Quantum Information Methods for Many-Body Physics · Notes Sec. 2.2.2
A Haar average over \(U^{\otimes\kk}\) lives on \(\kk\) copies of a quantum system, and permutations act there by shuffling the copies. This lesson fixes that action, shows that its traces count cycles, builds the projector onto the symmetric subspace, and introduces the wire diagrams that make all of it a matter of drawing.
We work with a \(\dd\)-dimensional Hilbert space \(\H=\C^\dd\) with orthonormal basis \(\{\ket x\}_{x=0}^{\dd-1}\). Its \(\kk\)-fold tensor power \(\H^{\otimes\kk}\) is the replica space. A permutation \(\sigma\in\SG_\kk\) acts by moving the content of slot \(j\) to slot \(\sigma(j)\):
The inverse is essential. For example, if \(\sigma=(1\;4\;3)(2)\), then \(1\mapsto4\), \(4\mapsto3\) and \(3\mapsto1\), therefore \(\PR_\sigma\ket{x_1,x_2,x_3,x_4}=\ket{x_3,x_2,x_4,x_1}\). With the usual convention \((\sigma\pi)(j)=\sigma(\pi(j))\), this defines a unitary representation:
\[ \PR_{\sigma\pi}=\PR_\sigma\PR_\pi,\qquad \PR_\iota=\one,\qquad \PR_{\sigma^{-1}}=\PR_\sigma^{-1}=\PR_\sigma^\dagger. \]Indeed,
\[ \begin{aligned} \PR_\sigma\PR_\pi\ket{x_1,\ldots,x_\kk}&=\ket{x_{\pi^{-1}(\sigma^{-1}(1))},\ldots,x_{\pi^{-1}(\sigma^{-1}(\kk))}}\\ &=\ket{x_{(\sigma\pi)^{-1}(1)},\ldots,x_{(\sigma\pi)^{-1}(\kk)}}=\PR_{\sigma\pi}\ket{x_1,\ldots,x_\kk}. \end{aligned} \]This calculation fixes the convention used in every later replica formula.
In practice we often work with \(\nn\) qudits of local dimension \(\ld\), \(\H=(\C^\ld)^{\otimes\nn}\). The \(\kk\)-fold copy \(((\C^\ld)^{\otimes\nn})^{\otimes\kk}\) is naturally a \(\kk\times\nn\) grid of qudits: each row is one replica of the full system, each column the \(\kk\) copies of a single qudit. Global unitaries \(U^{\otimes\kk}\) act row-wise, identically on each copy; permutations act column-wise, on the copies of each qudit in parallel. Reinterpreting \(((\C^\ld)^{\otimes\nn})^{\otimes\kk}\simeq((\C^\ld)^{\otimes\kk})^{\otimes\nn}\),
where \(\pr_\pi\) permutes the \(\kk\) replicas of one local qudit: it acts on \((\C^\ld)^{\otimes\kk}\), a space of dimension \(\ld^\kk\) (not \(\kk\)). This factorization is essential for efficient representations of randomized operations throughout the course.
Consider first the cyclic permutation \(\gamma=(\kk\;\,\kk{-}1\;\dots\;1)\). Then
\[ \begin{aligned} \tr(\PR_\gamma)&=\sum_{x_1,\dots,x_\kk=0}^{\dd-1}\braket{x_1,\dots,x_\kk}{x_2,\dots,x_\kk,x_1}\\ &=\sum_{x_1,\dots,x_\kk}\delta_{x_1,x_2}\delta_{x_2,x_3}\cdots\delta_{x_\kk,x_1}=\dd. \end{aligned} \]For an arbitrary \(\sigma\), choose \(\pi\) such that \(\pi\sigma\pi^{-1}\) has its cycles on consecutive elements (the cycle structures of \(\sigma\) and \(\pi\sigma\pi^{-1}\) agree). Then \(\PR_{\pi\sigma\pi^{-1}}\) is a tensor product of cyclic permutations, one per cycle, and
For the cyclic permutation, the same calculation gives
\[ \begin{aligned} \tr(A_1\otimes A_2\otimes\cdots\otimes A_\kk\,\PR_\gamma)&=\sum_{x_1,\dots,x_\kk}(A_1)_{x_1,x_2}(A_2)_{x_2,x_3}\cdots(A_\kk)_{x_\kk,x_1}\\ &=\tr(A_1A_2\cdots A_\kk). \end{aligned} \]For a general permutation the Kronecker deltas connect indices along each cycle independently. If \(\sigma=(a_1\,a_2\,\ldots\,a_\ell)\) is one cycle, the convention above gives \(\tr(A_{a_1}A_{a_\ell}\cdots A_{a_2})\); different cycles use disjoint summation indices, so their contributions multiply. With \(\mathcal C(\sigma)\) the set of cycles,
The apparent reversal inside each trace is only a consequence of our explicit permutation convention; it disappears when all factors are equal.
Take \(\sigma=(1\;4\;2)(3)\). Since \(\sigma^{-1}(1)=2\), \(\sigma^{-1}(2)=4\), \(\sigma^{-1}(3)=3\) and \(\sigma^{-1}(4)=1\),
\[ \begin{aligned} \tr\!\left[(A_1\otimes A_2\otimes A_3\otimes A_4)\PR_\sigma\right] &=\sum_{x_1,\ldots,x_4}(A_1)_{x_1x_2}(A_2)_{x_2x_4}(A_3)_{x_3x_3}(A_4)_{x_4x_1}\\ &=\tr(A_1A_2A_4)\,\tr(A_3). \end{aligned} \]The arrows are not an extra rule but a compact record of the index contractions; the order \(A_1A_2A_4\) is fixed by the chosen action of \(\PR_\sigma\), not by a reading order of the cycle notation.
For the important case \(A_1=\cdots=A_\kk=:A\),
Its orthogonal projector is the average over the group,
Proof. It is an orthogonal projector: substituting \(\tau=\sigma\pi\) and \(\pi=\sigma^{-1}\), respectively,
\[ P^2=\frac{1}{(\kk!)^2}\sum_{\sigma,\pi}\PR_{\sigma\pi}=\frac{1}{\kk!}\sum_{\sigma}\frac{1}{\kk!}\sum_{\tau}\PR_\tau=P,\qquad P^\dagger=\frac1{\kk!}\sum_\sigma\PR_{\sigma^{-1}}=P. \]Every \(\psi\in\Sym_{\kk,\dd}\) satisfies \(P\ket\psi=\frac1{\kk!}\sum_\sigma\ket\psi=\ket\psi\), so \(\Sym_{\kk,\dd}\) lies in the range of \(P\). Conversely, if \(P\ket\psi=\ket\psi\), then \(\PR_\pi\ket\psi=\PR_\pi P\ket\psi=\frac1{\kk!}\sum_\sigma\PR_{\pi\sigma}\ket\psi=P\ket\psi=\ket\psi\), so the range of \(P\) is exactly \(\Sym_{\kk,\dd}\). □
Its dimension follows from the trace formula and the Stirling numbers of the previous lesson:
The methods above are generic and straightforward, but the algebra is often cumbersome. A graphical notation helps: like Feynman diagrams for perturbative expansions, it is simply bookkeeping of the operations above. We draw a permutation by connecting input position \(j\) on the left to output position \(\sigma(j)\) on the right. The five-element permutation \(\tau=(1\;2)(3\;5)(4)\) is
The notation makes products easy. In the next diagram the lines of \(\sigma=(1\;2\;3)(4)(5)\) are met first and those of \(\tau=(1\;2)(3\;5)(4)\) second, so it represents \(\tau\cdot\sigma\):
Since the group structure is the same, the diagrams also stand for the operators \(\PR_\sigma\). Taking a trace adds curves that connect the initial and final endpoints; every cycle closes into a loop, and every loop is a free index sum worth \(\dd\):
PracticeQuBite unit 4 trains \(\PR_\sigma\) on product states, wire diagrams, traces and the symmetric subspace, ending with Exercise Sheet 1, Ex. 3.
Practice on QuBite