1"use strict"; 2!function(){try{var e="undefined"!=typeof window?window:"undefined"!=typeof global?global:"undefined"!=typeof self?self:{},n=(new Error).stack;n&&(e._sentryDebugIds=e._sentryDebugIds||{},e._sentryDebugIds[n]="f38a3b78-0c88-5a7f-a029-3d6f11216072")}catch(e){}}(); 3(self.webpackChunkpennylane_website=self.webpackChunkpennylane_website||[]).push([[6099],{32386:function(e,t,n){n.r(t),n.d(t,{Head:function(){return x},default:function(){return E}});var l=n(2784),a=n(45392);function r(e){const t=Object.assign({p:"p",hr:"hr",strong:"strong"},(0,a.ah)(),e.components);return l.createElement(l.Fragment,null,l.createElement(t.p,null,"Dirac notation is just a special kind of notation that is used in quantum mechanics for vectors."),"\n",l.createElement(t.hr),"\n",l.createElement(t.p,null,"A vector in Dirac notation is written with a vertical line on the left and a large angle bracket on the right, as below."),"\n",l.createElement(t.p,null,"$$\n|\\psi\\rangle\n$$"),"\n",l.createElement(t.p,null,"The components of this vector are complex numbers. It could be something like $|\\psi\\rangle = 0.8 |0\\rangle + 0.6 |1\\rangle$, where $|0\\rangle$ and $|1\\rangle$ are our basis."),"\n",l.createElement(t.p,null,"The ",l.createElement(t.strong,null,"conjugate transpose")," of this vector is written in the âreverseâ way, with a vertical line on the right and a large angle bracket on the left, as below."),"\n",l.createElement(t.p,null,"$$\n\\langle\\psi|\n$$"),"\n",l.createElement(t.p,null,"If vectors notated as $|\\psi\\rangle$ are column vectors, vectors notated as $\\langle\\psi|$ are row vectors. We can multiply a row vector by a column vector according to the usual rules of matrix multiplication, resulting in the ",l.createElement(t.strong,null,"inner product")," of the two vectors, as shown below."),"\n",l.createElement(t.p,null,"$$\n\\langle\\phi||\\psi\\rangle\n$$"),"\n",l.createElement(t.p,null,"Very often, the two vertical lines in the middle are simplified to just one vertical line."),"\n",l.createElement(t.p,null,"$$\n\\langle\\phi|\\psi\\rangle\n$$"),"\n",l.createElement(t.p,null,"The row vector on the left is called a ",l.createElement(t.strong,null,"bra"),", and the column vector on the right is called a ",l.createElement(t.strong,null,"ket"),", and, written together like this, they make a ",l.createElement(t.strong,null,"bra-ket"),"."),"\n",l.createElement(t.p,null,"If we want to apply a matrix $A$ to one of these kets, we just write"),"\n",l.createElement(t.p,null,"$$\nA|\\psi\\rangle\n$$"),"\n",l.createElement(t.p,null,"Itâs common in quantum mechanics to see the inner product of a bra with the result of a matrix multiplied by a ket, as below."),"\n",l.createElement(t.p,null,"$$\n\\langle\\phi|A|\\psi\\rangle\n$$"),"\n",l.createElement(t.p,null,"Here, we are just taking a vector, $\\psi$, applying a matrix to it to get a new vector, and then finding the inner product of that new vector and the vector $\\phi$."))}var i=function(e){void 0===e&&(e={});const{wrapper:t}=Object.assign({},(0,a.ah)(),e.components);return t?l.createElement(t,e,l.createElement(r,e)):r(e)},o=n(56694),s=n(82256),c=n(28546),u=n(2409),m=n(76741),d=n(9619),p=n(86165),h=n(95692),g=n(19948),v=n(45051),$=n(52322);const f={Notes:p.Z,Footnotes:u.Z,MarkdownText:d.Z,PermaLink:g.Z,a:m.Z,p:h.Z};function w(e){let{data:t,children:n}=e;const{mdx:l}=t,r=null==l?void 0:l.frontmatter,i=null==l?void 0:l.tableOfContents;return(0,$.jsxs)($.Fragment,{children:[(0,$.jsx)(o.Z,{pageTitle:(null==r?void 0:r.title)||""}),(0,$.jsx)(c.Z,{mainSection:(0,$.jsxs)("section",{className:"glossary-item-container w-[830px]",children:[(null==r?void 0:r.title)&&(0,$.jsxs)("h1",{children:[r.title," ",(0,$.jsx)(g.Z,{title:r.title})]}),(0,$.jsx)(a.Zo,{components:f,children:n})]}),sidebarSection:null!=i&&i.items&&i.items instanceof Array&&i.items.length>0?(0,$.jsx)(s.Z,{linksList:null==i?void 0:i.items,contentTitle:"Glossary"}):void 0})]})}function E(e){return l.createElement(w,e,l.createElement(i,e))}const x=e=>{var t,n,l,a,r,i,o,s,c,u;let{data:m}=e;return(0,$.jsx)(v.Z,{title:null!==(t=m.mdx)&&void 0!==t&&null!==(n=t.frontmatter)&&void 0!==n&&n.meta_title?m.mdx.frontmatter.meta_title:(null===(l=m.mdx)||void 0===l||null===(a=l.frontmatter)||void 0===a?void 0:a.title)||"",image:(null===(r=m.mdx)||void 0===r||null===(i=r.frontmatter)||void 0===i?void 0:i.meta_image)||"",description:(null===(o=m.mdx)||void 0===o||null===(s=o.frontmatter)||void 0===s?void 0:s.meta_description)||"",pathname:"/glossary/"+(null===(c=m.mdx)||void 0===c||null===(u=c.frontmatter)||void 0===u?void 0:u.slug)})}}}]); 4//# sourceMappingURL=component---src-templates-glossary-glossary-tsx-content-file-path-home-runner-work-pennylane-ai-react-pennylane-ai-react-apps-pennylane-website-content-glossary-what-is-dirac-notation-index-mdx-dceaf507d9cc7f9bfd33.js.map 5//# debugId=f38a3b78-0c88-5a7f-a029-3d6f11216072
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