Dependencies

A bra operator \(\left\langle \cdot \right\rvert \) on a Hilbert space \(\mathcal{H}\) is defined as the anti-linear map \(\mathcal{H}\to \mathcal{B}(\mathcal{H},\mathbb {C})\) and is given by \(x\mapsto (y\mapsto \left\langle x \mid y\right\rangle )\).

Given \(x,y\in \mathcal{H}\), we get \(\left\langle x\right\rvert \circ \left\lvert y\right\rangle (1)=\left\langle x \mid y\right\rangle \).

Given \(x\in \mathcal{H}\), we get \(\left\langle x\right\rvert ^*=\left\lvert x\right\rangle \).

Let \(f\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)\) and \(x\in \mathcal{H}_2\). Then \(\left\langle x\right\rvert \circ f=\left\langle f^*(x)\right\rvert \).

Given a quantum set \((\mathcal{A},m,\eta ,\sigma _r)\) and \(x\in \mathcal{A}\), we get \(\left\langle x\right\rvert ^{\operatorname {r}}=\left\langle \sigma _{-1}(x^*)\right\rvert \).

Given a quantum set \(B\), we have

where \((u_i)_i\) is an orthonormal basis of \(B\).

Let \((\mathcal{A},m,\eta )\) be a finite-dimensional algebra and Hilbert space. Then we can form a co-algebra by letting \(m^*\) be the co-multiplication and \(\eta ^*\) be the co-unit.

Let \(\mathcal{H}_1,\mathcal{H}_2\) be Hilbert spaces, and let \(a,c\in \mathcal{H}_1\setminus \{ 0\} \) and \(b,d\in \mathcal{H}_2\setminus \{ 0\} \). Then if \(\left\lvert a\right\rangle \! \! \left\langle b\right\rvert =\left\lvert c\right\rangle \! \! \left\langle d\right\rvert \), then there exists some \(0\neq \alpha ,\beta \in \mathbb {C}\) such that \(a = \alpha c\) and \(b=\alpha \beta d\).

Let \(\mathcal{H}\) be a Hilbert space, and let \(x,y\in \mathcal{H}\). Then if \(\left\lvert x\right\rangle \! \! \left\langle x\right\rvert =\left\lvert y\right\rangle \! \! \left\langle y\right\rvert \), then there exists some \(0\neq \alpha \in \mathbb {C}\) such that \(x=\alpha {y}\) (i.e., they are co-linear).

If \(p,q\in \mathcal{L}(E)\) such that \(q\) is idempotent, then

Given a Hilbert space \(\mathcal{H}\), we have \({\mathcal{B}(\mathcal{H})}^\prime =\{ \alpha \operatorname {id}:\alpha \in \mathbb {C}\} \).
In other words, \(x\in \mathcal{B}(\mathcal{H})\) commutes with all operators \(y\in \mathcal{B}(\mathcal{H})\) if and only if \(x=\alpha \operatorname {id}\) for some \(\alpha \in \mathbb {C}\).

Let \((\mathcal{A},m,\eta )\) be a finite-dimensional algebra and Hilbert space. Then the counit is exactly \(\left\langle 1\right\rvert \).

Given a quantum set \((\mathcal{A},m,\eta ,\sigma )\), we get

for any \(r\in \mathbb {R}\).

Given \(\alpha \in \mathbb {C}\), we get \(mm^*=\alpha \operatorname {id}\) if and only if \(\operatorname{Tr}(Q^{-1}_i)=\alpha \) for all \(i\in [\mathfrak {K}]\).

A linear functional \(f\) on \(A\) is said to be faithful if \(f(x^*x)=0\) if and only if \(x=0\) for all \(x\in A\).

The adjoint of \(psi\) on \((\bigoplus _iM_{n_i},\psi )\) and \(\mathbb {C}\) is given by \(\mathbb {C}\to {B}\colon {x\mapsto {x1}}\). In other words, \(\psi ^*=\left\lvert 1\right\rangle \).

Given a linear functional \(\phi \colon {M_n}\to \mathbb {C}\), we have
\(\phi \) is a positive and faithful map \(\, \Leftrightarrow \, \) \(\phi _Q\) is positive-definite.
Again, \(\phi _Q\) is the matrix given by \(\phi \) defined in Definition 4.1.1.

Given a linear functional \(\phi \colon {M_n}\to \mathbb {C}\), then the following are equivalent,

1. \(\phi \) is positive and faithful,

2. \(\phi _Q\) is pos-def and \(\forall {x}\in {M_n}:\phi (x)=\operatorname{Tr}(Qx)\),

3. \(M_n\times {M_n}\to \mathbb {C}\colon (x,y)\mapsto \phi (x^*y)\) defines an inner product on \(M_n\).

Given a linear functional \(\phi \) on \(M_n\), we have,
\(\phi \) is positive \(\, \Leftrightarrow \, \) \(0\leq \phi _Q\).
Here, \(\phi _Q\) is the matrix of \(\phi \) defined in Definition 4.1.1.

Given a linear functional \(\phi \colon {M_n}\to \mathbb {C}\), we get \(\phi \) is real (star-preserving) if and only if \(\phi _Q\) is self-adjoint.
Here, \(\phi _Q\) is the matrix given by \(\phi \) defined in Definition 4.1.1.

Given a linear functional \(\phi \colon M_n \to \mathbb {C}\), we define \(\phi _Q=\sum _{i,j}\phi (e_{ij})e_{ji}\).

Given a linear functional \(\phi \colon M_n\to \mathbb {C}\), we get \(\phi \) is given by \(x\mapsto \operatorname{Tr}(\phi _Q \, x)\).

For any finite-dimensional algebra \((B,m,\eta )\) that is also a Hilbert space, if \(\left\langle xy \mid z\right\rangle =\left\langle y \mid x^*z\right\rangle \) for all \(x,y,z\in B\), then \((B,m,\eta )\) is a Frobenius algebra.

Given an algebra and co-algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), then if

then we get the following equations (the Frobenius equations),

We say an algebra and co-algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\) is a Frobenius algebra when it satisfies the Frobenius equation condition,

For any \(x,y,z\in {B}\), we get \(\left\langle xy \mid z\right\rangle =\left\langle x \mid z\sigma _{-1}(y^*)\right\rangle \).

For any \(x,y,z\in {B}\), we get \(\left\langle xy \mid z\right\rangle =\left\langle y \mid x^*z\right\rangle \).

Let \((\mathcal{A}_1,m_1,\eta _1),(\mathcal{A}_2,m_2,\eta _2)\) be finite-dimensional algebras and Hilbert spaces, and let \(f\colon \mathcal{A}_1\to \mathcal{A}_2\) be a linear map. Then \(f\) is an algebra homomorphism if and only if \(f^*\) is a co-algebra homomorphism.

Given a positive map \(\phi \colon M_n \to A\), where \(A\) is a \(^*\)-algebra, we get \(\phi \) is star-preserving.

A ket operator \(\left\lvert \cdot \right\rangle \) on a Hilbert space \(\mathcal{H}\) is defined as the linear map \(\mathcal{B}(\mathcal{H},\mathcal{B}(\mathbb {C},\mathcal{H}))\) and is given by \(x\mapsto (\alpha \mapsto \alpha x)\).

Given a Hilbert space \(\mathcal{H}\) that is also a \(^*\)-algebra, and \(x\in \mathcal{H}\), we get \(\left\lvert x\right\rangle ^{\operatorname {r}}=\left\lvert x^*\right\rangle \).

Given quantum sets \((\mathcal{A}_1,m_1,\eta _1,\sigma _r)\) and \((\mathcal{A}_2,m_2,\eta _2,\vartheta _r)\), and \(f\colon \mathcal{A}_1\to \mathcal{A}_2\), we get

Given a finite-dimensional inner product space \(E\) over \(\mathbb {C}\) and \(T\in \mathcal{B}(E)\), we get
\(T\) is positive semi-definite \(\Leftrightarrow \) \(T=\sum _i\left\lvert v_i\right\rangle \! \! \left\langle v_i\right\rvert \) for some tuple \((v_i)\) in \(E\).

Given C\(^*\)-algebras \(A,C\), we say a linear map \(f\colon A \to C\) is positive when \(0\leq f(a)\) for all \(0\leq a\). In other words, \(f\) maps positive elements in \(A\) to positive elements in \(C\).

We say \(A\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)\) is real (star-preserving) when \(A(a^*)={A(a)}^*\) for each \(a\in \mathcal{H}_1\).

Let \(A\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)\). Then \(A\) is real if and only if \(A^{\operatorname {r}}=A\).

We define the map \(\cdot ^{\operatorname {r}}\) as the self-invertible anti-linear automorphism \((\mathcal{H}_1\to \mathcal{H}_2)\cong (\mathcal{H}_1\to \mathcal{H}_2)\) given by

Given \(A\in \mathcal{B}(\mathcal{H})\), we get \(\operatorname{Spectrum}({A}^{\operatorname {r}})=\overline{\operatorname{Spectrum}(A)}\).
In fact, \(x\in \ker (A-\lambda \operatorname {id})\) if and only if \(x^*\in \ker ({A}^{\operatorname {r}}-\bar{\lambda }\operatorname {id})\).

When \(f\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)\) and \(g\in \mathcal{B}(\mathcal{H}_3,\mathcal{H}_1)\), then we get \({(f\circ g)}^{\operatorname {r}}={f^{\operatorname {r}}}\circ {{g^{\operatorname {r}}}}\).

Let \(f\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)\) and \(x\in \mathcal{H}_1\). Then \(f\circ \left\lvert x\right\rangle =\left\lvert f(x)\right\rangle \).

Given a linear map \(T_1\in \mathcal{B}(E_2,E_3)\) and elements \(x\in E_2\), \(y\in {E_1}\), we get, \(T_1\circ \left\lvert x\right\rangle \! \! \left\langle y\right\rvert =\left\lvert T_1(x)\right\rangle \! \! \left\langle y\right\rvert \)

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given a Hermitian matrix \(x\in M_n\), we get \(\sqrt{x^2}\) and \(x\) commute.

Given a Hermitian matrix \(x\in M_n\), we define the positive square root of the square of \(x\) to be \(\sqrt{x^2}\).

This is clearly positive semi-definite.

Explicitly, given the decomposition \(x=UDU^*\) where \(D\) is the diagonal of the eigenvalues of \(x\), then we have \(\sqrt{x^2}=U\left\lvert {D}\right\rvert U^*\).

The square of the positive square root of a Hermitian matrix is equal to the square of the matrix, i.e., \(\left(\sqrt{x^2}\right)^2=x^2\).

If \(A\in {M_n}\). Then
\(0\leq {A}\) and is invertible \(\, \Leftrightarrow \, \) \(A\) is positive-definite.

Given \(t\in \mathbb {R}\), we define the algebra automorphism \(\sigma _t\colon B\cong B\) to be given by \(a\mapsto Q^{-t}aQ^{t}\) with inverse \(a\mapsto Q^taQ^{-t}\) (so \(\sigma _t^{-1}=\sigma _{-t}\)).

For any \(s,t\in \mathbb {R}\), we get \(\sigma _s\circ \sigma _t=\sigma _{s+t}\).

For any \(t\in \mathbb {R}\), we get \(\sigma _t=\operatorname {id}\) if and only if \(t=0\) or \(\psi \) is tracial.

For any \(t\in \mathbb {R}\), we get \(\sigma _t\) is also a co-algebra homomorphism.

For any \(t\in \mathbb {R}\), \(\sigma _t\) is self-adjoint.

Given a quantum set \((\mathcal{A},m,\eta ,\sigma _r)\), we get

Here, \(\varkappa \) is the identification \(\mathcal{A}\otimes \mathcal{A}\cong \mathcal{A}\otimes \mathcal{A}\) given by \(x\otimes y\mapsto y\otimes x\).

Given a quantum set \((\mathcal{A},m,\eta ,\sigma _r)\), we get

Here, \(\varkappa \) is the identification \(\mathcal{A}\otimes \mathcal{A}\cong \mathcal{A}\otimes \mathcal{A}\) given by \(x\otimes y\mapsto y\otimes x\).

For any \(x\in B\), we get \({\sigma _{t}(x)}^*=\sigma _{-t}(x^*)\).
In other words, \(\sigma _t^{\operatorname {r}}=\sigma _{-t}\).

We clearly get \(\sigma _0=\operatorname {id}\).

Given a quantum set \((\mathcal{A},m,\eta ,\sigma )\), we get

Given a quantum set \((\mathcal{A},m,\eta ,\sigma )\), we get

Given a non-unital \(^*\)-algebra homomorphism \(f\colon A \to B\), where there exists a star-isomorphism \(A\cong \bigoplus _iM_{n_i}\), we get \(f\) is a positive map if and only if \(f\) is star-preserving.

For all operators \(T,S\in \mathcal{B}(\mathcal{H})\), if \(TS=0\), then \(P_{\ker {T}}S=S\), where \(P_{\ker {T}}\) is the orthogonal projection onto \(\ker {T}\).

We define \(\Phi \) to be the linear isomorphism from \(\mathcal{B}(\mathcal{A}_1,\mathcal{A}_2)\) to the bimodule maps in \(\mathcal{B}(\mathcal{A}_1\otimes \mathcal{A}_2)\) given by \((\operatorname {rmul}\otimes \operatorname {lmul})\Upsilon \).

For each \(i\), we fix a faithful and positive linear functional \(\psi _i\) on \(M_{n_i}\), and we let \(Q_i\in {M_{n_i}}\) denote the positive definite matrix such that \(\psi _i\colon x\mapsto \operatorname{Tr}(Q_ix)\) (so each \(Q_i=\sum _{j,k}\psi _i(e_{jk})e_{kj}\)) – see Proposition 4.1.1.

Let \(\psi \) be a faithful positive linear functional on \(\bigoplus _iM_{n_i}\) given by \(\psi =\sum _i\psi _i\circ p_i\), where each \(p_i\) is the projection map \(\bigoplus _jM_{n_j}\to M_{n_i}\), and we let \(Q=\bigoplus _iQ_i\). So then, given \(x\in \bigoplus _i{M_{n_i}}\), we get \(\psi (x)=\operatorname{Tr}(Q x)\), where \(\operatorname{Tr}\) here is defined by the sum of the diagonals in each matrix block.

By Theorem 4.1.10, we define the inner product on each \(M_{n_i}\) by

for all \(x,y\in {M_{n_i}}\). We denote \((M_{n_i},\psi _i)\) to be the Hilbert space given by this inner product.
We define the inner product on \(\bigoplus _iM_{n_i}\) by

for all \(x,y\in \bigoplus _i{M_{n_i}}\), where \(\operatorname{Tr}\) here is defined as the sum of the diagonals in each matrix block.

We get \(\left[\iota _s(e_{ij}Q_s^{-1/2})_{i,j=1}^{n_s}\right]_{s=1}^{\mathfrak {K}}\) is an orthonormal basis of \((\bigoplus _iM_{n_i},\psi )\).
Here, \(Q=\psi _Q\) from Proposition 4.1.1.

Let \(f\) be the orthonormal basis from Proposition 4.2.3. Then for \(x\in \bigoplus _iM_{n_i}\), we get \(R_f(x)_{s,ij}=(xQ^{1/2})_{s,ij}\).
Here, \(Q=\psi _Q\) from Proposition 4.1.1.

Given a positive definite matrix \(Q\in {M_n}\), we have \(\operatorname{Tr}(Qx^*x)=0\) if and only if \(x=0\) for any \(x\in {M_n}\).

Given positive semi-definite matrices \(x,y\in M_n\), we have \(xy=yx\) if and only if \(0\leq {xy}\).

Given a Hermitian matrix \(x\in M_n\), there exists matrices \(a,b\in M_n\) such that \(x = a^*a - b^*b\).

Given a Hermitian matrix \(x\in M_n\), we get \(x=x_+-x_-\).

Given a Hermitian matrix \(x\in M_n\), we define \(x_+\) to be the matrix

Given a Hermitian matrix \(x\in M_n\), we get \(x_+x_-=0\).

Given a Hermitian matrix \(x\in M_n\), we get both \(x_+\) and \(x_-\) are positive semi-definite.

Given a Hermitian matrix \(x\in M_n\), we define \(x_-\) to be the matrix

For any \(r,t\in \mathbb {R}\) and linear maps \(f\) and \(g\) from quantum set \(A\) to quantum set \(B\), we get

A quantum graph is a pair \((B,A)\), where \(B\) is a quantum set and \(A\) is a linear map \(B\to B\) such that \(A\bullet A=A\).

A quantum graph \((B,A)\) is real if \(A\) is a real linear map.

Given a quantum graph \((B,A)\), we have \((B,A)\) is real if and only if \(A\) is a positive map.

Given a \(^*\)-algebra \((\mathcal{A},m,\eta )\) that is also a finite-dimensional Hilbert space, we say it is a quantum set when there is a modular automorphism \(\sigma _t\colon \mathcal{A}\cong \mathcal{A}\), which is an algebra automorphism for each \(t\in \mathbb {R}\), and that the following properties are satisfied (for a fixed \(k\in \mathbb {R}\)):

1. \(\sigma _t\circ \sigma _s=\sigma _{t+s}\),

2. \(\sigma _t^{\operatorname {r}}=\sigma _{-t}\),

3. \(\sigma _t\) is self-adjoint,

4. \(\forall {x,y,z}\in \mathcal{A}:\left\langle xy \mid z\right\rangle =\left\langle y \mid \sigma _{-k}(x)^*z\right\rangle \),

5. \(\forall {x,y,z}\in \mathcal{A}:\left\langle xy \mid z\right\rangle =\left\langle x \mid z\sigma _{-k-1}(y^*)\right\rangle \).

By Proposition 3.1.4, we see that \((\mathcal{A},m,\eta ,k)\) is a Frobenius algebra.

And by Proposition 3.1.3, we can see that for any \(t\in \mathbb {R}\), we get \(\sigma _t\) is also a co-algebra homomorphism.

We can easily see that we get \(\left\langle x \mid y\right\rangle =\eta ^*(x^*\sigma _k(y))\) for all \(x,y\in \mathcal{A}\).

It is clear that we get \(\sigma _0=1\) and \(\sigma _{t}^{-1}=\sigma _{-t}\) for \(t\in \mathbb {R}\).

Given quantum sets \((\mathcal{A}_1,m_1,\eta _1,\sigma _r)\) and \((\mathcal{A}_2,m_2,\eta _2,\vartheta _r)\), then for each \(t,r\in \mathbb {R}\), we define \(\Psi _{t,r}\) to be the linear isomorphism from \(\mathcal{B}(\mathcal{A}_1,\mathcal{A}_2)\) to \(\mathcal{A}_2\otimes \mathcal{A}_1^{\operatorname {op}}\) given by

with inverse given by

A ket-bra operator \(\left\lvert \cdot \right\rangle \! \! \left\langle \cdot \right\rvert \) is defined as the linear map from \(E_2\) to the anti-linear map \(E_1\to \mathcal{B}(E_1,E_2)\) and is given by

This is exactly \(\left\lvert \cdot \right\rangle \! \! \left\langle \cdot \right\rvert =\left\lvert \cdot \right\rangle \, \circ \, \left\langle \cdot \right\rvert \).

Given \(x\in E_2\) and \(y\in E_1\), we get \(\left\lvert x\right\rangle \! \! \left\langle y\right\rvert ^*=\left\lvert y\right\rangle \! \! \left\langle x\right\rvert \).

Given a linear map \(T_2\in \mathcal{B}(E_3,E_1)\) and elements \(x\in E_2\), \(y\in E_1\), we get \(\left\lvert x\right\rangle \! \! \left\langle y\right\rvert \circ T_2=\left\lvert x\right\rangle \! \! \left\langle T_2^*(y)\right\rvert \).

Given quantum sets \((\mathcal{A}_1,m_1,\eta _1,\sigma _r)\) and \((\mathcal{A}_2,m_2,\eta _2,\vartheta _r)\), and elements \(a\in \mathcal{A}_1\) and \(b\in \mathcal{A}_2\), we get

Given orthonormal bases \(b=(b_i)\) and \(c=(c_j)\) of finite-dimensional Hilbert spaces \(\mathcal{H}_1,\mathcal{H}_2\), and elements \(x\in \mathcal{H}_1\) and \(y\in \mathcal{H}_2\), we have \(\mathcal{M}_{c,b}(\left\lvert x\right\rangle \! \! \left\langle y\right\rvert )=R_b(x){R_c(y)}^*\).

Given \(x\) and \(y\) in a finite-dimensional Hilbert space \(A\), we get \(\operatorname {Tr}(\left\lvert x\right\rangle \! \! \left\langle y\right\rvert )=\left\langle y \mid x\right\rangle \).

Let \(e=(e_i)\) be an orthonormal basis of a finite-dimensional Hilbert space \(\mathcal{H}\). Then \(R_e^*=R_e^{-1}\).

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given a Frobenius algebra \((\mathcal{A},m,\eta ,\mu ,\varpi )\), we get,

Given quantum sets \((B_1,m_1,\eta _1,\sigma _r)\) and \((B_2,m_2,\eta _2,\vartheta _r)\), we define the Schur product operator \(\cdot \bullet \cdot \colon \mathcal{B}(\mathcal{B}(B_1,B_2),\mathcal{B}(\mathcal{B}(B_1,B_2),\mathcal{B}(B_1,B_2)))\) as the map \(m_2(x\otimes y)m_1^*\).

Given \(a,c\in (B_1,m_1,\eta _1,\sigma _r)\) and \(b,d\in (B_2,m_2,\eta _2,\vartheta _r)\), we get

Given linear maps \(A_1,A_2,A_3\colon X_1\to X_2\), we get

Given an orthonormal basis \((u_i)\) of a \(\mathbb {C}\)-inner product space \(E\), we get \(\sum _i\left\lvert u_i\right\rangle \! \! \left\langle u_i\right\rvert =\operatorname {id}\).

Given Hilbert spaces \(\mathcal{H}_1,\mathcal{H}_2,\mathcal{H}_3,\mathcal{H}_4\), and linear maps \(x\colon \mathcal{H}_1\to \mathcal{H}_2\) and \(y\colon \mathcal{H}_3\to \mathcal{H}_4\), we clearly get \({(x\otimes y)}^{\operatorname {r}}={x}^{\operatorname {r}}\otimes {y}^{\operatorname {r}}\).

We define the tensor swap map to be \(\varrho \colon A\otimes B^{\operatorname {op}}\cong B\otimes A^{\operatorname {op}}\), given by \(x\otimes y^{\operatorname {op}}\mapsto y\otimes x^{\operatorname {op}}\).

The adjoint of the unit map \(\eta \colon \mathbb {C}\to \mathcal{A}\) in an algebra and Hilbert space \(\mathcal{A}\) is \(\left\langle 1\right\rvert \).

Let \(\mathcal{A}\) be an algebra and a Hilbert space. Then the unit map \(\eta \colon \mathbb {C}\to \mathcal{A}\) (which is given by \(\alpha \mapsto \alpha 1\)) is exactly \(\left\lvert 1\right\rangle \).

We define \(\Upsilon \) to be the linear isomorphism from \(\mathcal{B}(\mathcal{A}_1,\mathcal{A}_2)\) to \(\mathcal{A}_1\otimes \mathcal{A}_2\) given by \((\operatorname {id}\otimes \operatorname {unop})\varrho \Psi _{0,1}\).