Hudson's Theorem: the Pure States of Non-Negative Wigner Function

Derives Lemma A50.4. Both integrals converge absolutely, since \(\abs{\ee^{-az^{2}+wz}} =\ee^{-(\operatorname{Re}a)z^{2}+(\operatorname{Re}w)z}\).

The case \(w=0\). Put \(I(a)=\int_{\R}\ee^{-az^{2}}\dd z\). Differentiating under the integral sign, which Theorem 11.109 licenses on any compact subset of the half-plane because the differentiated integrand is dominated there by \(z^{2}\ee^{-\varepsilon z^{2}}\) for some \(\varepsilon>0\), and checking the Cauchy–Riemann equations Equation (12.5) on the integrand, \(I\) is holomorphic with \(I'(a)=-\int z^{2}\ee^{-az^{2}}\dd z\). Integrating that by parts, \(\int z\cdot\bigl(z\ee^{-az^{2}}\bigr)\dd z =\int z\cdot\bigl(-\tfrac{1}{2a}\bigr) \tfrac{\dd}{\dd z}\ee^{-az^{2}}\dd z =\tfrac{1}{2a}\int\ee^{-az^{2}}\dd z\), the boundary terms vanishing. Hence

on the half-plane, which is connected; and \(\sqrt{1}\,I(1)=\sqrt{\pi}\) by the Gauss integral, itself the case \(\omega=0\) of Equation (21.42). So \(I(a)=\sqrt{\pi/a}\).

General \(w\). Put \(J(w)=\int_{\R}\ee^{-az^{2}+wz}\dd z\) at fixed \(a\). The same argument makes \(J\) entire with \(J'(w)=\int z\,\ee^{-az^{2}+wz}\dd z\), and the same integration by parts, now carrying the factor \(\ee^{wz}\), gives \(J'(w)=\tfrac{w}{2a}J(w)\). With \(J(0)=\sqrt{\pi/a}\) this integrates to Equation (A50.3).

Derives Lemma A50.5. Look for the inverse in the form \(X+\ii Y\) with \(X,Y\) real. The two equations \(MX-SY=\identity\) and \(MY+SX=0\) give \(Y=-M^{-1}SX\) and then \(\left(M+SM^{-1}S\right)X=\identity\). The matrix \(M+SM^{-1}S\) is real symmetric and positive definite — it is a sum of a positive definite matrix and the positive semi-definite \(SM^{-1}S\), since \(v\cdot SM^{-1}Sv=(Sv)\cdot M^{-1}(Sv)\geq0\) and \(M^{-1}>0\) — hence invertible, and \(X=(M+SM^{-1}S)^{-1}\) solves the pair. That exhibits an inverse, which is unique, and its real part is \(X\).

Derives Lemma A50.6. Invertibility and \(\operatorname{Re}(N^{-1})>0\) are Lemma A50.5 with \(M=N_{\mathrm{r}}\) and \(S=N_{\mathrm{i}}\). For the integral, let \(R\) be the real symmetric positive-definite square root of \(N_{\mathrm{r}}\), which exists by Theorem 9.84 — diagonalise \(N_{\mathrm{r}}\) orthogonally and take the positive square roots of its eigenvalues — and substitute \(\vect{x}=R^{-1}\vect{y}\). Then \(\vect{x}\cdot N\vect{x}=\vect{y}\cdot(\identity+\ii T)\vect{y}\) with \(T=R^{-1}N_{\mathrm{i}}R^{-1}\) real symmetric. Diagonalise \(T\) orthogonally, \(T=O\transpose DO\) with \(D=\diag(d_{1},\ldots,d_{n})\), and substitute \(\vect{y}=O\transpose\vect{z}\); the quadratic form becomes \(\sum_{j}(1+\ii d_{j})z_{j}^{2}\) and the linear form becomes \(\vect{w}'\cdot\vect{z}\) with \(\vect{w}'=OR^{-1}\vect{w}\). The integral factorises into \(n\) copies of Equation (A50.3) with \(a_{j}=1+\ii d_{j}\), each of which has \(\operatorname{Re}a_{j}=1>0\):

the Jacobian being \(\abs{\det(R^{-1}O\transpose)}=1/\det R\). The exponent reassembles as \(\vect{w}'\cdot(\identity+\ii D)^{-1}\vect{w}'/4 =\vect{w}\cdot R^{-1}O\transpose(\identity+\ii D)^{-1}OR^{-1}\vect{w}/4 =\vect{w}\cdot N^{-1}\vect{w}/4\), because \(N=R(\identity+\ii T)R\). The prefactor is a non-zero constant determined by \(N\), and for real \(N\) every \(d_{j}\) is zero and it is \(\pi^{n/2}(\det N)^{-1/2}\).

Derives Proposition A50.7. Put \(\vect{u}=\vect{x}+\vect{y}/2\) and \(\vect{v}=\vect{x}-\vect{y}/2\), so that the integrand of Equation (A50.1) is \(\exp\bigl(-\vect{u}\cdot A\vect{u}+\vect{b}\cdot\vect{u}+c -\vect{v}\cdot\overline{A}\vect{v} +\overline{\vect{b}}\cdot\vect{v}+\overline{c}\bigr)\) times \(\ee^{-\ii\vect{p}\cdot\vect{y}/\hbar}\). Expanding by symmetry of \(A\),

while \(\vect{b}\cdot\vect{u}+\overline{\vect{b}}\cdot\vect{v} =2\operatorname{Re}\vect{b}\cdot\vect{x} +\ii\operatorname{Im}\vect{b}\cdot\vect{y}\) and \(c+\overline{c}=2\operatorname{Re}c\). Collecting the terms carrying \(\vect{y}\) and adding the Fourier kernel, the whole \(\vect{y}\) dependence is \(-\tfrac{1}{2}\vect{y}\cdot A_{\mathrm{r}}\vect{y} +\ii\,\vect{y}\cdot\vect{w}\) with \(\vect{w}\) as stated, a real vector. The remaining integral is Equation (A50.6) with the real positive-definite \(N=A_{\mathrm{r}}/2\) and \(\vect{w}\to\ii\vect{w}\), giving \((2\pi)^{3/2}(\det A_{\mathrm{r}})^{-1/2} \exp(-\tfrac{1}{2}\vect{w}\cdot A_{\mathrm{r}}^{-1}\vect{w})\). That is Equation (A50.8). Every factor is real and positive: the exponential of a real number, and a positive constant. Since \(\vect{w}\) is an affine function of \((\vect{x},\vect{p})\), the exponent is a real quadratic in \((\vect{x},\vect{p})\) and \(W\) is a Gaussian on phase space.

Derives Lemma A50.8. For pure states \(\hat{\rho}_{1}=\ketbra{\psi_{1}}{\psi_{1}}\) and \(\hat{\rho}_{2}=\ketbra{\psi_{2}}{\psi_{2}}\) one has \(\tr(\hat{\rho}_{1}\hat{\rho}_{2}) =\braket{\psi_{1}}{\psi_{2}}\braket{\psi_{2}}{\psi_{1}} =\abs{\braket{\psi_{1}}{\psi_{2}}}^{2}\), so Equation (29.56) is Equation (A50.11).

For the second statement, note first that Equation (A50.10) is \(\varphi_{\vect{x}_{0},\vect{p}_{0}}(\vect{x}) =\ee^{\ii\vect{p}_{0}\cdot\vect{x}/\hbar}\, \varphi_{\vect{0},\vect{0}}(\vect{x}-\vect{x}_{0})\), and that substituting this into Equation (A50.1) makes the two exponential factors combine as \(\ee^{\ii\vect{p}_{0}\cdot\vect{y}/\hbar}\) and shifts the argument, so that

By Proposition A50.7 that function is a strictly positive Gaussian, and by Theorem 29.53 it integrates to \(1\) over phase space; hence \(\iint W_{\varphi_{\vect{x}_{0},\vect{p}_{0}}}(\vect{x},\vect{p}) \dd^{3}x_{0}\dd^{3}p_{0}=1\) for every \((\vect{x},\vect{p})\). Integrating Equation (A50.11) over \((\vect{x}_{0},\vect{p}_{0})\) and exchanging the order of integration — legitimate because every integrand in sight is non-negative, which is where the hypothesis \(W_{\psi}\geq0\) is used — gives \((2\pi\hbar)^{3}\int W_{\psi}\dd^{3}x\,\dd^{3}p\), which is \((2\pi\hbar)^{3}\) by the normalisation clause of Theorem 29.53.

Derives Proposition A50.9. Suppose the overlap vanishes for some \((\vect{x}_{0},\vect{p}_{0})\). By Equation (A50.11) the integral of \(W_{\varphi}W_{\psi}\) over phase space is then zero. Both factors are non-negative — the first strictly positive everywhere by Proposition A50.7, the second by hypothesis — so the product vanishes identically, and therefore \(W_{\psi}\) does. Integrating \(W_{\psi}\) over \(\vect{p}\) and using Equation (29.54) gives \(\abs{\psi(\vect{x})}^{2}=0\) for every \(\vect{x}\), contradicting \(\norm{\psi}=1\).

Derives Lemma A50.11. Write \(\zeta=\vect{\xi}+\ii\vect{\eta}\). Then \((\vect{x}-\zeta)\cdot(\vect{x}-\zeta) =\abs{\vect{x}-\vect{\xi}}^{2}-\abs{\vect{\eta}}^{2} -2\ii(\vect{x}-\vect{\xi})\cdot\vect{\eta}\), so the modulus of the kernel is \(\ee^{\abs{\vect{\eta}}^{2}/2s^{2}} \ee^{-\abs{\vect{x}-\vect{\xi}}^{2}/2s^{2}}\), a Gaussian in \(\vect{x}\); its product with \(\abs{\psi}\) is integrable by Cauchy–Schwarz (Proposition 16.4), and the bound is locally uniform in \(\zeta\). Differentiating under the integral sign in the real and imaginary parts of each \(\zeta_{k}\) — licensed by Theorem 11.109 on bounded regions together with the locally uniform domination just displayed, which controls the tails — shows that \(F\) has continuous first partial derivatives and that they satisfy the Cauchy–Riemann equations Equation (12.5) in each variable, because the integrand does. Hence \(F\) is holomorphic in each variable, and being continuous it is entire on \(\C^{3}\).

For Equation (A50.15), substitute \(\zeta=\vect{x}_{0}-\ii s^{2}\vect{p}_{0}/\hbar\), so that \(\vect{x}-\zeta=(\vect{x}-\vect{x}_{0}) +\ii s^{2}\vect{p}_{0}/\hbar\) and

The first two terms are the exponent of \(\overline{\varphi_{\vect{x}_{0},\vect{p}_{0}}}\) up to the constant \((\pi s^{2})^{-3/4}\) and the phase \(\ee^{\ii\vect{x}_{0}\cdot\vect{p}_{0}/\hbar}\), and the third is \(\abs{\operatorname{Im}\zeta}^{2}/2s^{2}\) because \(\operatorname{Im}\zeta=-s^{2}\vect{p}_{0}/\hbar\). That is Equation (A50.15). Finally \(\abs{\braket{\varphi}{\psi}}\leq\norm{\varphi}\norm{\psi}=1\) by Cauchy–Schwarz, which with \(\abs{\kappa}\) as displayed gives Equation (A50.16); and every \(\zeta\in\C^{3}\) arises from exactly one pair \((\vect{x}_{0},\vect{p}_{0})\), namely \(\vect{x}_{0}=\operatorname{Re}\zeta\) and \(\vect{p}_{0}=-\hbar\operatorname{Im}\zeta/s^{2}\).

Derives Lemma A50.12. Choose \(c_{0}\in\C\) with \(\ee^{c_{0}}=F(0)\), possible because \(F(0)\neq0\), and set

The integrand is continuous in \(t\) and holomorphic in \(\zeta\), since \(F\) is entire and non-vanishing, so \(g\) is entire by the same combination of Theorem 11.109 and Equation (12.5) used in Lemma A50.11. Fix \(\zeta\) and substitute \(u=vt\) in Equation (A50.18) written at the point \(t\zeta\); the result is \(g(t\zeta)=c_{0}+\int_{0}^{t}\sum_{j}\zeta_{j} (\pp_{j}F)(u\zeta)/F(u\zeta)\,\dd u\), so

the last step by the chain rule. Hence \(\frac{\dd}{\dd t}\bigl[F(t\zeta)\ee^{-g(t\zeta)}\bigr]=0\), so \(F(\zeta)\ee^{-g(\zeta)}=F(0)\ee^{-c_{0}}=1\).

Derives Lemma A50.13. By Theorem 12.20 the Taylor series \(u(\lambda)=\sum_{k\geq0}a_{k}\lambda^{k}\) converges on every disc. Fix \(r>0\). The series converges uniformly on the circle \(\abs{\lambda}=r\), so it may be integrated term by term against \(\ee^{-\ii k\vartheta}\), and for \(k\geq0\)

the first being Equation (12.13) written on the circle and the second holding because every term of the series then carries a strictly positive power of \(\ee^{\ii\vartheta}\). Conjugating the second identity and adding it to the first gives, for \(k\geq1\),

Write \(\mu(r)=A+Br^{2}\), so that \(\operatorname{Re}u(r\ee^{\ii\vartheta})\leq\mu(r)\) by hypothesis, and subtract the constant \(\mu(r)\) inside Equation (A50.21), which changes nothing because \(\int_{0}^{2\pi}\ee^{-\ii k\vartheta}\dd\vartheta=0\) for \(k\geq1\). Bounding the modulus of the integral by the integral of the modulus, and using that \(\mu(r)-\operatorname{Re}u\) is non-negative so that its modulus is itself,

the mean of \(\operatorname{Re}u\) over the circle being \(\operatorname{Re}a_{0}\) by the case \(k=0\) of Equation (A50.20). Hence for every \(k\geq1\)

and for \(k\geq3\) the right-hand side tends to \(0\) as \(r\to\infty\). So \(a_{k}=0\) for \(k\geq3\).

Derives Proposition A50.14. By Lemma A50.11 and Proposition A50.9 the entire function \(F\) has no zeros: \(F(\zeta)\) is a non-zero multiple of an overlap that never vanishes. So \(F=\ee^{g}\) with \(g\) entire, by Lemma A50.12, and Equation (A50.16) gives

since \(\abs{\operatorname{Im}\zeta}\leq\abs{\zeta}\).

Fix \(\vect{\beta}\in\C^{3}\) and consider \(u_{\vect{\beta}}(\lambda)=g(\lambda\vect{\beta})\), entire in the single variable \(\lambda\) because \(g\) is entire and \(\lambda\mapsto\lambda\vect{\beta}\) is holomorphic. By Equation (A50.25), \(\operatorname{Re}u_{\vect{\beta}}(\lambda) \leq\frac{3}{4}\log(\pi s^{2}) +\abs{\vect{\beta}}^{2}\abs{\lambda}^{2}/2s^{2}\), so Lemma A50.13 makes \(u_{\vect{\beta}}\) a polynomial of degree at most two:

exactly, with no remainder. The two derivatives are computed by the chain rule: \(\dd g(\lambda\vect{\beta})/\dd\lambda|_{0} =\sum_{j}\beta_{j}\,\pp_{j}g(0)\) and \(\dd^{2}g(\lambda\vect{\beta})/\dd\lambda^{2}|_{0} =\sum_{j,k}\beta_{j}\beta_{k}\,\pp_{j}\pp_{k}g(0)\). Setting \(\lambda=1\) and renaming \(\vect{\beta}\) as \(\zeta\), which is legitimate because \(\vect{\beta}\) was arbitrary,

which is Equation (A50.24) with \(\Gamma_{jk}=-\tfrac{1}{2}\pp_{j}\pp_{k}g(0)\), symmetric because mixed holomorphic partial derivatives commute, \(\lambda_{j}=\pp_{j}g(0)\) and \(\nu=g(0)\). Note that no theorem about Taylor expansions in several variables was used: the expansion was performed along one complex line at a time, and Lemma A50.13 guaranteed that on every such line it terminates.

Derives Lemma A50.15. If \(P_{22}>0\), complete the square: with \(\vect{\eta}'=\vect{\eta}+P_{22}^{-1}P_{12}\transpose\vect{\xi}\),

as one checks by expanding. The map \((\vect{\xi},\vect{\eta})\mapsto(\vect{\xi},\vect{\eta}')\) is a linear bijection of \(\R^{2n}\), so \(Q>0\) if and only if the right-hand side is positive for every non-zero \((\vect{\xi},\vect{\eta}')\), which is the stated pair of conditions. Conversely \(Q>0\) forces \(P_{22}>0\), by setting \(\vect{\xi}=\vect{0}\), after which the displayed identity applies.

Derives Proposition A50.16. By Equation (A50.15), \(\abs{\braket{\varphi_{\vect{x}_{0},\vect{p}_{0}}}{\psi}} =(\pi s^{2})^{-3/4}\abs{F(\zeta)} \ee^{-\abs{\operatorname{Im}\zeta}^{2}/2s^{2}}\) with \(\zeta=\vect{\xi}+\ii\vect{\eta}\), \(\vect{\xi}=\vect{x}_{0}\) and \(\vect{\eta}=-s^{2}\vect{p}_{0}/\hbar\). That substitution has Jacobian \(\dd^{3}x_{0}\,\dd^{3}p_{0} =(\hbar/s^{2})^{3}\dd^{3}\xi\,\dd^{3}\eta\), so Equation (A50.12) says that

Now compute the real part of the exponent from Equation (A50.24). Since \(\zeta\cdot\Gamma\zeta =\vect{\xi}\cdot\Gamma\vect{\xi} -\vect{\eta}\cdot\Gamma\vect{\eta} +2\ii\,\vect{\xi}\cdot\Gamma\vect{\eta}\),

with \(L\) real affine. The exponent of Equation (A50.31) is therefore \(-2Q(\vect{\xi},\vect{\eta})+2L\), with \(Q\) as in Equation (A50.29). An integral \(\int_{\R^{6}}\ee^{-2Q+2L}\) over the whole space is finite if and only if the quadratic part \(Q\) is positive definite: if some \((\vect{\xi}_{0},\vect{\eta}_{0})\neq0\) had \(Q(\vect{\xi}_{0},\vect{\eta}_{0})\leq0\) then along the ray \(t(\vect{\xi}_{0},\vect{\eta}_{0})\) the integrand would be at least \(\ee^{2tL_{0}}\) with \(L_{0}\) the value of the linear part, which is bounded below on the ray only if \(L_{0}\ge 0\) and in either case the integral over a solid cone about the ray diverges, \(Q\) being continuous and hence bounded above on a neighbourhood of the ray. So \(Q>0\), and Lemma A50.15 with \(P_{22}=\identity/2s^{2}-\Gamma_{\mathrm{r}}\) and \(P_{12}=-\Gamma_{\mathrm{i}}\) turns that into Equation (A50.30) together with \(P_{22}>0\).

Derives Proposition A50.17. Forward. Insert \(\psi_{0}\) into Equation (A50.14). Expanding \((\vect{x}-\zeta)\cdot(\vect{x}-\zeta) =\abs{\vect{x}}^{2}_{\ast}-2\zeta\cdot\vect{x}+\zeta\cdot\zeta\), where \(\abs{\vect{x}}^{2}_{\ast}=\vect{x}\cdot\vect{x}\), the exponent is \(-\vect{x}\cdot N\vect{x} +(\vect{b}+\zeta/s^{2})\cdot\vect{x}+c-\zeta\cdot\zeta/2s^{2}\), and Equation (A50.6) evaluates the integral as \(C_{N}\exp\bigl[(\vect{b}+\zeta/s^{2})\cdot N^{-1} (\vect{b}+\zeta/s^{2})/4+c-\zeta\cdot\zeta/2s^{2}\bigr]\). The part quadratic in \(\zeta\) is \(\zeta\cdot N^{-1}\zeta/4s^{4}-\zeta\cdot\zeta/2s^{2}\), which is \(-\zeta\cdot\Gamma\zeta\) with \(\Gamma\) as in Equation (A50.33); the rest is affine in \(\zeta\) and supplies \(\vect{\lambda}\) and \(\nu\).

Converse. Solving Equation (A50.33) for \(N\) gives \(N^{-1}=K\) with

The first condition of Proposition A50.16, \(\Gamma_{\mathrm{r}}<\identity/2s^{2}\), is exactly \(K_{\mathrm{r}}>0\), which by Lemma A50.5 makes \(K\) invertible; set \(N=K^{-1}\) and \(A=N-\identity/2s^{2}\), complex symmetric and uniquely determined. It remains to show that \(A_{\mathrm{r}}>0\) is equivalent to Equation (A50.30). By Equation (A50.5),

so, \(A_{\mathrm{r}}=N_{\mathrm{r}}-\identity/2s^{2}\) and both matrices being real symmetric with the inverse order-reversing on positive definite matrices,

Substituting Equation (A50.34), the left-hand side is \(2s^{2}\identity-4s^{4}\Gamma_{\mathrm{r}} +4s^{4}\Gamma_{\mathrm{i}} \bigl(\identity/2s^{2}-\Gamma_{\mathrm{r}}\bigr)^{-1} \Gamma_{\mathrm{i}}\), the factors \(4s^{4}\) cancelling against the inverse of \(K_{\mathrm{r}}\); so the last inequality of Equation (A50.36) is \(\Gamma_{\mathrm{i}}(\identity/2s^{2}-\Gamma_{\mathrm{r}})^{-1} \Gamma_{\mathrm{i}}<\Gamma_{\mathrm{r}}\), which is Equation (A50.30).

Proof of Theorem A50.3. Derives Theorem A50.3. The “if” half is Proposition A50.7, which also gives the last sentence of the theorem.

For the “only if” half, let \(W_{\psi}\geq0\) everywhere. By Proposition A50.14 the entire function \(F\) of Equation (A50.14) is \(\ee^{g}\) with \(g\) a quadratic polynomial, of matrix \(\Gamma\); by Proposition A50.16 that \(\Gamma\) satisfies the positivity conditions; and by Proposition A50.17 there is therefore exactly one complex symmetric \(A\) with \(A_{\mathrm{r}}>0\), together with a \(\vect{b}\) and a \(c\) matching the affine part, such that the Gaussian \(\psi_{0}=\exp(-\vect{x}\cdot A\vect{x}+\vect{b}\cdot\vect{x}+c)\) has \(F_{0}=F\). Because \(A_{\mathrm{r}}>0\), \(\psi_{0}\) lies in \(L^{2}(\R^{3})\).

Both \(\psi\) and \(\psi_{0}\) therefore have the same overlaps with every coherent state: by Equation (A50.15) applied to each of them, and with the same non-vanishing factor \(\kappa\),

for every \(\vect{x}_{0}\) and \(\vect{p}_{0}\). Hence \(\psi-\psi_{0}\in L^{2}(\R^{3})\) is orthogonal to every \(\varphi_{\vect{x}_{0},\vect{p}_{0}}\), that is to every \(W(\zeta) \Omega_{s}\), and those vectors are total in \(L^{2}(\R^{3})\) by Proposition A49.15. So \(\psi=\psi_{0}\) in \(L^{2}\), which is Equation (A50.2) almost everywhere.