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649 | 649 | "\n", |
650 | 650 | "If we have a pure state:\n", |
651 | 651 | "\n", |
652 | | - "$$|\\psi\\rangle = \\alpha |0 \\rangle + \\beta |1\\rangle$$\n", |
| 652 | + "$$\\left| \\psi\\right\\rangle = \\alpha \\left| 0 \\right\\rangle + \\beta \\left| 1 \\right\\rangle$$\n", |
653 | 653 | "\n", |
654 | 654 | "The corresponding density matrix is:\n", |
655 | 655 | "\n", |
656 | | - "$$\\rho = |\\psi \\rangle \\langle \\psi| = (\\alpha |0 \\rangle + \\beta |1\\rangle)(\\alpha^* \\langle 0 | + \\beta^* \\langle 1|) = \\\\= |\\alpha|^2|0 \\rangle \\langle 0| + \\alpha \\beta^*|0 \\rangle \\langle 1| + \\alpha^* \\beta|1 \\rangle \\langle 0|+ |\\beta|^2 |1 \\rangle \\langle 1| = \n", |
| 656 | + "$$\\rho = \\left| \\psi \\right\\rangle \\langle \\psi| = (\\alpha \\left| 0 \\right\\rangle + \\beta \\left| 1 \\right\\rangle)(\\alpha^* \\left\\langle 0 \\right| + \\beta^* \\left\\langle 1\\right|) = \\\\= |\\alpha|^2\\left| 0 \\right\\rangle \\langle 0| + \\alpha \\beta^*\\left| 0 \\right\\rangle \\left\\langle 1\\right| + \\alpha^* \\beta|1 \\rangle \\langle 0|+ |\\beta|^2 |1 \\rangle \\left\\langle 1\\right| = \n", |
657 | 657 | "\\begin{pmatrix} \n", |
658 | 658 | "|\\alpha|^2 & \\alpha \\beta^* \\\\ \n", |
659 | 659 | "\\alpha^* \\beta & |\\beta|^2 \n", |
660 | 660 | "\\end{pmatrix}$$\n", |
661 | 661 | "\n", |
662 | | - "$\\rho$ has the same descriptive abilities for the quantum state as the $\\left|\\psi \\right\\rangle$. If the state is not pure and let's say with $p_1$ probability we have $\\psi_1$ state and with $p_2$ probability we have $\\psi_2$ state (and this can be extended to $n$ cases: $p_n$ probability for $\\psi_n$ state):\n", |
| 662 | + "$\\rho$ has the same descriptive abilities for the quantum state as the $\\left|\\psi \\right\\rangle$. If the state is not pure and let's say with $p_1$ probability we have $\\left| \\psi_1\\right\\rangle$ state and with $p_2$ probability we have $\\left| \\psi_2\\right\\rangle$ state (and this can be extended to $n$ cases: $p_n$ probability for $\\left| \\psi_n\\right\\rangle$ state):\n", |
663 | 663 | "\n", |
664 | 664 | "\\begin{equation*}\n", |
665 | | - "\\rho = p_1 |\\psi_1 \\rangle \\langle \\psi_1| + p_2 |\\psi_2 \\rangle \\langle \\psi_2| = \n", |
| 665 | + "\\rho = p_1 \\left| \\psi_1 \\right\\rangle \\left\\langle \\psi_1\\right| + p_2 \\left| \\psi_2 \\right\\rangle \\left\\langle \\psi_2\\right| = \n", |
666 | 666 | "\\\\\n", |
667 | 667 | "= p_1 \n", |
668 | 668 | "\\begin{pmatrix} \n", |
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