24 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 ...
  • 2,363
23 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 ...
  • 48.1k
20 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 ...
  • 4,598
19 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 ...
  • 3,497
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 ...
  • 19.5k
12 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 ...
12 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 ...
  • 4,598
11 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) & = |...
  • 4,598
11 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 ...
10 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, it can be ...
  • 48.1k
9 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 ...
9 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. ...
9 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 $\mu=2$.
  • 4,598
9 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|...
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9 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 ...
  • 14.7k
8 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., $$ \sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ , $$ ...
8 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} & \...
  • 3,553
8 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 ...
  • 3,339
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, ...
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 ...
8 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 ...
  • 356
8 votes
Accepted

How to maximise over linear functionals of quantum channels?

For the specific linear function you are interested in, the solution turns out to be trivial: you can take the channel to be $N_{X\rightarrow Y}(\rho) = \operatorname{Tr}(\rho) |\psi\rangle\langle \...
  • 4,598
8 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 ...
  • 4,898
8 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 ...
8 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}^...
  • 14.7k
7 votes

What quantum channels are considered in quantum communication, and how does this choice affect the construction of error correction codes?

First let me mention a minor point concerning terminology. The type of channel you are suggesting is often called a Pauli channel; the term depolarizing channel usually refers to the case where $p_x = ...
  • 4,598
7 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 ...
  • 5,978
7 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 ...
  • 4,598
7 votes
Accepted

What does it mean to take the Choi-Jamiolkowski of a quantum channel?

Let me quote my answer from over at physics.SE: The intuition Let us consider a channel $\mathcal E$, which we want to apply to a state $\rho$. (This could equally well be part of a larger system.) ...
7 votes
Accepted

Can any channel be written as $\Phi(X)=\operatorname{Tr}_{\mathcal Z}[U(X\otimes \sigma)U^\dagger]$ for any state $\sigma$?

No, this is not always possible. A counterexample is given by $\sigma=I/d'$ and $\Phi(X)=\mathrm{tr}(X)|0\rangle\langle0|$. To see this, note that for $X=I/d$, \begin{align} 2(1-1/d) & = \|\,|0\...

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