一个Mathematica包，用于执行涉及狄拉克符号中的.zip

# BernDirac [](https://www.wolfram.com/language/) [](https://www.gnu.org/licenses/gpl-3.0) [](https://github.com/bernie-wu/BernDirac)  A [Wolfram Mathematica](https://www.wolfram.com/mathematica/) package for performing calculations involving matrices/vectors in the [Dirac notation](https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation) which is usually used in [quantum mechanics](https://en.wikipedia.org/wiki/Quantum_mechanics)/[quantum computing](https://en.wikipedia.org/wiki/Quantum_computing). It utilises the *built-in functions without predefined meanings*, namely `Ket[]`, `Bra[]`, and `CircleTimes[]`, along with their respective alias, <code>| ⟩ ↔ <kbd>esc</kbd>ket<kbd>esc</kbd></code>, <code>⟨ | ↔ <kbd>esc</kbd>bra<kbd>esc</kbd></code> and <code>⊗ ↔ <kbd>esc</kbd>c*<kbd>esc</kbd></code>. The basis which this package works in is {|0⟩,|1⟩}, which is also known as the *computational basis* or the *Z basis*. The package was written in Wolfram Mathematica version [12.2](https://www.wolfram.com/mathematica/quick-revision-history.html) in Windows 10 and it has zero dependencies. ## Contact Bernard Wo - bernardwu+BernDirac@outlook.my Project Link: [https://github.com/bernie-wu/BernDirac](https://github.com/bernie-wu/BernDirac) # How to use? Download [BernDirac.wl](https://github.com/bernie-wu/BernDirac/blob/main/BernDirac.wl) and place it wherever you like. Then, in your Mathematica notebook, run the following line to load the package into your current Mathematica session: ```wolframlanguage Get[<path-to-BernDirac.wl>]; ``` # Functions that this package provides After loading [BernDirac.wl](https://github.com/bernie-wu/BernDirac/blob/main/BernDirac.wl) into your Mathematica notebook session, the following additional functions become available to use: * [`Ket[]`](#Ket "Go-to Ket[]") * [`Bra[]`](#Bra "Go-to Bra[]") * [`CircleTimes[]`](#CircleTimes "Go-to CircleTimes[]") * [`DiracForm[]`](#DiracForm "Go-to DiracForm[]") * [`PartialTr[]`](#PartialTr "Go-to PartialTr[]") The special quantities, namely the [**Bell states**](https://en.wikipedia.org/wiki/Bell_state) also become available to use via [`Ket[]`](#Ket "Go-to Ket[]") and [`Bra[]`](#Bra "Go-to Bra[]") (see section [Bell states](#Bell-states "Go-to Bell states")). ## Ket[] `Ket[]` is used to denote a *column vector*. The alias `| ⟩` for `Ket[]` can be obtained with <code><kbd>esc</kbd>ket<kbd>esc</kbd></code>. The allowed input for `Ket[]` is either `0` or `1` and the output for each case is as shown here: >> ![In:Ket[0]](Image/Ket/ket0_in.svg "Ket[0]") >>  > >> ![In:Ket[1]](Image/Ket/ket1_in.svg "Ket[1]") >>  `Ket[]` also supports multiple inputs, as long as they are `0` and `1`. >> ![In:Ket[1,1,0]](Image/Ket/ket110_in.svg "Ket[1,1,0]") >> ![Out:Ket[1,1,0]](Image/Ket/ket110_out.svg "{{0},{0},{0},{0},{0},{0},{1},{0}}") Note that `Ket[1,1,0]` is equivalent to `Ket[1]⊗Ket[1]⊗Ket[0]` (see [`CircleTimes[]`](#CircleTimes "Go-to CircleTimes[]")). ## Bra[] `Bra[]` is used to denote a *row vector* (i.e. `| ⟩=(⟨ |)†` where `†` denotes conjugate transpose). The alias `⟨ |` for `Bra[]` can be obtained with <code><kbd>esc</kbd>bra<kbd>esc</kbd></code>. The allowed input for `Bra[]` is either `0` or `1` and the output for each case is as shown here: >> ![In:Bra[0]](Image/Bra/bra0_in.svg "Bra[0]") >>  > >> ![In:Bra[1]](Image/Bra/bra1_in.svg "Bra[1]") >>  Just like `Ket[]`, `Bra[]` also supports multiple inputs, as long as they are `0` and `1`. >> ![In:Bra[1,1,0]](Image/Bra/bra110_in.svg "Bra[1,1,0]") >> ![Out:Bra[1,1,0]](Image/Bra/bra110_out.svg "{{0,0,0,0,0,0,1,0}}") Note that `Bra[1,1,0]` is equivalent to `Bra[1]⊗Bra[1]⊗Bra[0]` (see [`CircleTimes[]`](#CircleTimes "Go-to CircleTimes[]")). ## CircleTimes[] The alias `⊗` for `CircleTimes[]`, is used to denote the [*Kronecker product*](https://en.wikipedia.org/wiki/Kronecker_product) (sometimes also called [*Tensor product*](https://en.wikipedia.org/wiki/Tensor_product)). Use <code><kbd>esc</kbd>c*<kbd>esc</kbd></code> to obtain the alias. Below, we show that `⊗` works for multiple column vectors, row vectors, and square matrices. **Column vector** >> ![In:Ket[1]⊗Ket[1]⊗Ket[0]](Image/Ket/ket110_tensor_in.svg "Ket[1]⊗Ket[1]⊗Ket[0]") >> ![Out:Ket[1]⊗Ket[1]⊗Ket[0]](Image/Ket/ket110_out.svg "{{0},{0},{0},{0},{0},{0},{1},{0}}") **Row vector** >> ![In:Bra[1]⊗Bra[1]⊗Bra[0]](Image/Bra/bra110_tensor_in.svg "Bra[1]⊗Bra[1]⊗Bra[0]") >> ![Out:Bra[1,1,0]](Image/Bra/bra110_out.svg "{{0,0,0,0,0,0,1,0}}") **Square matrix** >> ![In:Ket[0].Bra[0]⊗Ket[1].Bra[1]](Image/BraKet/braket00_11_tensor_in.svg "Ket[0].Bra[0]⊗Ket[1].Bra[1]") >> ![Out:Ket[0].Bra[0]⊗Ket[1].Bra[1]](Image/BraKet/braket00_11_tensor_out.svg "{{{0, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}") ## DiracForm[] `DiracForm[]` prints the vector or matrix using the Dirac notation. It works for column vectors, row vectors, and square matrices. **Column vector** >> ![In:αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]](Image/Ket/αket00_βket11_tensor_dirac_in.svg "αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]") >> ![Out:αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]](Image/Ket/αket00_βket11_tensor_dirac_out.svg "α|0,0⟩+β|1,1⟩") **Row vector** >> ![In:αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1]](Image/Bra/αbra00_βbra11_tensor_dirac_in.svg "αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1]") >> ![Out:αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1]](Image/Bra/αbra00_βbra11_tensor_dirac_out.svg "α⟨0,0|+β⟨1,1|") **Square matrix** >> ![In:(αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]).(αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1])](Image/BraKet/αket00_βket11_αbra00_βbra11_tensor_dirac_in.svg "(αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]).(αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1])") >> ![Out:(αKet[0]⊗Ket[0]+βKet[1]⊗Ket[1]).(αBra[0]⊗Bra[0]+βBra[1]⊗Bra[1])](Image/BraKet/αket00_βket11_αbra00_βbra11_tensor_dirac_out.svg "α²|0,0⟩.⟨0,0|+αβ|0,0⟩.⟨1,1|+αβ|1,1⟩.⟨0,0|+β²|1,1⟩.⟨1,1|") > >>  >>  ## Bell states By using the special letter capital dotted `Φ` and subscripts, we can access the four Bell states using [`Ket[]`](#Ket "Go-to Ket[]") and [`Bra[]`](#Ket "Go-to Ket[]"). Note that the special dotted `Φ` is known as [FormalCapitalPhi](https://reference.wolfram.com/language/ref/character/FormalCapitalPhi.html) in the documentation (for more information, see the formal letters section [here](https://reference.wolfram.com/language/tutorial/MathematicalAndOtherNotation.html)). The dotted `Φ` alias can be accessed with <code><kbd>esc</kbd>.CapitalPhi<kbd>esc</kbd></code>, and inputting subscript(s) can be achieved with <code><kbd>ctrl</kbd>+<kbd>_</kbd></code>. >> ![In:Ket[FormalCapitalPhi_00]](Image/Bell_states/phi00_in.svg "Ket[FormalCapitalPhi_00]") >> ![Out:Ket[FormalCapitalPhi_00]](Image/Bell_states/phi00_out.svg "|0,0⟩/√2+|1,1⟩/√2") > >> ![In:Ket[FormalCapitalPhi_01]](Image/Bell_states/phi01_in.svg "Ket[FormalCapitalPhi_01]") >> ![Out:Ket[FormalCapitalPhi_01]](Image/Bell_states/phi01_out.svg "|0,1⟩/√2+|1,0⟩/√2") > >> ![In:Ket[FormalCapitalPhi_10]](Image/Bell_states/phi10_in.svg "Ket[FormalCapitalPhi_10]") >> ![Out:Ket[FormalCapitalPhi_10]](Image/Bell_states