaussian states is *independent* of the energy of the unknown state, up to doubly logarithmic factors. This improves exponentially over previous works. 3/.

In DV systems, one is usually interested in understanding how performances scale with the number of constituents (qubits). Instead, in CV systems, one is interested not only in the scaling with the number of constituents (modes) — but also with the *energy* of the state. 2/.

New paper out on the exciting and rapidly-growing topic of *quantum learning theory with continuous-variable systems* 🔦⚛️🔦. In short, we show that Gaussian states can be learned in a time that, surprisingly, is basically *independent* of the energy of the unknown state. 1/

RT @jenseisert: Learning quantum states - aka #quantum state tomography - is actually strangely difficult and sample inefficient for #conti….

RT @QuAntonioMele: Super happy to see that our paper "Optimal estimates of trace distance between bosonic Gaussian states and applications….

RT @GerardoAdesso: We are hiring! Looking for a Post-Doc in Quantum Optical Metrology to work with @Quantumlife2005 and myself at the @Unio….

Thanks a lot to my great collaborators SFE Oliviero, @varunup11201, and @UlysseChabaud🥳. We are happy to receive feedback!.

We also find similar bounds for classical probability distributions:. If two probability distributions are ε-close in TV distance, how close are their mean vectors and their covariance matrices? . This is a natural question, but it has never been explored as far as we known. 5/

New bosonic trace distance bounds:. If two (possibly non-Gaussian) states are ε-close in trace distance, how close are their covariance matrices, first moments, and symplectic eigenvalues? . Our new trace distance bounds precisely answer to this question:. 4/

This decomposition is crucial to prove the monotonicity of the symplectic rank, and we believe it may find other applications!.3/.

New decomposition of Gaussian unitaries:.Any n-mode Gaussian unitary can be written, for all k<n/2, as a composition of an n-mode passive Gaussian unitary, a Gaussian unitary that acts only on the first 2k modes, and another Gaussian unitary acting on the last n-k modes. 2/

I’d like to advertise two cute technical results of today’s paper that may be of independent interest (I hope):. A new decomposition of Gaussian unitaries and new bosonic trace distance bounds (the saga of bosonic trace distance bounds never ends! 😍).1/

Many thanks to my fantastic coauthors SFE Oliviero, Varun Upreti, and @UlysseChabaud for this great collaboration! It all began a year ago at the BIRS conference in Granada — an entire conference dinner spent brainstorming these ideas (after a glass or two of Spanish wine. ).13/.