numerical-shadow:properties

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numerical-shadow:properties [2014/02/04 02:07] lpawela |
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====== Properties of numerical shadow ====== | ====== Properties of numerical shadow ====== | ||

- | - By construction the distribution $P_A(z)$ is supported on the [[numerical-range|numerical range]] of $W(A)$ and it is normalized, $$\int_{W(A)} P_A(z) d^2 z =1.$$ | + | - By construction the distribution $P_A(z)$ is supported on the [[:numerical-range|numerical range]] of $W(A)$ and it is normalized, $$\int_{W(A)} P_A(z) d^2 z =1.$$ |

- The (numerical) shadow is unitarily invariant, $P_A(z)=P_{UAU^*}(z)$. This is a consequence of the fact that the integration measure ${\rm d} \mu(\psi)$ is unitarily invariant. | - The (numerical) shadow is unitarily invariant, $P_A(z)=P_{UAU^*}(z)$. This is a consequence of the fact that the integration measure ${\rm d} \mu(\psi)$ is unitarily invariant. | ||

- For any normal matrix $A$, such that $AA^*=A^*A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to a projection of a regular $N$--simplex of classical states ${\cal C}_N$ (embedded in ${\mathbb R}^{N-1}$) onto a plane. | - For any normal matrix $A$, such that $AA^*=A^*A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to a projection of a regular $N$--simplex of classical states ${\cal C}_N$ (embedded in ${\mathbb R}^{N-1}$) onto a plane. | ||

- For a non--normal matrix $A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to an orthogonal projection of the complex projective manifold $\Omega_N={\mathbb C}P^{N-1}$ onto a plane. | - For a non--normal matrix $A$, its shadow covers the numerical range $W(A)$ with the probability corresponding to an orthogonal projection of the complex projective manifold $\Omega_N={\mathbb C}P^{N-1}$ onto a plane. | ||

- For any two matrices $A$ and $B$, the shadow of their tensor product does not depend on the order, $$ P_{A \otimes B}(z)=P_{B \otimes A} (z)\; .$$ To show this property define a unitary swap operator $S$ which acts on a composite Hilbert space and interchanges the order in the tensor product, $S(\ket{z} \otimes \ket{y})=\ket{y} \otimes \ket{z}$. Thus $ \bra{x} A \otimes B \ket{x} = \bra{x} S^{*} B \otimes A S \ket{x}$, and since $S$ is unitary it does not influence the numerical shadow induced by the unitarily invariant Fubini-Study measure on complex projective space. | - For any two matrices $A$ and $B$, the shadow of their tensor product does not depend on the order, $$ P_{A \otimes B}(z)=P_{B \otimes A} (z)\; .$$ To show this property define a unitary swap operator $S$ which acts on a composite Hilbert space and interchanges the order in the tensor product, $S(\ket{z} \otimes \ket{y})=\ket{y} \otimes \ket{z}$. Thus $ \bra{x} A \otimes B \ket{x} = \bra{x} S^{*} B \otimes A S \ket{x}$, and since $S$ is unitary it does not influence the numerical shadow induced by the unitarily invariant Fubini-Study measure on complex projective space. |

numerical-shadow/properties.txt · Last modified: 2014/02/04 02:07 by lpawela