A $d$ -mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in ${Z}_{d}$ and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the $d$ modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the $d$ observers; and they also change the momentum with which the entanglement propagates.A $d$-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in ${Z}_{d}$ and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the $d$ modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the $d$ observers; and they also change the momentum with which the entanglement propagates.

Physical Review AApr 26, 20052 citations

Squeezing and entanglement in a two-mode system: theory and experiment

Vincent Josse, Aurélien Dantan, Alberto Bramati +1 more

Jan 1, 20040 citations

Conditional-not Displacement: Fast Multioscillator Control with a Single Qubit

Asaf A. Diringer, Eliya Blumenthal, Avishay Grinberg +2 more

Bosonic encoding is an approach for quantum information processing, promising lower hardware overhead by encoding in the many levels of a harmonic-oscillator mode. Scaling to multiple modes requires weak interaction for independent control, yet strong interaction for fast control. Applying fast and efficient universal control on multiple modes remains an open problem. Surprisingly, we find that displacements conditioned on the state of a single-qubit ancilla coupled to multiple harmonic oscillators are sufficient for universal control. We present the conditional- operation concept, which can be used to reduce the duration of entangling gates. Within this guiding concept, we develop the conditional- displacement control method which enables fast generation and control of bosonic states in multimode systems weakly coupled to a single-ancilla qubit. Our method is fast despite the weak ancilla coupling. The weak coupling in turn allows for excellent separability and thus independent control. We demonstrate our control on a superconducting transmon qubit weakly coupled to a multimode superconducting cavity. We create both entangled and separable cat states in different modes of the multimode cavity, showing entangling operations at low crosstalk while maintaining independent control of the different modes. We show that the operation time is not limited by the inverse of the coupling rate, which is the typical timescale, and we exceed it by almost 2 orders of magnitude. We verify our results with an efficient method for measurement of the multimode characteristic function which employs our conditional- displacement. Our results inspire a new approach toward general entangling operations and allow for fast and efficient multimode bosonic encoding and measurement. Published by the American Physical Society 2024Bosonic encoding is an approach for quantum information processing, promising lower hardware overhead by encoding in the many levels of a harmonic-oscillator mode. Scaling to multiple modes requires weak interaction for independent control, yet strong interaction for fast control. Applying fast and efficient universal control on multiple modes remains an open problem. Surprisingly, we find that displacements conditioned on the state of a single-qubit ancilla coupled to multiple harmonic oscillators are sufficient for universal control. We present the conditional- operation concept, which can be used to reduce the duration of entangling gates. Within this guiding concept, we develop the conditional- displacement control method which enables fast generation and control of bosonic states in multimode systems weakly coupled to a single-ancilla qubit. Our method is fast despite the weak ancilla coupling. The weak coupling in turn allows for excellent separability and thus independent control. We demonstrate our control on a superconducting transmon qubit weakly coupled to a multimode superconducting cavity. We create both entangled and separable cat states in different modes of the multimode cavity, showing entangling operations at low crosstalk while maintaining independent control of the different modes. We show that the operation time is not limited by the inverse of the coupling rate, which is the typical timescale, and we exceed it by almost 2 orders of magnitude. We verify our results with an efficient method for measurement of the multimode characteristic function which employs our conditional- displacement. Our results inspire a new approach toward general entangling operations and allow for fast and efficient multimode bosonic encoding and measurement. Published by the American Physical Society 2024

arXiv:2301.09831Physical Review XMar 26, 202427 citations

Entanglement of pure two-mode Gaussian states (7 pages)

R. W. Rendell, A. K. Rajagopal

Physical Review AJan 1, 20051 citation

d

-mode system is considered and the concepts of mode-position operator and mode-momentum operator are introduced. Their eigenvalues are integers in

{Z}_{d}

and the mode phase space is ${Z}_{d}\ifmmode\times\else\texttimes\fi{}{Z}_{d}$. The exponentials of these operators are displacement operators in the mode phase space and they form a Heisenberg-Weyl group. The displacement operators perform a cyclic permutation of a state among the

d

modes; and they also change its mode momentum (i.e., the rate with which this cyclic permutation takes place). In the case of entangled states the mode displacement operators permute the entanglement among the

d

observers; and they also change the momentum with which the entanglement propagates.