# Tag Info

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### What is the difference between a qudit system with d=4 and a two-qubit system?

For qubits, we usually base all of our operators on the Pauli matrices. Our basic gate set consists of the Pauli matrices themselves, Clifford gates like $H$ and $S$ that map between Pauli matrices, ...
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### How is quantum teleportation generalized to qudits?

Let's define the shift and clock matrices (the generalisation of the Pauli X and Z matrices) as  X=\sum_{i=0}^{d-1}|i+1\text{ mod }d\rangle\langle i|\qquad Z=\sum_{i=0}^{d-1}\omega^i|i\rangle\langle ...

### How does the NOT gate generalize beyond binary?

There is no universally accepted CNOT (or even NOT) gate for qutrits or qudits with d>2, and the NOT operator conventionally turns 0 to 1 and 1 to 0 when the input is 1 binary bit or qubit. Three-...
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### How to show that an n-level system is entangled?

Determining whether a given state is entangled or not is NP hard. So if you include all possible types on entanglement, including mixed states and multipartite entanglement, there is never going to be ...

### Do any specific types of qudits other than qubits and qutrits have a name?

There is no standard name for a qudit for $d>3$. The community has mostly settled on the term qudit (but you will still find qunit or quNit, for example, using $n$ or $N$ instead of $d$ in some ...
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### Are qutrits more robust to decoherence?

The statement in Wikipedia is very generic, and only cites this paper as a reference. Quoting from the abstract of the paper: We demonstrate that decoherence of many-spin systems can drastically ...
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### Can a single qutrit in superposition be considered entangled?

To talk about entanglement, you have to first identify subsystems. In your $d=4$ example, you defined an isomorphism $\mathbb{C}^4\simeq \mathbb{C}^2\otimes\mathbb{C}^2$ via the identification of ...

### How does the NOT gate generalize beyond binary?

As the previous answer mentions, how a controlled qudit gate is defined is up to a choice of convention. This paper contains a few examples of intuitively appealing definitions for controlled qudit ...

### How does the NOT gate generalize beyond binary?

Many classical programming languages are equipped with a construct known as the conditional statement if (condition) { u(); } where ...

### What is the difference between a qudit system with d=4 and a two-qubit system?

A fundamental difference between the two kinds of systems is that a two-qubit system can actually be in an entangled state. On the other hand, a single d=4 dimensional system does not possess ...

### How to show that an n-level system is entangled?

As suggested in your Wiki link, the way to detect an entangled state is to find a hyperplane that separates it from the convex set of separable states. This hyperplane represents what is called an ...
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### Computing with Logical Qunits

A logical qubit is made out of many physical qubits (or qudits), simply selecting a particular two-dimensional subspace. So you can’t make it “exclusively” out of logical qubits because they sit on ...
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### $n$ qubit vs. a $d=2^n$ qudit states and measurements

Without additional assumptions or context, there is no fundamental difference between an "$2^n$-dimensional qudit" and "$n$ qubits". Any "qudit system" over $2^n$ modes for some integer $n$ can be ...
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### Uniformly distributed state in the Weyl basis

If $|\langle\psi| U_{ab}|\psi\rangle|$ is constant then $\{\frac{1}{d} U_{ab}|\psi \rangle \langle\psi| U_{ab}^\dagger\}$ is a SIC-POVM of Weyl-Heisenberg type. It's conjectured that such a structure ...
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### Hilbert space to accurately represent 3x3 Rubik's Cube

This question does not need to be phrased as a quantum question. One can equally ask what classical register can be used to store a string that uniquely identifies each different configuration of the ...

### Do any specific types of qudits other than qubits and qutrits have a name?

After a lot of searching, it appears that the word "quqrit" has indeed been used in one (but I found only one!) paper from 2011, and indeed it was used to describe a 4-level system. But the word "...

### What is the most economical and preferred basis for the qudit?

You may be confusing two uses of the word "base". One definition of "base" has to do with how many digits are used to represent a number. For example, base two uses the digits 0 and 1, and the number ...
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### Combining Different Qunits

Quantum walks are a simple case of quantum dynamics that involves a qubit (named coin in this context) interacting with a high-dimensional qudit (named walker in this context). Almost anything in ...

### Is the column vector of a uniformly sampled random unitary matrix a uniformly sampled random state vector?

Yes. A uniformly (Haar random) sampled state vector $|\psi\rangle$ is characterized by the fact that the probability measure is invariant under any $U$, i.e., colloquially, $U|\psi\rangle$ is just as ...

### Is the column vector of a uniformly sampled random unitary matrix a uniformly sampled random state vector?

Suppose that was not the case. Then taking the first column of a uniformly random unitary matrix gives you a nonuniformly random state. That means that there is some state, call it $|\psi\rangle$, ...
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### How to generalize the relationship HXH = Z for higher dimensions

The appropriate $d$-dimensional analogue of $H$ turns out to be the Quantum Fourier Transform. This is obscured by the fact that even though $(1)$ is conjugation the inverse is written implicitly ...
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### What is the most economical and preferred basis for the qudit?

The preferred basis problem is essentially something from the many worlds interpretation: If we are to interpret a superposition as representing many universes, what basis should we choose? Since this ...
Is quantum computing limited to a superposition of only two states? In theory, it is not. Keep in mind that a qubit is a quantum analogue of the classical "bit" which has only two states $0$ and $1$...