Questions tagged [quantum-operation]

In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. If appropriate, also use the [quantum-channel] tag. (Wikipedia)

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115 views

If a state is only “close to” an eigenstate of an operator, how many applications of the operator does it take to scramble the state?

Suppose we have an operator $U$, and a register $|\lambda\rangle$ in an eigenstate of $U$ with eigenvalue $\lambda=1$. Repeatedly applying $U$ to $|\lambda\rangle$ does not affect $|\lambda\rangle$ - ...
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What's the difference between observing in a given direction and operating in that same direction?

So starting with an up particle: $$ \lvert \uparrow \rangle = \begin{bmatrix} 1 \\ 0 \\ \end{bmatrix} $$ My understanding is that you can measure $\lvert \uparrow \rangle$ in $X$ and have ...
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Quantum operation that always produces output orthogonal to input

I'm trying to show/convince myself the following statement is correct (I haven't been able to find any similar posts): "There is no reversible quantum operation that transforms any input state to a ...
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How to add scalar to quantum string

I need a way to add scalar values to a quantum string. Say if $| v \rangle = |1 1 0 \rangle + |1 0 1 \rangle$ then $|v \rangle + 5 = |1 0 1 1 \rangle + |1 1 0 0\rangle$ Is there a known method to ...
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Can one operator commute with four other operators?

I want to know whether I can have a operator $A$ which commutes with four other operators $M_1$, $M_2$, $M_3$, and $M_4$ (for instance, drawing the operators $M_j$ from $\{H,I,X,Y\}$). When can we ...
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60 views

Find the reduced density matrix for a four-qubit system

I have the state vector $|p\rangle$ made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to find the reduced density matrix ...
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31 views

Find the local unitary that takes the bell state to a state phi that has an extractable bell state

I have a state $|p\rangle$ that has an extractable Bell state and I want to write it as a Bell state, $|b\rangle$, with a local unitary acting on one side. Basically I am trying to find a local ...
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57 views

Why can any LOCC operation be written as $\sum_k (A_k\otimes B_k)\rho(A_k^\dagger\otimes B_k^\dagger)$?

This statement can be found in Vedral et al. 1997, eq. (1). More precisely, given a bipartite state $\rho_{AB}$, they claim that any operation on $\rho_{AB}$ involving local operations plus classical ...
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Do the eigenvalues of the Choi matrix have any direct physical interpretation?

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map, and let $J(\Phi)$ be its Choi representation. As is well known, any such map can be written in a Kraus representation of the form $$\Phi(X)=...
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Do the Kraus operators of a CPTP channel need to be orthogonal?

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map. Any such channel admits a Kraus decomposition of the form $$\Phi(X)=\sum_a A_a X A_a^\dagger,$$ for a set of operators $A_a\in\mathrm{Lin}(\...
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Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?

If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that $$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$ in the ...
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What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary ...
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Are CPTP operators and unitary operators the same thing?

I am reading some quantum papers (In particular, this one page 34) . One of the theorem statement reads, "For every CPTP operator M, we have that .... " I ...
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(Nielsen & Chuang) derive phase damping quantum operation

I am reading about the phase damping quantum operation on page 384 of Nielsen&Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition). Nielsen&Chuang derived the ...
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Calculating power of a quantum computer — RSA

As discussed in this question, the expected security of 1024-bit RSA is 80-bits: NIST SP 800-57 §5.6.1 p.62–64 specifies a correspondence between RSA modulus size $n$ and expected security strength ...
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Kronecker notation of an operator

Suppose I have the state $|A\rangle=|x\rangle^l\otimes |y\rangle^l \otimes |z\rangle^l \otimes |0\rangle_x^l\otimes |0\rangle_y^l\otimes |0\rangle_z^l$. I perform the transformation between the $|x\...
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Structural Physical Approximation of Partial Transpose

To make the partial transpose a complete positive and therefore physical map, one has to mix it with enough of the maximally mixed state to offset the negative eigenvalues. The most negative ...
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58 views

Does the dilation in Naimark's theorem produce a state?

A POVM, as defined for example in (Peres and Wooters 1991), is defined by a set of positive operators $\mu(a)$ satisfying $\sum_a \mu(a)=\mathbb 1$. We do not require the $\mu(a)$ to be projectors, ...
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How does a map being “only” positive reflect on its Choi representation?

We know that a map $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ being completely positive is equivalent to its Choi representation being positive: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal ...
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Direct derivation of the Kraus representation from the natural representation, using SVD

$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL( \X)...
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How are witness operators physically implemented?

Let's take an example of an entanglement witness of the form $W = | \phi \rangle \langle \phi | ^{T_2}$ where $ | \phi \rangle $ is some pure entangled state. If I wanted to test some state $\rho$, I ...
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What is the relevance of preservation of trace in completely postive trace preserving (CPTP) maps?

Why is the trace preserving part necessary? Is it not enough if it can take all matrices to matrices of trace 1?
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Implementing a depolarizing channel for 2 qubits on IBM Q

I am trying to use IBM Q to perform the following depolarizing channel on a state of 2 qubits $\rho=|\psi \rangle \langle \psi |$: $$\rho \to (1-\lambda)\rho + \frac{\lambda}{4}I$$ This is within ...
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How should we interpret these quantum logic gates as physical observables?

In quantum mechanics each operator corresponds to some physical observable, but say we have the operators $X,Y,Z,H, \operatorname{CNOT}$. I understand how these gates act on qubits, but what do they ...
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What's the difference between Kraus operators and measurement operators?

It is said in a lecture note[1] by John Preskill that, Equivalently, we may imagine measuring system $B$ in the basis $\{|a\rangle\}$, but failing to record the measurement outcome, so we are ...
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Identifying a matrix with the fundamental quantum logic gates [closed]

I have the following matrix \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & ...
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Understanding classical vs. quantum channel capacities

The classical channel capacity ($C_{ea}$) and the quantum channel capacity ($Q$) as defined here (eqs. 1 and 2) are given by \begin{equation} C_{ea} = \text{sup}_{\rho} \Big[S(\rho) + S(\Phi_t \rho) -...
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What is Stinespring Dilation?

I've consulted Nielsen and Chuang to understand the Stinespring Dilation, but wasn't able to find anything useful. How does this operation relate to partial trace, Kraus operators, and purification?
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201 views

How does the vectorization map relate to the Choi and Kraus representations of a channel?

I know that the Choi operator is a useful tool to construct the Kraus representation of a given map, and that the vectorization map plays an important role in such construction. How exactly does the ...
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332 views

What is a complementary map?

I have a quantum map described by the following Kraus operators $$A_0 = c_0 \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}, \qquad A_1 = c_1 \begin{pmatrix} ...
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POVM three-qubit circuit for symmetric quantum states

I have been reading this paper but don't yet understand how to implement a circuit to determine in which state the qubit is not for a cyclic POVM. More specifically, I want to implement a cyclic POVM ...
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The meaning of measurements in different bases

There are other similar questions. But I don't understand the answers. Suppose I express $a|0⟩+b|1⟩$ in the form $\frac{c}{\sqrt2}(|0⟩+|1⟩)+\frac{d}{\sqrt 2}(|0⟩−|1⟩)$ where $a,b,c,d∈\mathbb C$. ...
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For 2x2 and 2x3 systems, is the partial transpose the only positive but not CP operation?

Question: For 2x2 and 2x3 systems, is the partial transpose the only positive but not completely positive operation that is possible? Why this came up: The criteria for detecting if a state $\rho$ is ...
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What do you specify when you physically apply a unitary?

In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the ...
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526 views

Depolarizing channel operator sum representation

In Nielsen and Chuang, it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity matrix with ...
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How many Kraus operators are required to characterise a channel with different start and end dimensions?

If we have a quantum channel mapping from a $d$-dimensional state to a $d$-dimensional state, it can be described by at most $d^2$ Kraus operators. Suppose our channel maps instead from a $d_1$-...
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Dephasing channels

I'm taking a quantum information course and one of my exercises says to find $p,p'$, for which there is a channel $\tilde\Lambda(\Lambda(\rho))=\Lambda'(\rho)$, where $\Lambda$ and $\Lambda'$ are ...
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Isometric Extension of an Erasure Channel

Show that an isometric extension of the erasure channel is $$U^N_{A\to BE} =\sqrt{1−\epsilon}\left(|0\rangle_B \langle 0|_A +|1\rangle_B \langle 1|_A \right)\otimes|e\rangle_E+ \sqrt{\epsilon}|e\...
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1answer
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Confusion on the definition of the phase-damping channel

I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action ...
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Quantum channel Holevo information additivity: proof approach

I have an interesting idea for a proof approach that someone might find useful. Here it is. Suppose we are given a quantum qubit channel $N$ (for example the amplitude damping channel) whose Holevo ...
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Three sender quantum simultaneous decoder conjecture

Recently I have started to read about network quantum information theory, where network problems are studied under the classical-quantum channel. For example, capacities of the cq-MAC, cq-broadcast or ...
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622 views

Deduce the Kraus operators of the dephasing channel using the Choi

I'm trying to deduce the Kraus representation of the dephasing channel using the Choi operator (I know the Kraus operators can be guessed in this case, I want to understand the general case). The ...
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How are Rigetti and IBM QX device parameters related to Kraus operators?

Rigetti reports the following parameters: (https://www.rigetti.com/qpu) T1, T2* times 1-qubit gate fidelity (F1q) 2-qubit gate fidelity (F2q) and, read-out fidelity (Fro) IBM QX reports the ...
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Depolarizing channel implementation on IBM Q

Given a single qubit in the computational basis, $|\psi\rangle =\alpha |0\rangle + \beta|1\rangle$, the density matrix is $\rho=|\psi\rangle\langle\psi|=\begin{pmatrix} \alpha^2 & \alpha \beta^*\\ ...
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Image of a sum of positive operators contains the images of each individual operator?

In the proof of Proposition 2.52 of John Watrous' QI book, there is the statement that $\text{im}(\eta(a))\subset\text{im}(\rho)$, where $\rho=\sum_{i=1}^{N}\eta(i)$ is a sum of positive operators and ...
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351 views

Advances in Quantum Channel Capacity

I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general. Quantum capacity is the highest rate at which quantum information can be ...
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218 views

Tensor product properties used to obtain Kraus operator decomposition of a channel

I work on a Quantum Information Science II: Quantum states, noise and error correction MOOC by Prof. Aram Harrow, and I do not understand which property of tensor products is used in one of the ...
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How to find the fidelity between two state when one is an operator?

I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states Show that the fidelity between the ...
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Quantum Channel Models

The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is $\rho\...
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Is acting with a positive map on a state not part of a larger system allowed?

In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators. I know that for a positive trace-preserving operator $\Lambda$ (e.g. the ...