26 votes
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Significance of The Church of the Higher Hilbert space

The church of the larger (or higher, or greater) Hilbert space is just a trick that some people like (myself included) for rewriting some operations. The most general operations that you can write ...
DaftWullie's user avatar
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26 votes

If all quantum gates must be unitary, what about measurement?

Unitary operations are only a special case of quantum operations, which are linear, completely positive maps ("channels") that map density operators to density operators. This becomes obvious in the ...
M. Stern's user avatar
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24 votes
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How does the vectorization map relate to the Choi and Kraus representations of a channel?

One way to understand the relationship between the Choi representation of a channel and its possible Kraus representations is to use the vectorization map. Suppose that we have two finite-dimensional ...
John Watrous's user avatar
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24 votes

Twirling Quantum Channels: Pauli and Clifford Twirling

Definitions Denoting the Haar measure of some function $f\left(x\right)$ over $d$-dimensional unitaries as $\int_{\mathrm U\left(d\right)}f\left(x\right)d\mu\left(x\right)$, twirling some arbitrary ...
Mithrandir24601's user avatar
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15 votes

If all quantum gates must be unitary, what about measurement?

Short Answer Quantum operations do not need to be unitary. In fact, many quantum algorithms and protocols make use of non-unitarity. Long Answer Measurements are arguably the most obvious example of ...
glS's user avatar
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13 votes

If all quantum gates must be unitary, what about measurement?

At risk of going off-topic from quantum computing and into physics, I'll answer what I think is a relevant subquestion of this topic, and use it to inform the discussion of unitary gates in quantum ...
Emily Tyhurst's user avatar
13 votes
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What is the "complementary map" of a channel with given Kraus decomposition?

Let's start by finding a complementary channel for any channel given by a Kraus representation $$ \Phi(X) = \sum_{k=1}^n A_k X A_k^{\dagger}. $$ To make the necessary equations clear, let us assume ...
John Watrous's user avatar
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12 votes
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Proof of an Holevo information inequality for a classical-classical-quantum channel

It appears that the statement is not true in general. Suppose $X = Y = \{0,1\}$, $\mathcal{H}$ is the Hilbert space corresponding to a single qubit, and $W$ is defined as \begin{align} W(0,0) & = |...
John Watrous's user avatar
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12 votes
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Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?

This question is posed, and answered positively, in Nielsen & Chuang in a subsection of chapter 8 entitled "System-environment models for and operator-sum representation". In my version, it can be ...
DaftWullie's user avatar
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12 votes

What is the "Stinespring Dilation"?

Not exactly sure what you find confusing, but the ultimate need for Stinespring dilation theorem is that in quantum mechanics the dynamics is in general defined by a completely positive trace ...
Amara's user avatar
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12 votes
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How to calculate the average fidelity of an amplitude damping channel

An elementary method is to simply carry out the integration $$ \begin{align} \overline{F} &= \int\langle\psi|\mathcal{N_\gamma}(|\psi\rangle\langle\psi|)|\psi\rangle d\psi\\ &=\int\langle\psi|...
Adam Zalcman's user avatar
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11 votes
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Do the Kraus operators of a CPTP channel need to be orthogonal?

There is an ambiguity in the choice of Kraus operators: If $\{E_a\}$ is a set of Kraus operators for a channel $\mathcal E$, so is $\{F_b\}$ with $F_b=\sum_a v_{ab} E_a$, with $(v_{ab})$ an isometry. ...
Norbert Schuch's user avatar
11 votes
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What does it mean "less than identity" in the operator sum representation?

Matrix inequalities of the form $A\ge B$ should be read as $$ A-B\ge 0\ , $$ which in turn means that all eigenvalues of $A-B$ are larger or equal than zero. In the given case, $M\le I$ means that ...
Norbert Schuch's user avatar
11 votes
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What is the relationship between Choi and Chi matrix in Qiskit?

( I copied some text from a previous answer of mine) Defining the Choi and $\chi$ matrix The Choi matrix is a direct result of the Choi-Jamiolkowski isomorphism. Some intuition on what this is can be ...
JSdJ's user avatar
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11 votes
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Inverting the depolarizing channel

The existence of the inverse of a linear map is independent of the way the map affects the trace. Moreover, if an invertible map preserves a property then its inverse necessarily also preserves the ...
Adam Zalcman's user avatar
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10 votes
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Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?

You cannot always find such a Kraus decomposition. Notice that any CPTP map $\mathcal E$ which does have a decomposition as a probabilistic mixture unitaries is unital, which is to say that it maps ...
Niel de Beaudrap's user avatar
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What does the adjoint of a channel represent physically?

The adjoint of a channel $\Phi$ represents how observables transform (in the Heisenburg picture), under the physical process for which $\Phi$ is the description of how states transform (in the ...
Niel de Beaudrap's user avatar
10 votes
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What is the rank of a quantum channel?

Every quantum channel has many Kraus representations that may differ in the number of Kraus operators. For example, for any positive integer $n$ and numbers $p_i$ with $i=1,\dots,n$ and $\sum_{i=1}^...
Adam Zalcman's user avatar
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10 votes

Why does the twirl of a quantum channel give a depolarizing channel?

I hope you do not mind if I zoom out a bit and talk about representation theory. I think a more general approach helps understanding the essential bits and will be helpful if you encounter similar ...
Markus Heinrich's user avatar
10 votes
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How does the number of copies affect the diamond distance?

TL;DR: If $\Phi$ and $\Psi$ are quantum channels (unitary or otherwise), then things are "too good to be true". Proposition. If $\|\Phi-\Psi\|_\diamond\le\varepsilon$ and $m=\max(\|\Phi\|_\...
Adam Zalcman's user avatar
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10 votes

What is the difference between quantum gates and quantum channels?

A quantum gate is a unitary operator on a Hilbert space, where typically this Hilbert space is associated with a system of qubits. In the case of a single qubit a quantum gate is a $2\times 2$ unitary ...
Condo's user avatar
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10 votes

Is there a comprehensive list of counterexamples in quantum information?

A few weeks ago I launched a little side project of mine: a google document on Counterexamples in Quantum Information The idea was to have a centralized document acting as a reference work for ...
Frederik vom Ende's user avatar
9 votes
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Confusion on the definition of the phase-damping channel

Let $\mathcal{N}$ be the channels which subscripts for which conventions. $$ \mathcal{N}_{N.C.} (\rho) = \begin{pmatrix} \rho_{00} & \rho_{01} \sqrt{1-\lambda}\\ \rho_{10} \sqrt{1-\lambda} & \...
AHusain's user avatar
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9 votes

Significance of The Church of the Higher Hilbert space

"Church of the higher hilbert space" is a term coined by John Smolin. According to quantiki it is: for the dilation constructions of channels and states, which [...] provide a neat characterization ...
auden's user avatar
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9 votes
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Degradable channels and their quantum capacity

A channel $\Phi$ is said to be degradable if there exists another channel $\Xi$ such that $\Xi\Phi$ is complementary to $\Phi$. The idea here is as follows. Suppose $\Phi$ is a channel and $\Psi$ is ...
John Watrous's user avatar
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9 votes
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Positive semidefinite relationship after partial trace

No, not necessarily. For example, take $\rho$ to be a GHZ state and let $\sigma$ be the completely mixed state of one qubit. We then have $\lambda=4$ and $\mu=2$.
John Watrous's user avatar
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What are examples of the correspondence between channels and their Stinespring dilations?

Obviously, there's infinitely many channels, so you cannot have an exhaustive table. I'll just give the Stinespring isometries corresponding to some notable ones. I'll use the examples from this post ...
glS's user avatar
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9 votes

Can a CPTP map increase the purity of a state?

Yes, some quantum channels can increase purity. For example the preparation channel $$ T(X) = \mathrm{Tr}[X] |\psi\rangle \langle \psi| $$ that can be thought of as throwing away your system and ...
Rammus's user avatar
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8 votes

Is acting with a positive map on a state not part of a larger system allowed?

Any map which is not Completely Positive, Trace Preserving (CPTP), is not possible as an "allowed operation" (a more-or-less complete account of how some system transforms) in quantum mechanics, ...
Niel de Beaudrap's user avatar
8 votes

If all quantum gates must be unitary, what about measurement?

There are several misconceptions here, most of them originate from exposure to only the pure state formalism of quantum mechanics, so let's address them one by one: All quantum operations must be ...
keisuke.akira's user avatar

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