27
votes
Accepted
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 ...
- 60.7k
27
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 ...
- 2,457
26
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 ...
- 3,756
25
votes
Accepted
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 ...
- 6,227
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♦
- 26.3k
14
votes
Accepted
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 ...
- 6,227
14
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 ...
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 ...
- 1,097
13
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 ...
- 431
13
votes
Accepted
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\|_\...
- 24.2k
12
votes
Accepted
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, ...
- 60.7k
12
votes
Accepted
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) & = |...
- 6,227
12
votes
Accepted
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 ...
- 7,673
12
votes
Accepted
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|...
- 24.2k
11
votes
Accepted
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.
...
- 7,673
11
votes
Accepted
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 ...
- 5,639
11
votes
Accepted
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 ...
- 12.3k
11
votes
Accepted
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 ...
- 24.2k
10
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 ...
- 3,469
10
votes
Accepted
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 ...
- 12.3k
10
votes
Accepted
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$.
- 6,227
10
votes
Accepted
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}^...
- 24.2k
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 ...
- 5,317
10
votes
Accepted
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♦
- 26.3k
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 ...
- 2,156
10
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|
$$
that can be thought of as throwing away your system and ...
- 6,226
9
votes
Accepted
How to find the operator sum representation of the depolarizing channel?
This really depends where you want to start from. For instance, you can construct the Choi state of $\mathcal E$, i.e.,
$$
\sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ ,
$$
...
- 7,673
9
votes
Accepted
Confusion on the definition of the phase-damping channel
Let $\mathcal{N}$ be the channels which subscripts for which conventions.
$$
\mathcal{N}_{N.C.} (\rho) = \begin{pmatrix}
\rho_{00} & \rho_{01} \sqrt{1-\lambda}\\
\rho_{10} \sqrt{1-\lambda} & \...
- 3,683
9
votes
Accepted
What's the difference between Kraus operators and measurement operators?
Quantum measurement (without results recording) is just a special case of quantum operation (quantum channel). So, yes, measurement operators (as in general measurement formalism) are indeed Kraus ...
- 7,612
9
votes
Accepted
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 ...
- 6,227
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