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1 vote

How to mathematically describe the action of CNOT on the control qubit alone?

In the ZX calculus, the CNOT gate factors into a Z type node for the control linked to an X type node for the target. The Z type node (the "control part of the operation") has three ports: $...
Craig Gidney's user avatar
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0 votes

How to mathematically describe the action of CNOT on the control qubit alone?

This depends on the question if you are allowing to use the information about the initial state of qubit 2. I am pointing this out because the information about qubit 2 is no longer contained in the ...
qubitzer's user avatar
  • 117
4 votes

How to mathematically describe the action of CNOT on the control qubit alone?

The action $\mathcal{E}$ of CNOT on just the control qubit is the composition of two functions: the action \begin{align} \mathcal{C}(\rho)=\text{CNOT}\,\rho\,\text{CNOT}\tag1 \end{align} of CNOT on ...
Adam Zalcman's user avatar
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4 votes
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What are kraus operators of a qubit interacting a thermal environment?

One way to model interaction with a thermal environment (of some temperature $T$) is through so-called thermal operations. Given some system's Hamiltonian $H_S$ they're all channels $\Phi$ of the form ...
Frederik vom Ende's user avatar
1 vote
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Can the spectral radius of a completely positive map exceed the spectral radius of its transition matrix?

While the inequality in quesiton does hold for all channels and all self-adjoint positive maps (as shown in the above question) perhaps surprisingly it fails for general completely positive maps. For ...
Frederik vom Ende's user avatar
1 vote
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Kraus decomposition of merging in lattice surgery

CP maps for lattice surgery merge with post-selection Post-selecting on $M_{ZZ}=0$, lattice surgery merge is described by the completely positive linear map with Kraus representation \begin{align} \...
Adam Zalcman's user avatar
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2 votes

Kraus decomposition of merging in lattice surgery

Lattice merge isn't faithfully captured by a CPTP map description (and therefore a Kraus representation). Tracking $M$, the measurement outcome, conditions what logical correction operation you will ...
rmehta's user avatar
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1 vote
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Is every quantum channel covariant with respect to some non-trivial Hamiltonian?

This quesiton passes all of Norbert Schuch's canonical channel examples except for the last one: consider the entanglement breaking channel $$ \Phi(\rho):={\rm tr}(F\rho)\sigma_1+{\rm tr}(({\bf1}-F)\...
Frederik vom Ende's user avatar
1 vote
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To what extent is the normal form of the Pauli transfer matrix unique?

It turns out that uniqueness can only be guaranteed if $\lambda_1>\lambda_2>|\lambda_3|$; otherwise one can construct counterexamples (as we will do below). In order to understand why ...
Frederik vom Ende's user avatar
1 vote

Solving optimization problems on real quantum hardware

Current quantum hardware based on a quantum circuit model has a limited number of qubits, which are prone to errors and decoherence. So, comparing solutions from classical solvers with those produced ...
MonteNero's user avatar
  • 2,646
1 vote

Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

The short answer is: yes the inclusion in question is true for all maps $\Phi$ which are completely positive and completely co-positive. In fact, it turns out to hold for the slightly more general ...
Frederik vom Ende's user avatar
2 votes
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IBM quantum computer backend cycle time and real gate duration

The integers in the InstructionDurations() are the multipliers of the real-system timestep dt. So for example, consider the ...
Shravan Patel's user avatar
4 votes
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Is the trace of a positive map always positive?

Consider the qubit map $\Phi(\rho):=\sigma_Y\rho^T\sigma_Y$, that is, $$ \Phi\begin{pmatrix}\rho_{11}&\rho_{12}\\\rho_{21}&\rho_{22}\end{pmatrix}=\begin{pmatrix}\rho_{22}&-\rho_{12}\\-\...
Frederik vom Ende's user avatar
2 votes
Accepted

If states are close together does there always exist a channel close to the identity mapping one to the other?

We shall see that for general mixed states no such upper bound can exist in the following precise sense: For no continuous function $c:[0,2]\to[0,2]$ with $c(0)=0$ does it hold that for all $\|\rho-\...
Frederik vom Ende's user avatar
6 votes

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 ...
4 votes

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 ...
3 votes

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 ...
5 votes

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. ...
4 votes

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. ...
3 votes
Accepted

How to view operator norms on open-system representation of quantum channels

First, let us prove that every quantum channel is trace-norm contractive using the unitary Stinespring representation. For this we need the following lemma. Lemma. For all $m,n\in\mathbb N$ and all $...
Frederik vom Ende's user avatar
9 votes

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 ...
2 votes
Accepted

Can any separable $\rho=\sum_i p_i\sigma_i\otimes\tau_i$ be written as $\rho=(I\otimes T)(\sum_ip_i\sigma_i\otimes|i⟩\!⟨i|)$ for some channel $T$?

You can just use $T(|i\rangle\!\langle i|)=\tau_i$.
glS's user avatar
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3 votes

Does monotonicity of diamond distance hold for intermediate channels?

TL;DR: No. The suggested bound fails to hold for any norm. Briefly, if we choose $\mathcal{A}=\mathcal{F}$ to be an idempotent channel then the right-hand side vanishes. However, if we choose $\...
Adam Zalcman's user avatar
  • 22.9k
4 votes
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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^\...
Adam Zalcman's user avatar
  • 22.9k
3 votes

$M(\rho)=\operatorname{Tr}_2[U(\rho\otimes\rho_2)U^{\dagger}]$ 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 ...
Frederik vom Ende's user avatar

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