Counterexamples in Quantum Information
Compiled by Frederik vom Ende. Last updated: 2025, Mar 27th
Introduction
This document is inspired by the famous Counterexamples in …-series from the mathematics literature: there, books list statements from a certain area of mathematics which one may intuitively think are true but then a counterexample is provided. Personally, I am making occasional use of the book “Counterexamples in Analysis” by Gelbaum & Olmsted which is why I am quite fond of it.
As far as I am aware there is no such documentation yet for quantum physics in general or quantum information theory in particular (cf. also this physics.SE post) which is why I decided to compile this list. Statements are grouped by subfield (e.g., “Quantum channels”, “Quantum states”, …)—see also the Outline on the left side—for a better overview. If at some point this document becomes broader in scope I wouldn’t mind turning it into a preprint or even a book; but for now such a simple document is probably the best way to do this.
How to use this document
This list is intended to be a reference work: Say you have a statement or a question which you think may be true (e.g., “Is the composition of mixing channels again mixing?”). Then you would either look through the “Quantum channels” section or you would search (Ctrl+F or ⌘+F) for a buzzword, e.g., “mixing”, and, ideally, find the statement you are interested in. In our case we find:
Under every question there is a link to a counterexample highlighted in purple—which links to a paper, some handwritten note, a stackexchange post, etc.—and sometimes there is some additional context or even a slightly more general statement which turns out to be true instead.
I did not find my statement here / I have a counterexample which is not on this list, but I think it should be on here
Given how huge the field of quantum information is, I do not make any claims of this document being even close to complete. The initial set of counterexamples largely overlaps with things I found useful or even discovered during my own research which is why it only covers a small portion of what could potentially be on this list. If you know of a counterexample which you should should be on here that
- disproves a concise statement (i.e. can be formulated in one line) and
- you found useful for your own research (e.g., because it defies expectation)
please tell me about it! You can either submit a contribution through this google form, write me a message on Slack, send me an e-mail (frederik.vomende[at]gmail.com), or contribute to the corresponding quantumcomputing.stackexchange post. For the acknowledgments please refer to the end of the document. In fact here are hyperlinks to this document’s main sections:
Quantum computing & quantum complexity
And now, without further ado: the list!
Quantum channels
Quantum channels: general properties
Every positive map is completely positive
One may argue that this is the mother of all counterexamples in quantum information: the transposition map. Counterexamples go 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 x n matrices; for n=2 this—in today’s notation—equals the map ρ↦σxρTσx. To my knowledge the transposition map was first used as an explicit counterexample 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 diag(1,a1,a2,a3) where a1,a2,a3∈[-1,1] are chosen such that the Fujiwara-Algoet conditions | ai ± aj | ≤ | 1 ± ak | are violated, cf. also the comments below this qc.SE answer. Note that Fujiwara-Algoet is equivalent to (a1,a2,a3) 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).
Every quantum channel is diagonalizable
To clarify, a channel is said to be diagonalizable if its representation matrix is diagonalizable in the sense that it can be written as SDS-1 for some invertible S and some diagonal D. Either way, a number of counterexamples can be found in this math.SE post. It turns out, however, that every quantum channel can be approximated by diagonalizable channels arbitrarily well as is shown in this paper
Every unital channel is mixed unitary
As shown in Theorem 1 in this paper this is true for qubits, but for three and more dimensions one can construct counterexamples, cf. Example 4.3 in Watrous’ book (alt link) and also this qc.SE answer
A channel is an isometry (i.e., trace-norm preserving) if and only if it is isometric (i.e., of the form V(.)V†
While it is easy to see that every isometric channel is an isometry, the converse does not always hold as this counterexample shows. However, there are special cases where this equivalence does in fact hold, e.g., if the dimensions of the channel input and the channel output are the same.
Every positive map can be written using the transpose
What is true is that every positive linear map Φ:ℂm x m→ℂn x n for (m,n)=(2,2), (2,3), and (3,2) can be written as Φ=Λ1 + Λ2 ◌ T for some Λ1 and Λ2 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 Λ1 + Λ2 ◌ T for any Λ1 and Λ2 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 is a mixing channel
A channel Ψ is said to be mixing if there exists a unique state ω such that for all ρ one has Ψ n(ρ)→ω as n→∞. With this an of two mixing qubit channels such that their product is not mixing anymore can be found here
The Hilbert-Schmidt distance is monotonic under channels
A counterexample is constructed in this paper
Every channel in the convex hull of bijective channels is bijective
Check this qc.SE answer for a counterexample
The composition of extremal channels is again extremal
Check this qc.SE answer for a counterexample
Every channel is covariant with respect to some non-trivial Hamiltonian
Check this qc.SE answer for counterexamples
Every trace-preserving qubit map that sends the Bloch ball into the Bloch ball is a channel
This condition only guarantees that the map is positive but not completely positive. Two counterexamples are given in this qc.SE answer & subsequent comments
The diamond norm is submultiplicative, even when defined as a sup over just all states
Check this qc.SE answer for a counterexample
The diamond norm is monotonic under intermediate channels
An example of channels A,B,C such that ‖A∘B∘C - A∘B‖◇ is larger than ‖A∘C - A‖◇ is given in this qc.se answer
Every completely positive map can be made trace-preserving via Kraus rank-1 maps
A simple counterexample is given in this qc.SE answer
The dual of a strictly positive map is strictly positive
While it is true that a map is (completely) positive if and only if its dual is (completely) positive—where the dual is defined as usual via tr(Ψ*(B)A)=tr(BΨ(A))—duality does not transfer strict positivity, which is the property that all positive definite states ρ>0 are mapped to something positive definite. A counterexample is given in this qc.SE answer
Every extreme point of the unital channels is an extreme point of the set of all channels
A counterexample is given in Section II of this paper
The p-operator norm of channels is multiplicative under tensor products
See this and this paper for counterexamples
Every quantum channel is mean ergodic
This is true in finite dimensions, but wrong in infinite dimensions (see Example 2.3.19 in here)
The norm of every Hermitian preserving map is attained on a pure state
While this is true for positive maps, a counterexample for Hermitian-preserving maps can be found here
Quantum channels are either degradable or anti-degradable
Counterexamples, showing that there are channels that are neither degradable nor anti-degradable, can be found in this paper.
Quantum channels and eigenvalues
A quantum channel has real eigenvalues if and only if it admits Hermitian Kraus operators
While the latter is sufficient for real eigenvalues, it is far from necessary, cf. this qc.SE answer
Every symmetric subset of the unit disk (of n² elements which contains 1) is the spectrum of some channel
For qubits this is almost true, the additional condition needed are the Fujiwara-Algoet conditions (cf. Thm. 1 in this paper or, alternatively, Eq. (35) in this paper). As a counterexample to our statement, the qubit map with the Pauli transfer matrix diag(1 , ⅔ , ⅔ , 0) satisfies our conditions but is not completely positive because it violates Fujiwara-Algoet. Interestingly, in higher dimensions the statement is true if the set of desired eigenvalues has only “few” non-zero entries (Thm. 2 in this paper).
The peripheral eigenvalues of a completely positive map are always semisimple
While this is true for positive maps which are either trace-preserving or unital, just positivity (or a version thereof, e.g., complete or strict positivity) are not enough to draw this conclusion, cf. this qc.se answer
A completely positive map with all eigenvalue on the unit circle is a unitary channel
Counterexamples can be found in this as well as this qc.SE answer
Every quantum channel has 2→2 norm at least 1
This is true if input and output of the channel are of the same size because then every channel has a fixed point (cf. Theorem 4.24 in Watrous’ book)—in this case the 2→2 norm of a channel is 1 if and only if the channel is unital, cf. this paper or Theorem 4.27 in Watrous’ book. However, if input and output of the channel are of different sizes, then this need not be true anymore. A simple counterexample for this is the reset channel Ψ : ℂm×m→ℂn×n, X↦tr(X) 1/n for which ‖Ψ‖2→2=(m/n)1/2 which is of course <1 whenever m<n.
A completely positive map always has non-negative determinant
For a counterexample cf. Example 4 in this paper. Interestingly, Theorem 5 therein shows that completely positive maps with Kraus rank at most 2 always have non-negative determinant.
A positive map always has non-negative trace
This is indeed true for completely positive maps, but may fail for general positive maps, cf. this qc.SE answer
The spectral radius of a channel is never smaller than the spectral radius of its classical action, i.e. of the associated transition matrix
A counterexample can be found in this qc.se answer. This may be even more surprising given this statement is in fact true for quantum channels as well as self-adjoint positive maps, cf. here
Representations & extensions of channels
The (sorted) normal form of the Pauli transfer matrix is unique
For a counterexample see this qc.SE answer
The unitary Stinespring representation works for every ancilla state
While every channel can be written as trE(U((.)⊗|0⟩⟨0|)U*) for some unitary U, there exist states ω and channels 𝛟 such that 𝛟 ≠ trE(U((.)⊗ω)U*) for all unitaries U. An example where ω is any qubit state is given in this paper, and an example for the more special case where ω the maximally mixed state of any dimension can be found here. Moreover, this paper gives an example of a d-dimensional channel which cannot be dilated via any d-dimensional mixed environment, and an example
Every unital channel can be written in Stinespring form with maximally mixed ancillary state
While this paper shows that every unital (𝛟(1)=1) qubit channel can be written as trE(U((.)⊗( 1/dim(E) ))U*) for some finite-dimensional environment E and some joint unitary U, counterexamples for qutrits and beyond can be found in this paper.
Two Stinespring unitaries which give rise to the same channel are related by a local unitary
While this is true for Stinespring isometries, this fails for Stinespring unitaries as soon as the common auxiliary state is not pure anymore, cf. this qc.SE answer. 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 auxiliary state.
Dilating a unitary channel requires the Stinespring unitary to be of tensor product form
Of course if U=U1⊗U2, then trE(U((.)⊗ω)U*) is a unitary channel. The converse, however, does not hold: it can happen that trE(U((.)⊗ω)U*) is a unitary channel but U is not of product form (see John Watrous’ comment under this qc.SE question). In fact, the latter conclusion can only be drawn if and only if the auxiliary state ω is full rank, cf. this qc.SE answer.
The extension of a pure operation to a multi-partite system is again pure
An operation is called “pure” if pure states are mapped to pure states. With this, a counterexample can be found in this qc.SE answer. This example also shows that there exist rank non-increasing channels 𝛟 for which id⊗𝛟 can in fact increase the rank of certain inputs.
A completely positive map defined on a Hermitian subspace has a completely positive extension
Such a channel need not even have a positive extension, cf. Example 2 in this paper
The distance between any two Stinespring isometries is upper bounded by the square root of the diamond distance of the respective channels
This is the original Kretschmann-Schlingemann-Werner conjecture, and strictly speaking it is false as Example 1 in this paper shows. However, the latter paper conjectures that this inequality holds if the right-hand side is multiplied by an additional factor of √2
If Stinespring isometries are more than √2 apart, then the fidelity of the corresponding channels is zero
For a numerical counterexample see Appendix B in this paper
Every positive map defined on an operator system can be extended to a positive map defined on the full space
Already in his original paper Arveson gave a counterexample with qubit output. Interestingly, even if the positive map satisfies the stronger conditions of being unital and norm 1 this is still wrong; two counterexamples to this stronger version can be found in this paper
Every permutation covariant channel admits a permutation covariant Stinespring dilation
A counterexample is given in this qc.SE answer
Quantum channels and dynamics
Any evolution that decreases state distinguishability is (time-dependent) Markovian
Theorem 17 ff. in this paper gives a counterexample (which is worked out in more detail in Example 4 in this paper)
The purity is convex under Markovian dynamics
A counterexample is given in this qc.SE answer
The convex hull of (time-dependent) Markovian channels is the set of all channels
While this is true for qubits, the Holevo-Werner channel shows that this is wrong in higher dimensions, cf. footnote [19] in this paper.
Every Lindblad generator is diagonalizable
The counterexample from the question “Every quantum channel is diagonalizable” carries over because given any channel 𝛟, the map 𝛟-id is a valid Lindblad generator (as shown in Lemma 1 of this paper). Thus any example where 𝛟 is not diagonalizable carries over as then, the generator 𝛟-id is not diagonalizable either. However, every Lindbladian can be approximated by diagonalizable Lindbladians arbitrarily well as is shown in this paper
For all Lindblad generators L there exists t > 0 such that id+tL is again a channel
This is wrong whenever L has eigenvalues on the imaginary axis, cf. this phys.SE post for a counterexample.
If eL is a quantum channel, then L is a Lindblad generator
As explained in this qc.SE post any non-Markovian bijective channel is a counterexample
If etL is a quantum channel for all t large enough, then L is a Lindblad generator
Some “pancake channels” make for counterexamples as described in this qc.SE answer; given a Pauli-diagonal generator L with suitably chosen entries—which is not of Lindblad form as it fails to be conditionally completely positive—there exists t0>0 (actually, t0≈0.61) such that etL is a channel if and only if t>t0.
Every autonomous unitary dilation of a Markovian open system evolution can be dynamically decoupled
It turns out that every Markovian open system evolution admits an autonomous unitary dilation which cannot be dynamically decoupled, cf. this paper (arXiv)
If a Lindblad generator Lt is self-dual at all times, then the corresponding dynamics are self-dual
While this is of course true for time-independent L, or if Lt commutes with itself at any two different times, a counterexample for the general case can be found in Section IV.A in this paper
Quantum states
Quantum states: general properties
If two states are ε-close their purifications are ε-close, as well
A counterexample can be found here
The fidelity depends only on the difference of states
While this is true for the trace distance, a counterexample for the fidelity is given in this qc.SE answer
The closest diagonal state to a given state is always the dephased original state
While this is true for a single qubit, this already fails in three dimensions, cf. this qc.SE answer
All states that are useful for teleportation violate Bell inequalities
While the converse of this statement is always true, the statement itself only holds for pure states. Indeed, in the case of mixed states Werner states are counterexamples as shown in this paper
Vanishing quantum discord is necessary for completely positive reduced dynamics
Originally, Shabani and Lidar claimed in this paper (arXiv) that the initial system-environment state having vanishing quantum discord is necessary (and sufficient) 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.
If two states have the same purity, they have the same rank
A qutrit counterexample is given here.
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 which was originally shown in this paper
If two bipartite states are close, then the corresponding Schmidt bases have large overlap
Counterexamples are given as answers to this qc.SE question
Every state which violates the separability criterion of the local uncertainty relations violates the realignment criterion for separability
A counterexample for two qubits as well as two qutrits is given on p.3 of this paper
Positive partial transpose implies separability beyond qubit-qutrit systems
Examples of entangled 2x4 and 3x3 states with positive partial transpose are given in this paper
State transformations
If two states are close together, then there exists a channel which maps one state to the other and which is close to the identity
While this is true if one only considers pure states, a counterexample for the general case can be found in this qc.SE answer; this example relies on the fact that the only channel which can map a full-rank state to a pure state is the reset channel
The Alberti-Uhlmann theorem holds beyond qubits
A counterexample for two qutrit states is given in Proposition 6 of this paper, and a qutrit-qubit counterexample is the subject of this paper
Local unitary equivalent stabilizer states are also local Clifford equivalent
This was known as the LU-LC conjecture to which a counterexample—followed by a family of counterexamples—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 can be identified with H⊗H*
In general, H⊗H* is isomorphic to the Hilbert-Schmidt operators (cf. here). In finite dimensions, this is not a restriction 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
Some counterexamples that, in general, (YXY)+≠YX+Y can be found in this math.SE post
General quantum information
Entropies
The relative entropy of entanglement is additive
A counterexample can be found in Section V.B of this paper
The minimal output entropy is additive
Counterexamples are known to exist as first proven in this paper by means of a randomized construction. This proof has been further refined (i.e. in terms of bounds) and simplified here and here.
The quantum conditional entropy is non-negative
A counterexample to this is the pure state 2-½(|01⟩-|10⟩) as first observed in this paper
The data processing inequality also holds for all sandwiched Rényi relative entropies
Violation of the data processing inequality for sandwiched Rényi relative entropies with α<½ is proven in this paper
The quantum capacity of the tensor product of two zero-quantum-capacity channels is zero
A counterexample was first provided in this paper. 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 upper bounded by the quantum relative entropy of the traced out states
A counterexample is given in this qc.se answer, where it is even shown that adding the dimension of the traced out subsystem to the bound does not help.
Entanglement
Separable states with respect to every bipartition are fully separable
An example of a three-qubit mixed state which is biseparable yet overall entangled can be found in this paper
Entangled states never admit a hidden-variable model
While this is true for pure states, it is false for mixed states: a counterexample to this—which nowadays is known as Werner state—was first given in this paper
Bell nonlocality is equivalent to distillability of entanglement
A counterexample is constructed in this paper
If an entanglement witness’ hyperplane touches the separable states, then the witness is optimal
Counterexamples can be found under this qc.SE 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)
An entanglement witness is optimal if and only if it has the spanning property
While witnesses that have the spanning property (i.e.,{|x⟩⊗|y⟩:⟨x⊗ y|W|x⊗y⟩=0} spans the full space) are known to be optimal, there exist optimal witnesses—and even optimal decomposable witnesses—which do not have the spanning property, cf. this qc.SE 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 entanglement witness based on the Choi matrix of a positive map Φ is always weaker than the positive map Φ itself
For a given positive map Φ, the transpose of its Choi matrix is a "corresponding" witness in the sense that tr(C(Φ)ᵀ ρ) < 0 always implies (id ⊗ Φ)(ρ) ≱ 0. Skipping the transpose, however, may result in scenarios where C(Φ) witnesses certain entangled states that Φ itself does not, as this counterexample shows.
The structural physical approximation of an optimal entanglement witness is separable
This was known as the SPA conjecture and has since been disproven by means of explicit counterexamples, cf. this paper (arXiv) and this paper (arXiv)
Bell nonlocality is equivalent to distillability of entanglement.
A counterexample can be found in this paper
All bound entangled states admit a hidden state model
The construction of a counterexample is at the heart of this paper
The square of a separable state is separable
A counterexample is given is this qc.se answer
Measurements
Every separable POVM can be written as a LOCC POVM
Counterexamples can be found in this paper
If the Lüders measurements of discrete observables do not disturb them mutually, then the observables commute
See Section IV in this paper for a counterexample
Quantum thermodynamics
The set of thermal operations is (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. The most prominent counterexample then is the so-called beta-swap, see bottom of p.4 of said paper.
Gibbs-preserving operations can be implemented via thermal operations and some (finite) amount of coherence
A family of Gibbs-preserving maps 𝛟 which cannot be written as 𝛟=Λ( (.)⊗η ) for any state η and any thermal operation Λ is constructed in Theorem 3 ff. in this paper
Every state transformation carried out by enhanced thermal operations can also be approximated by thermal operations
While these two classes of channels are known to (approximately) coincide for qubits, this paper features 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
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 [𝛟,adH]=0 whereas general Gibbs-preserving channels do not.
Every state transformation carried out by thermal operations can also be carried out by elementary thermal operations
Two quasi-classical states which can be transformed into each other via thermal operations but not via any convex combination of sequences of elementary thermal operations are constructed in Corollary 5 of this paper
Thermomajorization is a partial order if the system's Hamiltonian is non-degenerate
A qutrit counterexample is given in Remark 1 (iv) of this paper, which also falsifies a claim made in Remark 4.2 of this paper. 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 here
Given a convex set of states, the subsequent set thermomajorized states is convex, as well
For a qutrit counterexample see Figure D.1 in Appendix D of this paper
The second law of thermodynamics faithfully characterizes thermalization out of equilibrium
See section II.B and Figure 1 in this paper
Quantum computing & quantum complexity
Entanglement and interference are sufficient for significant quantum speedups
Clifford circuits can (1) create superposition such as with Hadamard gates, (2) create entanglement such as with CNOT gates, and (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
Quantum error correction
If a quantum error correcting code can correct every single-qubit X and Z error, then it can correct every single-qubit Y error
A code serving as a counterexample is constructed in this qc.SE answer
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.
Acknowledgments
I would like to thank the following people for contributing to this list: