Computational Aspects of the Trace Norm Contraction Coefficient

summary

The gist

The gist The paper shows that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard, which implies that determining the optimal success

In short

The paper proves that approximating a quantum channel's trace norm contraction coefficient within a constant factor is NP-hard, which means determining the optimal success probability for encoding bits in noisy quantum systems is also NP-hard. It establishes that deciding if the contraction coefficient equals 1 (perfect bit preservation) is hard and provides a converging hierarchy of semidefinite programming bounds to detect when this coefficient is less than 1.

Key concepts

Trace Norm Contraction Coefficient ($\eta_{tr}(\Phi)$): This coefficient measures how much a quantum channel $\Phi$ shrinks the trace norm (a measure of distinguishability) of quantum states. If it's close to 1, the channel is nearly isometric, meaning it preserves information well. Approximating this value within a constant factor is computationally difficult.
Optimal Success Probability ($P_{succ}(\Phi, k)$): This represents the maximum probability of successfully transmitting a bit through a quantum channel $\Phi$ using $k$ copies of encoding. It is directly related to the trace norm contraction coefficient by the formula $P_{succ}(\Phi, 2) = 1/2 + 1/2 \eta_{tr}(\Phi)$. Finding this optimal probability is equivalent to solving hard problems.
Quantum Confusability Graph ($G_{\Phi}$): This graph is a mathematical tool used to relate the properties of a quantum channel $\Phi$ to classical graph theory problems. Specifically, deciding if the independence number of this graph is at least 2 is shown to be NP-hard, linking quantum information theory to hard combinatorial problems.
Semidefinite Programming (SDP) Upper Bounds: SDP methods are used here to create increasingly tight upper bounds on the trace norm contraction coefficient. These bounds converge toward the true value, and the convergence speed is quantified by a polynomial function of the dimension $d$ and related parameters, providing a practical way to detect when $\eta_{tr}(\Phi) < 1.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Sample Complexity — explained in 18 episodes
Federated Learning — explained in 17 episodes
Retrieval-Augmented Generation (RAG) — explained in 16 episodes
Interpretability — explained in 15 episodes
Quantum Fisher Information (QFI) — explained in 15 episodes

This episode discusses

The paper

Computational Aspects of the Trace Norm Contraction Coefficient · Read on arXiv

Idris Delsol, Omar Fawzi, Jan Kochanowski, Akshay Ramachandran

inria · ENS Lyon · UCBL · LIP

We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard. This contrasts with the classical analogue of this problem that can clearly be solved efficiently. We also establish the NP-hardness of deciding if the contraction coefficient is equal to 1, i.e., the channel can perfectly preserve a bit. As a consequence, deciding if a non-commutative graph has an independence number of at least 2 is NP-hard. In addition, we establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.

DOI: 10.1109/TIT.2026.3743741

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Computational Aspects of the Trace Norm Contraction Coefficient".

Mira: The gist The paper shows that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So to summarize the paper "Computational Aspects of the Trace Norm Contraction Coefficient," they focus on this central question in information theory: how to optimally encode information when you have a specific noise model.

Mira: The paper sets out to study this question from an algorithmic point of view, given a classical description of a quantum channel Φ representing the noise and an integer k.

Lev: They start by establishing hardness results first, specifically showing that approximating the trace norm contraction coefficient within some factor one over sqrt(two) plus epsilon is NP-hard for any epsilon greater than zero <ref:2507.16737#pg1,that approximating the trace norm contraction coefficient>.

Kai: And they prove this even when the channel Φ is promised to be unital, which means it preserves the identity operator.

Mira: Furthermore, in a restricted setting where the channel is quantum input and classical output and unital, they show that the contraction coefficient is NP-hard to approximate to a factor p over pi plus epsilon for any epsilon greater than zero.

Lev: That hardness result comes from a reduction from the Little Grothendieck Problem, which is what they use to prove the general hardness result.

Kai: They also mention that for some special classes of channels, like if the channel Φ is composed t times where t goes to infinity, then the optimal success probability and hence the contraction coefficient can be computed efficiently.

Mira: That efficient case is something you see in specific scenarios rather than the general case they are proving hardness for.

Lev: The paper also defines some technical terms like the Kraus operators of the channel Φ and how they relate to its Choi state J(Φ), which is a bipartite state on the compound system.

Kai: They use these definitions to precisely define something called the optimal success probability Psucc(Φ, k) for transmitting k messages over that channel.

Mira: They define this success probability using a Positive Operator Valued Measure or POVM, which is a collection of positive operators on the space B(H).

Conclusion: Kai: So to wrap up "Computational Aspects of the Trace Norm Contraction Coefficient," they've shown that approximating the trace norm contraction coefficient within a constant factor is NP-hard.

Mira: And this hardness implies that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard as well.

Lev: It’s interesting because it contrasts with what we know about classical problems, which they say are solvable efficiently.

Kai: They also proved the NP-hardness of deciding if the contraction coefficient is equal to one which means the channel can perfectly preserve a bit <ref:2507.16737#pg1,the NP-hardness of deciding if the contraction coefficient is equal to>.

Mira: And as a consequence of that perfect preservation question, deciding if a non-commutative graph has an independence number of at least two is NP-hard.

Lev: The paper concludes by establishing a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient, showing how these bounds get tighter with more resources.

Kai: So essentially, they prove that approximating the trace norm contraction coefficient within a constant factor is hard, and they give us a way to detect when that coefficient is less than one using these SDPs.

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