Counterexamples in quantum information theory

Question

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 similar 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.

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This question is an attempt to fill this gap: a centralized post acting as a reference work for statements that one might think are true but are indeed false, as can be shown by means of a counterexample. There are several answers below, each of which contains all kinds of counterexamples that relate to a certain topic:

• Quantum channels
• Quantum states
• Quantum error correction
• General quantum information
• Quantum thermodynamics
• Quantum computing & quantum complexity theory

Importantly, all these answers are community wiki answers meaning everyone with at least 100 reputation can edit them. Hence if anyone is browsing these lists and they

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

they can just edit the corresponding answer themselves and add the counterexample. An alternative way is to use the associated Google form—for example if one is unsure whether an example fits this format—so it can be reviewed and, if approved, implemented into the body of counterexamples.

Answer 1

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 transposition map. This goes as far back as the original paper of Stinespring on complete positivity where he gave an example of a positive, but not completely positive map on $n \times n$ matrices; for $n=2$ this—in today’s notation—equals the map $\rho\mapsto\sigma_x\rho^T\sigma_x$. Explicitly, the transposition map was first used as in Arveson’s paper “Subalgebras of $C^*$-algebras”. After that, more advanced examples of maps which are $n-1$ but not $n$-positive were given by Choi as well as Takasaki & Tomiyama. A general class of qubit maps which are positive (but not completely positive) and trace-preserving is given by the Pauli transfer matrix ${\rm diag}(1,a_1,a_2,a_3)$ where $a_1,a_2,a_3\in[-1,1]$ are chosen such that the Fujiwara-Algoet conditions (arXiv) $|a_1\pm a_2|\leq|1\pm a_3|$ are violated, cf. also the comments below this answer. Note that Fujiwara-Algoet is equivalent to $(a_1,a_2,a_3)$ being in the Fujiwara-Algoet tetrahedron which is defined as the convex hull of the points $(1,1,1)$ (identity), $(1,-1,-1)$ (Pauli-X), $(-1,1,-1)$ (Pauli-Y), and $(-1,-1,1)$ (Pauli-Z), cf. Chapter 10.7 in the book "Geometry of Quantum States" by Bengtsson & Życzkowski (alt link).
• Not every quantum channel is diagonalizable. It turns out, however, that every quantum channel can be approximated by diagonalizable channels arbitrarily well as is shown in this paper
• Not every unital channel is mixed unitary. While every mixed unitary channel is unital—and while unital qubit channel is in fact mixed unitary—for three and more dimensions one can construct counterexamples to the converse implication, cf. this post, Theorem 1 in this paper, or Example 4.3 in the book of Watrous (alt link)
• Not every channel which is an isometry (i.e., trace-norm preserving) is also isometric. This is true in some cases, e.g., if the dimensions of the channel input and the channel output coincide, but if these dimensions differ then there exist trace-norm preserving channels which are not isometric, i.e., not of the form $V(\cdot)V^\dagger$ as this counterexample shows.
• Not every positive map is decomposable. What is true is that every positive linear map $\Phi:\mathbb C^{m\times m}\to\mathbb C^{n\times n}$ for $(m,n)=(2,2)$, $(2,3)$, and $(3,2)$ can be written as $\Phi=\Lambda_1 + \Lambda_2 \circ {}^T$ for some $\Lambda_1,\Lambda_2$ completely positive by the Størmer-Woronowicz theorem. In general, positive maps which can be written in this form are called "decomposable", and for all other pairs $(m,n)$ there exist examples of positive maps which are indecomposable, i.e., they cannot be written as $\Lambda_1 + \Lambda_2 \circ {}^T$ for any $\Lambda_1,\Lambda_2$ completely positive. The earliest and most famous examples here are by Choi for $(m,n)=(3,3)$ and by Woronowicz for $(m,n)=(2,4)$. For more examples of indecomposable maps see p.301 in the book "Geometry of Quantum States" by Bengtsson & Życzkowski (alt link).
• The composition of mixing channels need not be mixing.
• The Hilbert-Schmidt distance is not monotonic under channels. (arXiv). Indeed this property is what characterizes unital channels as shown in this paper (arXiv)
• Unlike with positive matrices, there may be non-invertible channels in-between invertible channels, where "invertible" is meant in the mathematical way, i.e. "bijective"
• The composition of extremal channels need not be extremal.
• Not every channel is covariant with respect to some non-trivial Hamiltonian
• Not every trace-preserving qubit map that sends the Bloch ball into the Bloch ball is a channel. Sending the Bloch ball into the Bloch ball only guarantees that the map is positive. Indeed, two counterexamples are given in this answer & subsequent comments
• The diamond norm is not monotonic under intermediate channels.
• There exist completely positive maps for which $K_2\Psi(K_1^\dagger(⋅)K_1)K_2^\dagger$ is not trace preserving (i.e. a quantum channel) regardless of how $K_1,K_2$ were chosen.
• The dual of a strictly positive map need not be strictly positive. While it is true that a map is (completely) positive if and only if its adjoint is (completely) positive the adjoint does not transfer strict positivity as this counterexample shows.
• There exist extreme points of the unital channels which are not an extreme point of the set of all channels, cf. Section II of this paper
• The $p$-operator norm of channels is not multiplicative under tensor products; cf. this (arXiv) and this (arXiv) paper
• Not every quantum channel is mean ergodic. While in finite dimensions every channel is mean ergodic an example of a non-ergodic infinite-dimensional channel can be found in Example 2.3.19 in this PhD thesis (arXiv)
• The norm of a Hermitian-preserving map need not be attained on a pure state.
• There exist quantum channels which are neither degradable nor anti-degradable. (arXiv)

Quantum channels and eigenvalues

• Not every quantum channel has a fixed point. While, in finite dimensions, every positive trace-preserving map has a fixed point by Brouwer's fixed-point theorem, for infinite-dimensional Hilbert spaces this need no longer be true, cf. Remark 5 in this paper. However, under the physical assumption that the channel leaves states with "bounded preparation cost" invariant, one can again establish the existence of a fixed point, cf. Theorem 1 in the aforementioned paper.
• A quantum channel with real eigenvalues need not admit Hermitian Kraus operators.
• There exist symmetric subsets of the unit disk (of $n^2$ elements which contains $1$) which are not the spectrum of any channel. For qubits symmetry is almost (but not quite) enough to be the spectrum of some channel: the additional condition needed are the Fujiwara-Algoet conditions (cf. Theorem 1 in this paper or, alternatively, Eq. (35) in this paper). An explicit counterexample would, e.g., be the qubit map with Pauli transfer matrix ${\rm diag}(1,\frac23,\frac23,0)$ which has symmetric spectrum but is not completely positive because it violates Fujiwara-Algoet.
• The peripheral eigenvalues of a completely positive map need not be semisimple.
• A completely positive map with eigenvalues on the unit circle need not be a unitary channel. Counterexamples can be found here and here
• There exist quantum channels with $2\to 2$ norm strictly less than $1$. If input and output of a channel $\Phi$ are of the same size then $\|\Phi\|_{2\to 2}\geq 1$ with equality if and only if the channel is unital, cf. this paper (arXiv) or Theorem 4.27 in Watrous’ book (alt link). An example of a channel $\Phi$ with input and output of different sizes such that $\|\Phi\|_{2\to 2}<1$ is given by the reset channel $\Phi:\mathbb C^{m\times m}\to\mathbb C^{n\times n}$, $X\mapsto{\rm tr}(X)\frac{\bf1}n$ as $\|\Phi\|_{2\to 2}=\sqrt{m/n}$ which is of course $<1$ whenever $m<n$.
• A completely positive map may have negative determinant. For a counterexample cf. Example 4 in this paper (arXiv). Interestingly, Theorem 5 in this paper shows that completely positive maps with Kraus rank at most 2 always have non-negative determinant.
• A positive map may have negative trace. Indeed a non-negative trace can only be guaranteed for completely positive maps.
• The spectral radius of a channel may be smaller then the spectral radius of the corresponding transition matrix/classical action. Special cases where such an inequality does hold are quantum channels as well as self-adjoint positive maps.

Representations & extensions of channels

• The (sorted) normal form of the Pauli transfer matrix is not always unique.
• The unitary Stinespring representation does not work for every ancilla state. While every channel can be written as ${\rm tr}_E(U((\cdot)\otimes|0\rangle\langle 0|)U^*)$ for some unitary $U$ and some large enough environment $E$ (as is proven in any decent book on quantum information) there exist states $\omega$ and channels $\Phi$ such that $\Phi\neq{\rm tr}_E(U((\cdot)\otimes\omega)U^*)$ for all unitaries $U$; an example where $\omega$ is any qubit state is given in this paper (arXiv) (note that in this example the environment state $|0\rangle\langle 0|$ does not work either because one qubit is too small an environment for that). An example for the more special case where $\omega$ the maximally mixed state of any dimension can be found in this answer. Moreover, this paper (arXiv) gives an example of a $d$-dimensional channel which cannot be dilated via any $d$-dimensional mixed environment.
• There exist unital channels which cannot be written in Stinespring form with maximally mixed ancillary state. (arXiv). Be aware that unital qubit channels can always be written in that way, cf. this paper (arXiv)
• Stinespring unitaries which give rise to the same channel are not necessarily related via local unitaries, even if the environment state is full rank. However, at the end of the linked answer it is shown that a weaker form of local unitary equivalence can be recovered when going to the purification of the environment state.
• Dilating a unitary channel does not require the Stinespring unitary to be of tensor product form. Such a conclusion can be drawn if and only if the environment state has full rank, cf. this answer.
• There exist pure operations $\Phi$ for which ${\rm id}\otimes \Phi$ is not pure anymore. – an operation is called "pure" if pure states are mapped to pure states. This counterexample also shows that there exist rank non-increasing channels $\Phi$ for which ${\rm id}\otimes\Phi$ can in fact increase the rank of certain inputs.
• Completely positive maps defined on a Hermitian subspace may not admit a (completely) positive extension, cf. Example 2 in this paper (arXiv)
• The distance between any two Stinespring isometries may be larger than the square root of the diamond distance of the respective channels. An example—which thus disproves the original Kretschmann-Schlingemann-Werner conjecture (arXiv)—can be found in this paper as Example 1. However, the latter paper conjectures that this inequality holds if the right-hand side is multiplied by an additional factor of $\sqrt2$.
• Stinespring isometries can be more than $\sqrt2$ apart without the fidelity of the corresponding channels being zero, cf. Appendix B in this paper
• Not every positive map defined on an operator system can be extended to a positive map defined on the full space. Interestingly, even if the positive map satisfies the stronger conditions of being unital and norm $1$ this is still wrong as shown in this paper (arXiv)
• Not every permutation covariant channel admits a permutation covariant Stinespring dilation.

Quantum channels and dynamics

• There exist (time-dependent) Markovian evolutions which do not decrease state distinguishability, cf. Theorem 17 ff. in this paper (arXiv) & Example 4 in this paper (arXiv)
• There exist Lindbladians for which the purity is not convex in time
• There exist channels which cannot be written as the convex combination of (time-dependent) Markovian channels, cf. footnote 19 in this paper (arXiv)
• Not every Lindblad generator is diagonalizable. An example of a non-diagonalizable Lindblad generator can be found in Example 6 of this paper (arXiv). Moreover, the previously linked examples of non-diagonalizable channels carry over to the Lindblad case because, given any channel $\Phi$ the map $\Phi-{\rm id}$ is always a valid Lindblad generator (Lemma 1 of this paper). However, as with channels, every Lindblad generator can be approximated by diagonalizable Lindbladians arbitrarily well as is shown in this paper
• Given a Lindblad generator $L$ it can happen that ${\rm id}+\tau L$ is not a channel for any $\tau> 0$.
• If $e^L$ is a quantum channel for some linear map $L$ one cannot conclude that $L$ is of Lindblad form
• If $e^{tL}$ is a quantum channel for some linear map $L$ and all $t\geq t_0$ (for some $t_0>0$) one cannot conclude that $L$ is of Lindblad form
• Not every autonomous unitary dilation of a Markovian open system evolution can be dynamically decoupled. (arXiv)
• Lindblad generators $L_t$ which are self dual at all times can give rise to dynamics which are not self dual anymore. While this implication does hold if $L_t\equiv L$ is time-independent, or if $[L_t,L_{t'}]=0$ for all $t\neq t'$, a counterexample for the general case is given in Section IV.A of this paper.

Answer 2

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

• The one clean qubit DQC1 complexity class can estimate values of the Jones polynomial (arXiv) which is not known to be efficient classically.
• The commuting Hamiltonian problem is not obviously in NP: they can have highly entangled ground states (such as for topologically ordered systems, e.g. the Toric Code), and thus, it is not clear how to provide an efficiently checkable classical description of the ground state.

Answer 3

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.
• The closest diagonal state to a given state need not be the dephased original state.
• Not all states that are useful for teleportation violate Bell inequalities.
• Vanishing quantum discord is not necessary for completely positive maps. Originally, Shabani and Lidar claimed in this paper (arXiv) that the initial system-environment state having vanishing quantum discord is necessary (and sufficent) for the reduced dynamics of a system to be completely positive. A few years later Brodutch et al. presented a counterexample, cf. Section III in this paper (arXiv). Since then, Shabani and Lidar have published an erratum to their original paper.
• Two states with the same purity can have different ranks.
• Not every approximate Markov chain is close to a Markov chain. The existence of a counterexample is proven, e.g., in Proposition 5.9 of this book (arXiv) as originally shown in this paper (arXiv)
• Even if two bipartite states are close the corresponding Schmidt bases need not have large overlap.
• Not every state which violates the separability criterion of the local uncertainty relations violates the realignment criterion for separability. (arXiv)
• Positive partial transpose does not imply separability beyond qubit-qutrit systems. (arXiv)

State transformations

• Even if two states are close together, there need not exist a channel close to the identity mapping one to the other state; note that this is true for pure states but fails for mixed states
• The Alberti-Uhlmann theorem does not hold beyond qubits. A counterexample for two qutrit states is given in Proposition 6 of this paper (arXiv), and a qutrit-qubit counterexample is the subject of this paper
• Local unitary equivalent stabilizer states need not be local Clifford equivalent. This was known as the LU-LC conjecture to which a counterexample (arXiv)—followed by a family of counterexamples (arXiv)—of pairs of graph states (and, by extension, stabilizer states) have been found that are equivalent under local unitary operations, but not under local Clifford operations.

... 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.

Answer 4

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. While there are no explicit counterexamples, such examples have to exist as first proven in this paper by means of a randomized construction. The proof has been further refined (i.e. in terms of bounds) and simplified here (arXiv) and here (arXiv).
• The quantum conditional entropy may be negative; a counterexample to this is the pure state $(|01\rangle-|10\rangle)/\sqrt2$ as first observed in this paper (arXiv)
• The data processing inequality does not hold for all sandwiched Rényi relative entropies. (arXiv)
• The quantum capacity of the tensor product of two zero-quantum-capacity channels may not be zero. (arXiv). This phenomenon, in which two zero-capacity channels can combine to have non-zero capacity, is called superactivation.
• The quantum relative entropy of bi-partite states is not upper bounded by the quantum relative entropy of the traced out states, even when adding the dimension of the traced out subspace

Entanglement

• There exist entangled states which are separable with respect to every bipartition. (arXiv)
• Entangled states may admit a hidden-variable model. While entangled states never admit a hidden-variable model for pure states, a counteraxmple for mixed states—which nowadays is known as "Werner state"—was first given in this paper
• Entanglement witnesses can become zero on some separable states without being optimal. Counterexamples can be found under this post. Geometrically, this means that the hyperplane generated by some witness $W$ can touch the set of separable states without $W$ being optimal; such witnesses are also called "weakly optimal", cf. Section 2.5.2 in "Entanglement detection" by Gühne and Tóth (arXiv)
• Entanglement witnesses can be optimal without having the spanning property. While witnesses that have the spanning property (i.e., ${\rm span}\{|a_i\rangle\otimes|b_i\rangle:\langle a_i\otimes b_i|W|a_i\otimes b_i\rangle=0\}$ has dimension $\dim\mathcal H_A\cdot\dim\mathcal H_B$) are known to be optimal, there exist optimal witnesses—and even optimal decomposable witnesses—which do not have the spanning property, cf. this answer for a counterexample and further references. In particular, this means that a comment made in the review "Entanglement Detection" by Gühne and Tóth (arXiv) which claims equivalence of these two properties is incorrect.
• The adjoint cannot be waived when relating a positive map to "its" entanglement witness. For a given positive map $\Lambda$, it is known that applying the usual Choi–Jamiołkowski isomorphism $\mathsf C$ to the adjoint channel $\Lambda^\dagger$ yields a "corresponding" witness in the sense that ${\rm tr}(\mathsf C(\Lambda^\dagger)\rho)<0$ always implies $({\rm id}\otimes\Lambda)(\rho)\not\geq 0$. Skipping the adjoint, however, may result in scenarios where $\mathsf C(\Lambda)$ witnesses certain entangled states that ${\rm id}\otimes\Lambda$ does not, as this counterexample shows.
• The structural physical approximation of an optimal entanglement witness need not be separable. The first counterexamples were given in this paper (arXiv) and this paper (arXiv).
• Bell nonlocality is not equivalent to distillability of entanglement.
• Not all bound entangled states admit a hidden state model. (arXiv)
• The square of a separable state is not necessarily separable.

Measurements

• Not every separable POVM can be written as a LOCC POVM. (arXiv)
• Even if the Lüders measurements of discrete observables do not disturb them mutually, the observables may not commute, see Section IV in this paper for a counterexample

Answer 5

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 a color code. A counterexample to this is the Penrose tiling: while it is 3-colorable (as shown in this paper) the number of overlapping vertices is 1 between a lot of faces that meet in a high-degree vertex. This renders the corresponding X and Z-type stabilizer generator candidates non-commuting.

Answer 6

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)