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Efficient optimization of the quantum relative entropy
Hamza Fawzi, Omar Fawzi
TL;DR
Many quantum information quantities are formulated as optimizations involving quantum relative entropy. The paper uses semidefinite programming approximations to compute them, reporting greater speed and flexibility than tailored algorithms and a numerical counterexample to a proposed inequality.
Problem
Many quantum information quantities can be formulated as optimization problems involving quantum relative entropy.
Source: §1 · p. 1Method
The paper computes quantum information quantities using semidefinite programming constraints built from rational approximations of the logarithm that preserve operator concavity.
Source: §1 · pp. 1–2Results
Takeaways & Limitations
The framework is applied to classical-quantum channel capacity, entanglement-assisted classical capacity, degradable-channel quantum capacity, and relative entropy of entanglement.
Source: §1 · p. 1Larger channel-environment dimensions can produce much larger SDPs, though channel symmetry may reduce their size.
Source: §3.2 · p. 6
Abstract
from arXiv · showhide
Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.
1 Introduction
Quantum information measures often require optimization involving quantum relative entropy, which standard conic solvers do not natively support. The paper presents an SDP-based framework for computing several such quantities and applies it to a proposed inequality involving relative entropy of recovery.
1 Introduction
- The framework addresses channel capacities, including classical-quantum, entanglement-assisted classical, and degradable-channel quantum capacities.These are among the problems the paper considers using semidefinite programming.Source: §1 · p. 1
- The paper also considers relative entropy of entanglement and relative entropy of recovery.Source: §1 · pp. 1–2
- The SDP-based approximations are generally much faster and more flexible than previously tailored algorithms, and yield a numerical counterexample for a proposed inequality involving relative entropy of recovery.The paper identifies the inequality as a proposed lower bound on quantum conditional mutual information in terms of relative entropy of recovery.Source: §1 · p. 2
- The approximations are implemented in CvxQuad, with accompanying Matlab code for the paper's computations.Some code also requires quantinf for density-matrix manipulations.Source: §1 · p. 2
- For each function, the SDP has blocks whose sizes depend on the input dimension and whose number is k + m, parameters controlling approximation accuracy.CvxQuad defaults to k = m = 3, which the authors observed gives good accuracy for most practical situations.Source: §1 · p. 2
- CvxQuad uses rational approximations of the logarithm that preserve operator concavity, expressing supported functions as semidefinite constraints usable with off-the-shelf solvers.The approximation is obtained by quadrature on an integral representation of log.Source: §1 · p. 2
2 Notations
This section establishes notation for density operators, entropy, quantum relative entropy, and quantum channels.
2 Notations
- D(A) denotes density operators: Hermitian positive semidefinite matrices on A with unit trace.Source: §2 · p. 2
- The von Neumann entropy is H(ρ) = −Tr[ρ log ρ], and quantum relative entropy is D(ρ∥σ) = Tr[ρ(log ρ−log σ)].Source: §2 · p. 3
- For a bipartite state, conditional entropy is H(A|B)ρ := H(AB)ρ − H(B)ρ.Source: §2 · p. 3
- A quantum channel from A to B is a completely positive, trace-preserving linear map between their operator spaces.Source: §2 · p. 3
3 Capacities of quantum channels
The section formulates several quantum-channel capacities as convex optimization problems using CvxQuad’s semidefinite approximations of entropy-related functions. It evaluates classical-to-quantum capacity accuracy and computation time, and shows how the framework applies to entanglement-assisted capacity and quantum capacity for degradable channels.
3.1 Classical to quantum channel
- The classical-to-quantum channel capacity is optimized over input probabilities using quantum entropy functions in a CVX formulation.The illustrated code uses a two-symbol input alphabet.Source: §3.1 · p. 3
- For random binary-input channels with pure-state outputs, CvxQuad’s computed capacity has absolute error of order 10^-6 against the known value.The known capacity is h2((1 + ϵ)/2), where ϵ is the overlap magnitude of the output states.Source: §3.1 · p. 3, Table 2 · p. 4
- CvxQuad’s SDP approach scales better than ’s tailored first-order method for channels with large output dimension.Table 3 compares their computation times for classical-to-quantum channel capacity.Source: §3.1 · p. 4, Table 3 · p. 4
3.2 Entanglement-assisted classical capacity
- Entanglement-assisted classical capacity is computed by maximizing channel mutual information, whose concavity in the input state supports convex optimization.The section illustrates the computation for an amplitude-damping channel and plots capacity against γ.Source: §3.2 · pp. 4–5
- The entanglement-assisted-capacity SDP can become much larger for channels with larger environment spaces, while channel symmetry may reduce its size.For amplitude damping, dim(E) = 2; the text notes phase-damping and depolarizing examples with environment dimensions 3 and 4.Source: §3.2 · p. 6
3.3 Quantum capacity of degradable channels
- For degradable channels, the quantum-capacity limit reduces to n = 1 and coherent information is concave in the input state, enabling CvxQuad formulation.The section illustrates this computation for amplitude damping over γ ∈ [0, 1/2], where the channel is degradable.Source: §3.3 · p. 7, Figure 2 · p. 7
- For general non-degradable channels, an approximately degradable approach can yield an upper bound through a similar convex optimization problem.The approach is attributed to [SSWR14].Source: §3.3 · p. 7
4 Relative entropy of entanglement
The section computes a PPT-relaxed relative entropy of entanglement because the separable-state constraint is difficult to describe. CvxQuad is reported to be faster and to scale better than a tailored cutting-plane method on the tested states.
4 Relative entropy of entanglement
- Replacing the separability constraint with a positive-partial-transpose constraint gives a lower bound on the relative entropy of entanglement.The separable-state set is difficult to describe in general; higher levels of the SDP hierarchy provide other constraint replacements.Source: §4 · p. 8
- CvxQuad’s quantum_rel_entr function computes the resulting PPT-relaxed quantity REE(1)(ρ).The paper provides CVX code for this computation.Source: §4 · p. 8
- For randomly generated bipartite states of different sizes, CvxQuad was much faster than the cutting-plane method and scaled better for large states.In all tested cases, its returned value lay within the cutting-plane method’s guaranteed interval of length 10^-3 (ϵ = 10^-3).Source: §4 · p. 8, Table 4 · p. 9
5 Relative entropy of recovery and conditional mutual information
The section tests a proposed lower bound relating the relative entropy of recovery to quantum conditional mutual information. Numerical examples, including an explicit pure state, show that the inequality can be violated, although there is also a range of θ where it holds.
5 Relative entropy of recovery and conditional mutual information
- Quantum Markov chains have zero conditional mutual information and allow C to be generated from the AB marginal by acting only on B.This motivates relating conditional mutual information to how well a state is recoverable.Source: §5 · p. 9
- The order-1/2 measure ˜D1/2 is smaller than quantum relative entropy; improved inequality (11) by replacing it with the larger measured relative entropy DM, and [LW14] and ask whether it can be further improved.Source: §5 · p. 10
- The relative entropy of recovery is a quantum relative entropy optimization, allowing the proposed inequality to be tested numerically on a given state.The paper reports a range of θ where the relative entropy of recovery is greater than or equal to the conditional mutual information.Source: §5 · pp. 10–11
- Some of 2000 randomly generated rank-one pure states violate the proposed inequality between relative entropy of recovery and conditional mutual information.The tested systems have dim(A) = dim(B) = dim(C) = 2.Source: Figure 3 · p. 10, §5 · p. 10
- The paper gives an explicit pure tripartite state that violates the proposed inequality.For the state defined by Equations (13)–(14), the violation occurs for small θ.Source: §5 · p. 10, Figure 4 · p. 11