# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
Given a single qubit in the computational basis, $|\psi\rangle =\alpha |0\rangle + \beta|1\rangle$, the density matrix is $\rho=|\psi\rangle\langle\psi|=\begin{pmatrix} \alpha^2 & \alpha \beta^*\\ ... 2answers 126 views ### Image of a sum of positive operators contains the images of each individual operator? In the proof of Proposition 2.52 of John Watrous' QI book, there is the statement that$\text{im}(\eta(a))\subset\text{im}(\rho)$, where$\rho=\sum_{i=1}^{N}\eta(i)$is a sum of positive operators and ... 1answer 351 views ### Advances in Quantum Channel Capacity I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general. Quantum capacity is the highest rate at which quantum information can be ... 1answer 218 views ### Tensor product properties used to obtain Kraus operator decomposition of a channel I work on a Quantum Information Science II: Quantum states, noise and error correction MOOC by Prof. Aram Harrow, and I do not understand which property of tensor products is used in one of the ... 1answer 63 views ### How to find the fidelity between two state when one is an operator? I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states Show that the fidelity between the ... 1answer 311 views ### Quantum Channel Models The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state$\rho$is$\rho\...
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 ...