# Tag Info

### Counterexamples in quantum information theory

Quantum Channels Quantum channels: general properties Not every positive map is completely positive. One may argue that this is the mother of all counterexamples in quantum information: the ...

### Can a CPTP map increase the purity of a state?

Yes, some quantum channels can increase purity. For example the preparation channel $$T(X) = \mathrm{Tr}[X] |\psi\rangle \langle \psi|$$ that can be thought of as throwing away your system and ...
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### Why is the coefficient-squared the probability, and not just the coefficient itself?

Scott Aaronson describes quantum mechanics as "statistics but with the L2-norm". States are L2-norm unit vectors (sum of squared amplitudes is 1) instead of L1-norm unit vectors (sum of ...
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### How can the depolarizing channel be a quantum operation?

The correct linear form of the depolarizing channel is $$\varepsilon(\rho) = (1-p)\rho + p\frac{I}{2}{\rm Tr}(\rho).$$ For density matrices ${\rm Tr}(\rho)=1$, so you can usually see the form ...
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### What channels preserve the purity of all pure inputs?

TL;DR: The unitary and reset channels are the only ones that return pure output for every pure input. That's because under Stinespring dilation the requirement that $\Phi$ return pure output for every ...
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### Does a quantum channel always preserve the identity matrix?

No, such maps are referred to as unital maps. A counterexample is the replacement map $$\mathcal{E}(X) = \mathrm{tr}(X)\sigma$$ defined for some density matrix $\sigma$.
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### Counterexamples in quantum information theory

General quantum information Entropies The relative entropy of entanglement is not additive, see Section V.B of this paper (arXiv) for a counterexample The minimal output entropy is not additive. ...
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### Existence of Hamiltonians such that the time evolution unitary becomes identity

I don't believe that this is always possible. For instance, what if my set of $\{H_i\}$s comprise a single term that I can construct to be arbitrarily awkward? The key feature will be gaps between ...
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### Can any channel be represented as $A\rho A^\dagger$ for some $A$?

No, this is not possible in general. To see it, consider for example what happens taking the trace of that expression. You'd get: \operatorname{tr}(A^\dagger A\rho)=\operatorname{tr}\left(\sum_j K_j^...
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### Qiskit reverse_bits is not equivalent to swapping qubits

The issue is that the qubit values are being mixed together throughout the circuit, so swapping at the end is not enough. However, if you also swap at the beginning of the circuit, the qubit values ...
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### Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This solution uses the spectral theorem and some elementary linear algebra computations. If you let $P = \sum_{i = 1}^{d'} \lambda_i \vert{\psi_i}\rangle\langle{\psi_i}\vert$ be the spectral ...

### Resources for understanding non-unitary channels and operators

Lecture Notes on the Theory of Open Quantum Systems by Lidar The Theory of Quantum Information by Watrous Principles of Quantum Communication Theory: A Modern Approach by Khatri & Wilde
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### Counterexamples in quantum information theory

Quantum states Quantum states: general properties The purifications of two $\varepsilon$-close states need not be $\varepsilon$-close. The fidelity depends on more than just the difference of states. ...

### Counterexamples in quantum information theory

Quantum error correction A quantum error correcting code that corrects every single-qubit X and Z error need not correct every single-qubit Y error. Not any 3-colorable lattice can be used to create ...

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### Is the "unitary twirling operation" physically realizable?

Question 1: I guess it depends what your understanding of "physical" is. In my understanding, everything you can do in the lab is physical. Thus, twirling is perfectly physical. Note that ...
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