Questions tagged [quantum-operation]

For questions about completely positive (CP) linear maps between quantum states. Can also be used for trace-preserving CP maps (quantum channels). For questions about unitary operations, please use quantum-gate instead.

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48 views

How does the extremality of a POVM reflect on its Naimark dilation isometry?

Let $\mu:\Sigma\to\mathrm{Pos}(\mathcal X)$ be some POVM, with $\Sigma$ the finite set of possible outcomes, and $\mathrm{Pos}(\mathcal X)$ the set of positive semidefinite operators on a finite-...
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1answer
36 views

How do successive operators act in the Heisenberg picture?

In the Schrodinger picture, it is clear how write a single gate for two operators. For example if operators $A$ then $B$ act on a state $\vert \psi \rangle$, this gives $BA\vert \psi \rangle$, (noting ...
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1answer
22 views

What is the correct notation to denote operations conditional on a measurement outcome?

What is the correct mathematical notation to describe the following setup? I have classical state in register $A$ which I can think of as $\sum_i p_i \vert i\rangle\langle i \vert_A$. I measure this ...
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3answers
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How to describe the evolution of a density matrix using the Choi matrix?

How do I apply the Choi matrix on a Density matrix. Say my process is a Hadamard gate, and my input state is the ground state on 1 qubit (qubit id 0). $U = H = \dfrac{1}{\sqrt{2}} \begin{bmatrix}1&...
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1answer
31 views

Equivalent statement of the unitary freedom of Kraus operator?

There is a well-known form of the unitary freedom of Kraus operators, which can be found in Nielsen's book, stating that two sets of Kraus operators describe the same physical process of the system(...
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294 views

How are eavesdroppers detected when using BB84 in the presence of noise?

I would like to expand upon this question: What is the probability of detecting Eve's tampering, in BB84? Let's say that when the receiver (colloquially referred to as Bob) receives a qubit and ...
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What does Z-local mean for operator?

I am new to quantum and was reading a paper and found the following sentence is not quite clear to me. I wonder if I can see some clear examples to help me understand the meaning of "Z-local&...
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1answer
47 views

Is the trace norm monotone with respect to quantum operations?

The trace norm is defined to be $$\| K \| = \mathrm{tr}\sqrt{K^\dagger K}.$$ Is it true that we have $$\| \mathcal E(K) \|\leq \|K\|,$$ for any quantum operation $\mathcal{E}: A\otimes B \to A\otimes ...
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Determining whether there exists an equivalent set of unitary Kraus operators

I have a CPTP quantum channel $\mathcal{E}$ that I've characterized by an operator sum representation $\{E_i\}$ for $i=1, \dots, m$ which acts on an input state like $$ \mathcal{E}(\rho) = \sum_{i=1}^...
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78 views

Do sequences of operations (including measurements) applied to different halves of an entangled pair always commute?

Let us say $A$ has one half of an entangled qubit pair, and $B$ has the other half. $A$ may be able to perform any type of operation on their half of the pair, such as unitary operations, entangling ...
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2answers
141 views

Can Kraus operators change a mixed state into a pure state?

It seems that Kraus operators cannot change a pure state into a mixed one (wrong). For any pure state can be written as $|\psi\rangle\langle\psi|$, so after the Kraus operators. It becomes $$\sum_l\...
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62 views

Is the map $\rho\rightarrow Tr(\sigma\rho)$ completely positive?

Let $\sigma$ be a fixed positive semidefinite matrix (edit: need unit trace too as pointed out if we want trace nonincreasing). Is the map $$N:H\rightarrow\mathbb{C}$$ given by $N(\rho) = Tr(\sigma\...
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80 views

How to use the Kraus operators to represent the total density matrix instead of the reduced one?

In Nielsen's book, the Kraus operator can be attained by trace out the enviroment: $$\operatorname{Tr}_{\rm env}[\hat{U}(|\psi\rangle\otimes|0\rangle)(\langle\psi|\otimes\langle 0|)\hat{U}^\dagger]. $$...
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1answer
106 views

Permutation covariant channels and their Stinespring dilations

I am interested in a quantum channel from $A^{\otimes n}$ to $B^{\otimes n}$ denoted as $N_{A^{\otimes n} \rightarrow B^{\otimes n}}(\cdot)$. Let $\pi(\cdot)$ be a permutation operation among the $n$ ...
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2answers
81 views

Prove that a Bell state is invariant under the single-qubit gate acting on both qubits

I have a Bell state ${\Psi}^{-}= \frac{1}{\sqrt2} (|01\rangle - |10\rangle).$ How can I prove that this state is invariant (up to a global phase), when doing the same unitary $U$ on each qubit? That ...
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1answer
61 views

For a bipartite operator $M\in L(H_{AB})$, suppose $0\leq M\leq \mathbb{I}$. Prove $M^{AB}\leq M^A\otimes \mathbb{I}$

As stated in the title, let $M$ be a linear operator on a finite bipartite Hilbert space. Suppose $0\leq M^{AB}\leq \mathbb{I}$ and $0\leq M^A,M^B\leq\mathbb{I}$, where $M^A=\mathrm{Tr}_B\left(M^{AB}\...
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Does the von Neumann entropy equal the smallest accessible Shannon entropy?

I've been reading about the von Neumann entropy of a state, as defined via $S(\rho)=-\operatorname{tr}(\rho\ln \rho)$. This equals the Shannon entropy of the probability distribution corresponding to ...
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1answer
51 views

Umambiguous discrimination using POVM with highest discriminate probability

I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it. Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
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1answer
183 views

What are examples of extremal non-projective POVMs?

Fix some finite-dimensional space $\mathcal X$. Define a POVM as a collection of positive operators summing to the identity: $\mu\equiv \{\mu(a):a\in\Sigma\}\subset{\rm Pos}(\mathcal X)$ such that $\...
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2answers
85 views

In quantum process tomography, how does $\chi$ characterize a quantum process?

I'm working through Nielsen and Chuang and I'm pretty confused by the discussion of quantum process tomography. I'm trying to work through an example of 1-qubit state tomography given by N&C (box ...
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4answers
138 views

Can a qubit be entangled with an arbitrary quantum state, without altering it?

For example, if an adversary were to get hold of one half of an entangled 2-qubit quantum state, $|\psi \rangle$, travelling along a channel, would they be able to entangle one of their own qubits ...
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What is the best quantum process tomography method?

This question is somewhat related to this question. What is currently the best method for quantum process tomography? By best I mean, the one that can achieve the best accuracy of estimation per qubit ...
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What is the best method for estimating average channel fidelity?

This thesis shows an efficient way to estimate average channel fidelity (in chapter 4). However, it is somewhat old (from 2005). Are there any better methods out there? By better I mean: are there ...
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1answer
197 views

Quantum capacity for serial composition of quantum channels

Recently, I have been working with quantum channel capacity for quantum-quantum channels and I was wondering if there exist some results for channel compositions. Specifically, I have been looking for ...
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39 views

Kraus representation of a convex combination of CPT maps

Let $\Phi_1,\Phi_2$ be CPT maps with Kraus decomposition \begin{equation} \Phi_1=\sum_{k=1}^{d_1}M_k\rho M_k^\dagger, \quad \Phi_2=\sum_{k=1}^{d_2} N_k\rho N_k^\dagger, \quad \text{s.t.} \quad \sum_{k=...
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Can the CCNR entanglement criterion be seen as a “natural” statement about entanglement breaking channels?

(The CCNR criterion) The computable cross-norm or realignment (CCNR) entanglement criterion, as discussed in (Gühne and Toth 2008), is based on the observation that any bipartite state $\rho$ can be ...
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1answer
51 views

Can Gate Set Tomography work on Quantum Channels?

I stumbled across a new paper on gate set tomography. Can gate set tomography be applied to a quantum channel or multiple quantum channels? Will the same advantages still apply of not having to 'rely ...
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1answer
49 views

Understanding the quantum circuit for the quantum adder Toffoli gate

I am trying to understand the toffoli operation for the quantum adder below: (especially for the second toffoli gate) but I am stuck in understanding the calculation to get the correct outputs. The ...
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Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

Let us consider two quantum channels $\Phi:M_d\rightarrow M_{d_1}$ and $\Phi_c:M_d\rightarrow M_{d_2}$ that are complementary to each other, i.e., there exists an isometry $V:\mathbb{C}^d\rightarrow \...
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Computing $e^x$ on a quantum computer

Does anyone know how to make a quantum circuit to compute exponentials where the exponent can be a fraction? To be more precise, I'm looking for a fixed point quantum arithmetic circuit that does the ...
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1answer
124 views

Can quantum error correction work on any type of channel?

It says on wikipedia that quantum error correction can (at best) correct phase flips and bit flips. A popular form of representing a quantum channel is in its Kraus representation (scroll down to ...
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1answer
34 views

Ranges of quantum states that are related via a quantum channel

Let $\rho\in M_n$ and $\sigma\in M_m$ be two quantum states. We denote the orthogonal projections onto $\text{range}(\rho)$ and $\text{range}(\sigma)$ by $P_\rho$ and $P_\sigma$, respectively. Now, if ...
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1answer
44 views

Operation conditioned on measurement result

I have a 2 qubit circuit where I wish to measure the first qubit and the measurement outcome determines what operation to implement on qubit 2. The whole process can be simulated using the following ...
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3answers
146 views

Why are entanglement breaking channels, defined as $\Phi(\rho)=\sum_a \operatorname{Tr}(\mu(a)\rho)\sigma_a$, entanglement breaking?

Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form $$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$ for some POVM $\{\mu(a)\}_a$ and states $\...
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2answers
79 views

Can you perform quantum process tomography using an orthonormal basis the contains non Hermitian matrices?

In the thesis "Efficient Simulation of Random Quantum States and Operators" on page 25 there is a portion of text explaining a method for quantum process tomography. It claims that states ...
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How do I derive Stinespring and Kraus representations of a map such that $\Lambda(\rho)=|0\rangle\langle0|$ for all $\rho$?

Can't find any info on Stinespring dilation so I thought I could post here. If I have a qubit complete positive map $\Lambda$, that maps all inputs to the output $|0\rangle$, $\Lambda(\rho)=|0\rangle\...
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1answer
87 views

What is the most general quantum operation that preserves the marginal?

Suppose I have two states $\rho_{AB}$ and $\sigma_{AB}$ such that the marginals $\rho_A = \sigma_A$. What is the most general operation that could have acted on $\rho$ to output $\sigma$? For example, ...
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3answers
109 views

What is the Kraus representation of the quantum channel with Choi $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$?

This matrix $$c_{\lambda} = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$ is the Choi–Jamiołkowski matrix of a quantum channel for any $\lambda \in [0,1]$. The ...
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1answer
46 views

Show that $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$ is the Choi–Jamiołkowski matrix of a quantum channel

I'm curious how to show how this matrix: $$c = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$ is the Choi–Jamiołkowski matrix of a quantum channel for any $\...
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What are examples of zero capacity quantum channels with Choi rank less than $d$?

All the currently known examples of quantum channels with zero quantum capacity are either PPT or anti-degradable. These notions can be conveniently defined in terms of the Choi matrix of the given ...
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1answer
68 views

Minimal output dimension of a quantum channel

Consider quantum channels $\Phi : M_n \rightarrow M_{d_1}$ and $\Psi : M_n \rightarrow M_{d_2}$ with $d_1\leq d_2$. We say that $\Phi$ is isometrically extended by $\Psi$ (denoted $\Phi \leq_{\text{...
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Does a quantum channel map the maximally mixed input state into an output state with maximal rank?

Consider a quantum channel $\Phi : M_n \rightarrow M_m$ and let $\frac{\mathbb{I}_n}{n}$ be the maximally mixed input state. For all input states $\rho\in M_n$, is it true that $$\quad \text{rank} \, \...
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119 views

In Stinespring dilation, can we always use a mixed state as the ancilla?

The Stinespring dilation theorem states that any CPTP map $\Lambda$ on a system with Hilbert space $\mathcal{H}$ can be represented as $$\Lambda[\rho]=tr_\mathcal{A}(U^\dagger (\rho\otimes |\phi\...
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1answer
40 views

Dimensionality and value of $\mathbb{I}_A$ in Quantum operations

I was checking this question answer but I still can't get what is the value and dimension of $\mathbb{I}_A$ in this question and on the answer. Is it an identity matrix or some vector? It also appears ...
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1answer
60 views

Why does $\sum_n \langle n|M_m\rho M_m^\dagger|n\rangle$ simplify to $\langle \psi|M_m^\dagger M_m|\psi\rangle$?

I was trying to derive the formula for $p(m)$ in exercise 8.2 on page 357 in Nielsen & Chuang. But I am wondering what rule I can apply to simplify this $$\mathrm{tr}(\mathcal{E}_m(\rho) )= \...
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1answer
79 views

Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358) $$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
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Is there some notion of work associated with performing a measurement?

Let a measurement be described by POVM elements $M_i$ such that probability $p(i) = Tr[\rho M_i]$ for some state $\rho$. I want to know whether there is some notion of work associated with such ...
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1answer
127 views

How to calculate the average fidelity of an amplitude damping channel

An answer to this question shows how to calculate the average fidelity of a depolarizing channel. How would one go about calculating this for an amplitude dampening channel? I tried working out the ...
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1answer
64 views

What additional conditions are there to make POVM measurement same as projective measurement?

We know that POVMs are applied in the more general cases where the system is not necessarily closed. So mathematically, how does going from open to closed system change the scenario in case of POVM so ...
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1answer
52 views

What is the general form of a classical-quantum state?

In the literature, one comes across the following situation: Alice holds two registers $X$ and $A$ and it is given that $X$ is a classical register. What is the most general way to write down Alice's ...