What Does the "\oplus" Symbol Represent?

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In summary, O Plus symbol is used to denote the "direct sum" or "binary exclusive or" of two vector spaces.
  • #1
DukeLuke
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"O Plus" symbol

I keep running across this symbol in a introduction to quantum computing text and I can't figure out what it stands for. If anyone could help me out I would appreciate it. The symbol is [tex] \oplus [/tex].
 
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  • #2


An example of usage would help.

I've seen that symbol used primarily in two ways.

The first way is the "direct sum" of two vector spaces, or similar objects. If you have a vector space V and a vector space W, then [itex]V \oplus W[/itex] is the vector space of ordered pairs whose first element is in V and whose second element is in W. With chosen bases, you have [itex]\mathbf{C}^m \oplus \mathbf{C}^n \cong \mathbf{C}^{m+n}[/itex].


The other way is binary exclusive or a.k.a. addition modulo two. The commutative, associative operation whose values are given by the table
[tex]
\begin{array}{c|cc}
& 0 & 1 \\
\hline
0 & 0 & 1 \\
1 & 1 & 0
\end{array}
[/tex]
 
  • #3


Here's two examples of usage

1.
(Note irreversible NAND gate is [itex] (a, b) \rightarrow ¬(a \Lambda b) [/itex]
We can actually construct a reversible version
of the NAND gate that preserves all the information about the input: For
example, the (Toffoli) gate

[tex] (a, b, c) \rightarrow (a, b, c \oplus a \Lambda b) [/tex]

is a reversible 3-bit gate that flips the third bit if the first two both take
the value 1 and does nothing otherwise.

2.
Now suppose we have a quantum black box that computes f(x). Of course
f(x) might not be invertible, while the action of our quantum computer is
unitary and must be invertible, so we’ll need a transformation Uf that takes
two qubits to two:

[tex]U_f : \mid x \rangle \mid y \rangle \rightarrow
\mid x \rangle \mid y \oplus f(x)\rangle [/tex]

(This machine flips the second qubit if f acting on the first qubit is 1, and
doesn’t do anything if f acting on the first qubit is 0.) We can determine if
f(x) is constant or balanced by using the quantum black box twice.
 

Related to What Does the "\oplus" Symbol Represent?

1. What does the "\oplus" symbol represent in mathematics?

The "\oplus" symbol is a mathematical symbol that represents the exclusive or operation. This means that it is used to represent a logical operator that is true only when exactly one of the two conditions it connects is true.

2. How is the "\oplus" symbol used in logic and Boolean algebra?

In logic and Boolean algebra, the "\oplus" symbol is used to represent the exclusive or operation between two logical expressions. It is often used in conjunction with other logical operators, such as AND and OR, to form more complex expressions.

3. Is the "\oplus" symbol the same as the plus sign?

No, the "\oplus" symbol is not the same as the plus sign. The plus sign is a basic mathematical symbol used to indicate addition, while the "\oplus" symbol is a logical operator used in Boolean algebra to represent the exclusive or operation.

4. Can the "\oplus" symbol be used in other fields of science?

Yes, the "\oplus" symbol can be used in other fields of science, such as computer science and electrical engineering. In computer science, it is used to represent the bitwise exclusive or operation, while in electrical engineering, it is used to represent the XOR gate in digital circuits.

5. Are there other symbols that have a similar meaning to the "\oplus" symbol?

Yes, there are other symbols that have a similar meaning to the "\oplus" symbol, such as the "⊕" symbol and the "⊻" symbol. These symbols are often used interchangeably to represent the exclusive or operation in different contexts.

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