The aim of this thesis is to develop a framework for assessing performance in quantum information processing with continuous variables. In particular, we focus on quantifying the fundamental limitations on communication and computation over bosonic Gaussian systems. Due to their infinite-dimensional structure, we make a realistic assumption of energy constraints on the input states of continuous-variable (CV) quantum operations. Our first contribution is to show that energy-constrained distinguishability measures can be used to establish tight upper bounds on the communication capacities of phase-insensitive, bosonic Gaussian channels -- thermal, amplifier, and additive-noise channels. We then prove that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. Next, we develop theoretical and numerical tools based on energy-constrained distinguishability measures to quantify the accuracy in implementing Gaussian unitary operations. Finally, we propose an optimal test for the performance of CV quantum teleportation in terms of the energy-constrained channel fidelity between ideal CV teleportation and its experimental implementation. Here we prove that the optimal state for testing CV teleportation is an entangled superposition of twin-Fock states. These results are relevant for experiments that make use of Gaussian unitaries and CV teleportation.
The aim of this thesis is to develop a framework for assessing performance in quantum information processing with continuous variables. In particular, we focus on quantifying the fundamental limitations on communication and computation over bosonic Gaussian systems. Due to their infinite-dimensional structure, we make a realistic assumption of energy constraints on the input states of continuous-variable (CV) quantum operations. Our first contribution is to show that energy-constrained distinguishability measures can be used to establish tight upper bounds on the communication capacities of phase-insensitive, bosonic Gaussian channels -- thermal, amplifier, and additive-noise channels. We then prove that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. Next, we develop theoretical and numerical tools based on energy-constrained distinguishability measures to quantify the accuracy in implementing Gaussian unitary operations. Finally, we propose an optimal test for the performance of CV quantum teleportation in terms of the energy-constrained channel fidelity between ideal CV teleportation and its experimental implementation. Here we prove that the optimal state for testing CV teleportation is an entangled superposition of twin-Fock states. These results are relevant for experiments that make use of Gaussian unitaries and CV teleportation.
Quantum teleportation is a primitive in several important applications, including quantum communication, quantum computation, error correction, and quantum networks. In this paper, we propose an optimal test for the performance of continuous-variable (CV) quantum teleportation in terms of the energy-constrained channel fidelity between ideal CV teleportation and its experimental implementation. Work prior to ours considered suboptimal tests of the performance of CV teleportation, focusing instead on its performance for particular states, such as ensembles of coherent states, squeezed states, cat states, etc. Here, we prove that the optimal state for testing CV teleportation is an entangled superposition of twin Fock states. We establish this result by reducing the problem of estimating the energy-constrained channel fidelity between ideal CV teleportation and its experimental approximation to a quadratic program and solving it. As an additional result, we obtain an analytical solution to the energy-constrained diamond distance between a photodetector and its experimental approximation. These results are relevant for experiments that make use of CV teleportation and photodetectors.
Continuous-variable (CV) teleportation is a fundamental protocol in quantum information science. A number of experiments have been designed to simulate ideal teleportation under realistic conditions. In this paper, we detail an analytical approach for determining optimal input states for quantifying the performance of CV unidirectional and bidirectional teleportation. The metric that we consider for quantifying performance is the energy-constrained channel fidelity between ideal teleportation and its experimental implementation, and along with this, our focus is on determining optimal input states for distinguishing the ideal process from the experimental one. We prove that, under certain energy constraints, the optimal input state in unidirectional as well as bidirectional teleportation is a finite entangled superposition of twin-Fock states saturating the energy constraint. Moreover, we also prove that, under the same constraints, the optimal states are unique; that is, there is no other optimal finite entangled superposition of twin-Fock states.
We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the ‘data-processing bound,’ is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the ‘ ε -degradable bound’ and the ‘ ε -close-degradable bound,’ are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
Discriminating between noisy quantum processes is a central primitive for quantum communication, metrology, and computing. While discrimination limits for finite-dimensional channels are well understood, the continuous-variable setting, particularly under experimentally relevant energy constraints, remains significantly less developed. In this work, we establish an energy-constrained chain rule for the Belavkin-Staszewski channel divergence, which yields a fundamental upper bound on the error exponents achievable by fully adaptive, energy-constrained quantum channel discrimination protocols. We then derive efficiently computable bounds on asymmetric error exponents for energy-constrained discrimination of bosonic dephasing and loss-dephasing channels. Specifically, we show that three operationally relevant quantities -- the measured relative entropy, the Umegaki relative entropy, and the geometric Renyi divergence -- admit semidefinite program (SDP) formulations when the input energy is bounded and the Hilbert space is suitably truncated. Applying these tools, we demonstrate that optimal probes for these channels under energy constraints are Fock-diagonal, and we also enable numerically precise evaluation of bounds on achievable error exponents across discrimination strategies ranging from separable to fully adaptive. The resulting SDPs provide practical benchmarks for quantum-limited sensing in low-energy bosonic platforms.
We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the 'data-processing bound,' is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the 'ϵ-degradable bound' and the 'ϵ-close-degradable bound,' are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.
We establish three different upper bounds on the energy-constrained quantum and private capacities of bosonic thermal channels. The first upper which we call the bound, is obtained by decomposing a thermal channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing can be at most 1.45 bits larger than a known lower on these capacities of the thermal channel. The other two upper bounds, which we call the bound and the bound, are established using the notion of approximate degradability along with energy constraints. A comparison of these three bounds shows that the data-processing is near to a known lower for both low and high thermal noise and is very near to the ε-close-degradable only for low thermal noise. Moreover, for certain parameter regimes, we show that the ε-degradable is tighter than all other bounds. All three upper bounds are very near to a known lower for the case of low thermal noise and high transmissivity. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the R\'enyi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.