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### Counterexamples in quantum information theory

Quantum Computing / Quantum Complexity Theory Requirements for exponential speedup Clifford circuits can (1) create superposition such as with Hadamard gates, (2) create entanglement such as with ...

### 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 ...

### Counterexamples in quantum information theory

Quantum thermodynamics The set of thermal operations is not (topologically) closed. In the qubit case, the set of channels which lie arbitrarily close to the thermal operations is characterized in ...

### 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. ...

### 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. ...
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Accepted

### What is the meaning of $\sum_i K_iK_i^\dagger$ for a quantum channel with Kraus operators $K_i$?

Yes, the sum $\sum_iK_iK_i^\dagger$ is (a scalar multiple of) the quantum state output by the channel on the maximally mixed input \begin{align} N\left(\frac{I}{d}\right)=\frac{1}{d}\sum_iK_iK_i^\...
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### $M(\rho)=\operatorname{Tr}_2[U(\rho\otimes\rho_2)U^{\dagger}]$ is unitary $\iff U=U_1\otimes U_2$, a product of $2$ unitary operators?
To add to the other nice answers as well as John Watrous' great counterexample: Interestingly one can characterize when "$\operatorname{Tr}_2(U(\rho\otimes\omega)U^{\dagger})$ is a unitary ...