Petz recovery map

Key takeaways

Petz recovery map belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Petz recovery map to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Petz recovery map from memory before moving on to harder problems.

Reference excerpt

In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of a quantum analog of Bayes' theorem. Proposed by Dénes Petz, the Petz recovery map is a quantum channel associated with a given quantum channel and quantum state. (A quantum channel is a completely positive trace-preserving (CPTP) map.) This recovery map is designed in a manner that, when applied to an output state resulting from the given quantum channel acting on an input state, it enables the inference of the original input state. In essence, the Petz recovery map serves as a tool for reconstructing information about the initial quantum state from its transformed counterpart under the influence of the specified quantum channel. The Petz recovery map finds applications in various domains, including quantum retrodiction, quantum error correction, and entanglement wedge reconstruction for black hole physics.

Definition Suppose we have a quantum state which is described by a density operator σ {\displaystyle \sigma } and a quantum channel E {\displaystyle {\mathcal {E}}} , the Petz recovery map is defined as

P σ , E ( ρ ) = σ 1 / 2 E † ( E ( σ ) − 1 / 2 ρ E ( σ ) − 1 / 2 ) σ 1 / 2 . {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}(\rho )=\sigma ^{1/2}{\mathcal {E}}^{\dagger }({\mathcal {E}}(\sigma )^{-1/2}\rho {\mathcal {E}}(\sigma )^{-1/2})\sigma ^{1/2}.}

Notice that E † {\displaystyle {\mathcal {E}}^{\dagger }} is the Hilbert-Schmidt adjoint of E {\displaystyle {\mathcal {E}}} . Then, we have that

P σ , E ( E ( σ ) ) = σ . {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}({\mathcal {E}}(\sigma ))=\sigma .}

Thus, the Petz recovery map perfectly recovers the state σ {\displaystyle \sigma } from the channel output E ( σ ) . {\displaystyle {\mathcal {E}}(\sigma ).}

The Petz map has been generalized in various ways in the field of quantum information theory.

Properties of the Petz recovery map A crucial property of the Petz recovery map is its ability to function as a quantum channel in certain cases, making it an essential tool in quantum information theory.

The Petz recovery map is a completely positive map, since (i) sandwiching by the positive semi-definite operator E ( σ ) − 1 / 2 ( ⋅ ) E ( σ ) − 1 / 2 {\displaystyle {\mathcal {E}}(\sigma )^{-1/2}(\cdot ){\mathcal {E}}(\sigma )^{-1/2}} is completely positive; (ii) E † {\displaystyle {\mathcal {E}}^{\dagger }} is also completely positive when E {\displaystyle {\mathcal {E}}} is completely positive; and (iii) sandwiching by the positive semi-definite operator σ 1 / 2 ( ⋅ ) σ 1 / 2 {\displaystyle \sigma ^{1/2}(\cdot )\sigma ^{1/2}} is completely positive. It's also clear that P σ , E {\displaystyle {\mathcal {P}}_{\sigma ,{\mathcal {E}}}} is trace non-increasing, since

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Worked examples

Example 1 — a first encounter with Petz recovery map

Start with the simplest possible case. Write down what Petz recovery map claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Petz recovery map before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Petz recovery map ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Petz recovery map

In research: Petz recovery map appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Petz recovery map in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Petz recovery map is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum information theory, Von Neumann algebras, so understanding it makes those chapters shorter.
In everyday life: Look for Petz recovery map outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Petz recovery map in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Petz recovery map means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Petz recovery map out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Petz recovery map in simple terms?

In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of a quantum analog of Bayes' theorem. Proposed by Dénes Petz, the Petz recovery map is a quantum channel associated with a given quantum channel and quantum state.

Why does Petz recovery map matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Petz recovery map?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Petz recovery map.