Comparing quantum channels using Hermitian-preserving trace-preserving linear maps: A physically meaningful approach

This paper establishes a physically meaningful preorder for comparing quantum channels by demonstrating that one channel can be derived from another via a Hermitian-preserving trace-preserving (but not necessarily positive) linear map if their output statistics are distinguishable, thereby providing a framework to quantify implementation difficulty and analyze device incompatibility.

The Gist

Here is an explanation of the paper using simple language, everyday analogies, and metaphors.

The Big Picture: Comparing "Messy" Machines

Imagine you have two machines that process information. In the quantum world, these are called Quantum Channels.

• Machine A takes a delicate, perfect crystal (a quantum state) and passes it through.
• Machine B takes the same crystal and passes it through.

Usually, these machines are a bit "noisy." They might scratch the crystal or change its shape slightly. The goal of this paper is to answer a simple question: Is Machine A "better" or "more powerful" than Machine B?

In the past, scientists had a strict rule: Machine A is better than Machine B only if you could take the output of A, run it through another perfect machine, and turn it into the output of B. This is like saying, "I can turn a raw potato into a french fry, but I can't turn a french fry back into a raw potato."

The authors of this paper say: "Wait, let's loosen the rules a bit."

They propose a new way to compare machines. They ask: If I look at the output of Machine A, can I figure out exactly what the output of Machine B would have been, even if I can't physically turn A into B?

The New Rule: The "Magic Translator"

The authors introduce a special kind of "translator" called a Hermitian-Preserving Trace-Preserving (HPTP) map.

To understand this, imagine you are trying to translate a book from English to French.

• Standard Translation (Quantum Channel): You translate word-for-word. The grammar is perfect, and the meaning is preserved. This is what we usually allow.
• The "Magic Translator" (HPTP Map): This translator is a bit wild. It might use words that don't strictly exist in the dictionary or rearrange sentences in weird ways. It might even use "negative words" (like saying "I didn't not like it" to mean "I liked it").

The Paper's Discovery:
The authors prove that if you can use this "wild" Magic Translator to turn the output of Machine A into the output of Machine B, then Machine A is as powerful as Machine B.

Even though the translator uses "weird math" (negative probabilities or non-physical steps) that you can't build in a real lab, the information is still there. If you have enough copies of Machine A's output, you can mathematically reconstruct Machine B's output.

The "Potato" Analogy: Why This Matters

Let's use a food analogy to see why this is a big deal.

1. Machine A (The Depolarizing Channel): Imagine a machine that takes a fresh potato and mashes it into a smooth, uniform paste. It's very messy. You can't tell if the potato was red or white anymore.
2. Machine B (The Identity Channel): Imagine a machine that just hands you the fresh, whole potato.

The Old Rule: Can you turn the mashed potato (A) back into a whole potato (B)? No. So, under old rules, A is "worse" than B. You can't get B from A.

The New Rule: The authors say, "Hold on. If I have a lot of mashed potatoes, and I use my 'Magic Translator' (which involves some impossible math tricks), I can calculate exactly what the original potato looked like."

So, under the new rule, Machine A is just as powerful as Machine B because the information hasn't been lost; it's just hidden in the noise. You just need the right "decoder" to get it back.

The Catch: The "Impossible" Machine

Here is the twist. The paper shows that while Machine A can be mathematically turned into Machine B using this "Magic Translator," you cannot build a real physical machine to do it.

• The "Magic Translator" is not a real machine. It's like a recipe that requires you to subtract 5 eggs from a bowl that only has 3. You can write the math, but you can't actually do it in a kitchen.
• The Conclusion: Machine A is "as powerful as" Machine B in terms of information, but it is harder to physically implement Machine B if you only have Machine A.

The "Cost" of Doing the Impossible

The paper introduces a way to measure how hard it is to use Machine A to mimic Machine B. They call this "Physical Implementability."

• If the "Magic Translator" is close to a real machine, the cost is low.
• If the translator requires wild, impossible math (like huge negative numbers), the cost is high.

The Takeaway:
If you have a noisy machine (Machine A) and you want to simulate a perfect machine (Machine B), you can do it, but it will cost you. The "noisier" your starting machine is, the more "magic" (and cost) you need to simulate the perfect one.

Why Should You Care? (The "Incompatibility" Lesson)

The paper ends with a cool lesson about incompatibility.

Imagine you have two tools: a hammer and a screwdriver.

• Sometimes, you can use a hammer to drive a screw (it's messy, but it works).
• But sometimes, just because you can mathematically describe how to turn a hammer into a screwdriver, doesn't mean the hammer and screwdriver are "compatible" tools for the same job.

The authors show that just because you can mathematically convert the output of one quantum device to another, it doesn't mean the two devices work well together in the real world. It helps scientists understand the limits of quantum computers and how much "noise" they can tolerate before the information is truly gone.

Summary in One Sentence

This paper proves that even if a quantum machine is very noisy, it might still hold all the information of a perfect machine, provided you have a "magic math decoder" to unlock it—even though building that decoder in the real world might be impossible or very expensive.

Technical Summary

Here is a detailed technical summary of the paper "Comparing quantum channels using Hermitian-preserving trace-preserving linear maps: A physically meaningful approach" by Arindam Mitra and Jatin Ghai.

1. Problem Statement

In quantum information theory, Quantum Channels (Completely Positive Trace-Preserving or CPTP maps) describe the physical evolution of quantum systems. A fundamental question is how to compare two quantum channels, $\Lambda_1$ and $\Lambda_2$, to determine if one is "more powerful" or "more informative" than the other.

Traditionally, this comparison is done via post-processing: $\Lambda_1$ is considered superior to $\Lambda_2$ if $\Lambda_2$ can be obtained by concatenating $\Lambda_1$ with another quantum channel ($\Lambda_2 = \Theta \circ \Lambda_1$, where $\Theta$ is CPTP). However, this relation is restrictive. The authors investigate a broader, physically motivated scenario: Given an unknown input state, if the output statistics of $\Lambda_2$ can be uniquely reconstructed from the output statistics of $\Lambda_1$ using quantum measurements, does this imply a specific mathematical relationship between them?

The paper addresses whether this "statistical recoverability" implies that $\Lambda_2$ is a post-processing of $\Lambda_1$ via a standard quantum channel, or if it requires a broader class of maps, specifically Hermitian-Preserving Trace-Preserving (HPTP) maps, which are not necessarily positive (and thus not physically realizable as standalone channels).

2. Methodology

The authors employ a rigorous mathematical framework combining quantum measurement theory, linear algebra, and resource theory:

• Asymptotic Power Definition: They define a channel $\Lambda_1$ as "at least as powerful as" $\Lambda_2$ in the asymptotic sense ($\Lambda_1 \succeq_{asymp} \Lambda_2$) if there exist measurements $M_1$ and $M_2$ such that the probability distribution of outcomes for $\Lambda_2$ can be perfectly reconstructed from the outcomes of $\Lambda_1$ for any input state $\rho$.
• Informationally Complete (IC) Measurements: The analysis relies heavily on IC measurements, where the output statistics uniquely determine the quantum state. The authors prove that if the relation holds for any IC measurement, it holds specifically for minimal IC measurements.
• Choi-Jamiolkowski Isomorphism: Used to represent linear maps as matrices (Choi matrices) to analyze properties like positivity and trace preservation.
• Kernel Analysis: The authors utilize the kernel (null space) of linear maps to establish necessary and sufficient conditions for the ordering relation.
• Physical Implementability Quantifier: They utilize the quantifier $\nu(\Theta)$ (introduced in Ref. [5]) to measure the "cost" of simulating a non-physical HPTP map $\Theta$ using physical CPTP maps. This is defined as the logarithm of the minimum sum of coefficients in a linear decomposition of $\Theta$ into CPTP maps.

3. Key Contributions and Results

A. Equivalence of Asymptotic Power and HPTP Concatenation

The central theoretical result is Theorem 1, which establishes a fundamental equivalence:
$$\Lambda_1 \succeq_{asymp} \Lambda_2 \iff \exists \Theta \in \text{HPTP} \text{ such that } \Lambda_2 = \Theta \circ \Lambda_1$$

• Significance: This proves that if one can statistically recover the output of $\Lambda_2$ from $\Lambda_1$, the mathematical link between them is an HPTP map. Crucially, this map $\Theta$ does not need to be positive (i.e., it is not a valid quantum channel).
• Implication: This distinguishes "statistical recoverability" from "physical post-processing."

B. Hierarchy of Preorders

The paper establishes a strict hierarchy among different ways of comparing channels (illustrated in Fig. 1):

1. Post-processing Preorder ($\succeq_{postproc}$): $\Lambda_2 = \Theta \circ \Lambda_1$ where $\Theta$ is CPTP.
2. Positive Map Preorder ($\succeq_{P}$): $\Lambda_2 = \Theta \circ \Lambda_1$ where $\Theta$ is Positive Trace-Preserving (PTP).
3. HPTP/Asymptotic Preorder ($\succeq_{asymp} \equiv \succeq_{HP}$): $\Lambda_2 = \Theta \circ \Lambda_1$ where $\Theta$ is HPTP.

The inclusion relation is:
$$C^{CP} \subseteq C^{P} \subseteq C^{asymp} = C^{HP}$$
Example 2 demonstrates that $\Lambda_1 \succeq_{asymp} \Lambda_2$ does not imply $\Lambda_1 \succeq_{postproc} \Lambda_2$. Specifically, a depolarizing channel can be "asymptotically more powerful" than the identity channel (in terms of reconstructability via specific measurements), yet the identity channel cannot be obtained from the depolarizing channel via any physical quantum channel.

C. Characterization via Kernel and Trace Distance

The authors provide alternative characterizations for $\Lambda_1 \succeq_{asymp} \Lambda_2$:

• Theorem 2: The relation holds if and only if $\ker(\Lambda_1) \subseteq \ker(\Lambda_2)$.
• Theorem 3: The relation holds if and only if the condition $\|\Lambda_1(\rho) - \Lambda_1(\sigma)\|_1 = 0$ implies $\|\Lambda_2(\rho) - \Lambda_2(\sigma)\|_1 = 0$. This links the ordering to the distinguishability of states.

D. Physical Implementability Quantifier ($R_{\Lambda_1}(\Lambda_2)$)

To quantify the difficulty of implementing $\Lambda_2$ given that $\Lambda_1$ is already available, the authors define:
$$R_{\Lambda_1}(\Lambda_2) := \log \min_{\Theta} \{ 2^{\nu(\Theta)} \mid \Lambda_2 = \Theta \circ \Lambda_1, \Theta \in \text{HPTP} \}$$

• Properties: They prove $R_{\Lambda_1}(\Lambda_2) = 0$ if and only if $\Lambda_1 \succeq_{postproc} \Lambda_2$ (i.e., no extra cost).
• Monotonicity (Theorem 4): If $\Lambda_1$ is post-processed to become $\Lambda'_1$ (making it noisier), the cost $R$ decreases. Conversely, if $\Lambda_2$ becomes noisier, the cost to implement it from $\Lambda_1$ also decreases.

E. Implications for Quantum Incompatibility

The paper addresses Example 3, showing that the incompatibility of two channels cannot be simply reduced to the incompatibility of a measurement and a channel. Even if a measurement and a channel are compatible, the channels themselves might be incompatible. This clarifies the limits of using measurement-channel compatibility to infer channel-channel compatibility.

4. Significance and Conclusion

• Physically Meaningful Generalization: The work provides a rigorous physical justification for using HPTP maps (which are generally unphysical) as a tool for comparing quantum channels. It shows that while HPTP maps cannot be implemented directly, they represent the mathematical "bridge" required to explain statistical recoverability between channels.
• Resource Theory: By introducing $R_{\Lambda_1}(\Lambda_2)$, the paper quantifies the "resource cost" of simulating a target channel using a source channel when the transformation requires non-physical (non-positive) operations.
• Hierarchy Clarification: It resolves the distinction between "can we get the statistics?" (HPTP) and "can we get the state via a physical device?" (CPTP), showing these are distinct concepts with a clear hierarchy.
• Future Directions: The authors suggest exploring the physical meaning of positive maps (intermediate between CPTP and HPTP) in channel concatenation and investigating connections to information-theoretic and thermodynamic properties.

In summary, the paper establishes that statistical recoverability of quantum channel outputs is mathematically equivalent to the existence of an HPTP map connecting them, a relation that is strictly broader than physical post-processing, and provides a quantitative framework to measure the cost of bridging this gap.