As was already asked about in this phys.SE question many years ago—which, sadly, got closed and never received an answer—is there a collection of counterexamples in quantum information theory, "in the spirit of books like [...] Counterexamples in analysis"? Alternatively: assuming I have a statement I want to prove and Google didn't help me, is there a comprehensive list or document I can check to see whether there already exists a counterexample to it? The fantastic post of Norbert Schuch on canonical examples of quantum channels is in the spirit of this question but is, of course, only meant as a starting point to falsify conjectures about quantum channels so there is much to be found outside of what his list covers. Thus the idea is to have a centralized post acting as a reference work for statements that one might think are true but are, indeed, false & to list/link a counterexample that disproves the statement.

Edit: As per this meta post the answers below this question have been ported from the Counterexamples in Quantum Information google doc that I launched a few weeks ago. As of today, April 28th, 2024 there are several answers each of which deals with a separate class of counterexamples:

The initial set of counterexamples are mostly concerned with quantum channels and general state transformations, an overlap which comes from the fact that these are what I found useful or even discovered during my own research; of course, that doesn't mean that the list has to be limited to these topics.

Importantly, all these answers are community wiki answers meaning everyone with at least 100 reputation can edit them. So if anyone reading this

did not find their statement / has a counterexample which is not on the list, but they think it should be

you can just edit the corresponding answer yourself and add the counterexample. Alternatively, you can use the associated Google form—for example if you're not sure whether your example fits this format—so it can be reviewed and, if approved, implemented into the body of counterexamples.


6 Answers 6


Quantum Channels

Quantum channels: general properties

Quantum channels and eigenvalues

Representations & extensions of channels

Quantum channels and dynamics


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 CNOT gates, (3) cause interference such as with Z gates. But the Gottesman-Knill theorem shows that circuits composed solely of Clifford gates can be classically simulated.

  • For decision problems in NP, the Aaronson-Ambainis conjecture (arXiv) proposes constraints on the amount of structure needed for exponential speedups. But, the Yamakawa-Zhandry problem (arXiv) provides a (black-box) exponential speedup for an NP search problem.

Restricted models of quantum computing


General quantum information





Quantum error correction


Quantum states

Quantum states: general properties

State transformations

... from the perspective of operator theory

  • The bounded operators on a Hilbert space $\mathcal H$ may be larger than $\mathcal H\otimes\mathcal H^*$. The reason for this is that $\mathcal H\otimes\mathcal H^*\simeq\mathcal B^2(\mathcal H)$ with the latter being the Hilbert-Schmidt operators, cf. this phys.SE answer; but in infinite dimensions there are bounded operators which are not Hilbert-Schmidt (a simple example here is the identity operator).
  • Taking the positive part commutes with conjugation with a state.

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 Theorem 10 of this paper (arXiv). The most prominent counterexample then is the so-called beta-swap, see bottom of p.4 of said paper.
  • There exist Gibbs-preserving operations which cannot be implemented via thermal operations and some (finite) amount of coherence.
  • Not every state transformation carried out by enhanced thermal operations can be approximated by thermal operations. While these two classes of channels are known to (approximately) coincide for qubits, cf. this paper (arXiv) a pair of qutrit states which can be transformed into each other via an enhanced thermal operation, but no thermal operation can get one state even close to the other state is given in this paper (arXiv).
  • Not every Gibbs-preserving channel is a thermal operation. The first counterexample of a Gibbs-preserving map which is not a thermal operation (and, in fact, not even an enhanced thermal operation) was given in this paper. The core of the argument is that diagonal initial states remain diagonal under (enhanced) thermal operations because the latter have to obey the time-translation symmetry $[\Phi,{\rm ad}_H]=0$ whereas general Gibbs-preserving channels do not, cf. this paper
  • Not all state transformations carried out by thermal operations can also be carried out by elementary thermal operations, cf. Corollary 5 in this paper
  • Thermomajorization is not a partial order, cf. Remark 1 (iv) in this paper (arXiv). Indeed, in the quasi-classical realm the necessary and sufficient condition for thermomajorization being a partial order is that distinct sets of eigenvalues of the Gibbs states always add up to something different, cf. Theorem 1 in this paper
  • Given a convex set of states, the subsequent set thermomajorized states need not be convex, cf. Figure D.1 in Appendix D of this paper (arXiv)
  • The second law of thermodynamics does not characterize thermalization out of equilibrium, cf. Section II.B and Figure 1 in this paper (arXiv)

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