Definitions for Inequalities for Quantum Entropy: A Review with Conditions for Equality

QAlgorithms.Papers.Independent.Intakethird100.P51.FaithfulDensity

For any finite type $A$ and any complex matrix $X$ indexed by $A\times A$, this defines a faithful density matrix to mean that $X$ is positive definite and its complex trace is exactly $1$. No nonemptiness assumption is imposed on $A$; when $A$ is empty, its trace is $0$, so this predicate cannot hold.

QAlgorithms.Papers.Independent.Intakethird100.P51.JC

This proposition says that for every $n,k\in\mathbb N$, every family of real weights $w_0,\ldots,w_k$ with $w_j\ge0$ and $\sum_jw_j=1$, and every pair of families of positive semidefinite complex $(n+1)\times(n+1)$ matrices $X_j,Y_j$ satisfying $\ker Y_j\subseteq\ker X_j$ for every $j$, the matrices $\overline X=\sum_jw_jX_j$ and $\overline Y=\sum_jw_jY_j$ satisfy

$$\ker\overline Y\subseteq\ker\overline X, \qquad D(\overline X\|\overline Y)\le\sum_jw_jD(X_j\|Y_j).$$

For a supported pair, $D(P\|Q)=\operatorname{Re}\operatorname{tr}(f(P)-P\log Q)$, where $f(x)=x\ln x$ and the logarithm are applied by continuous functional calculus, with their total real values at zero. The relative entropy definition assigns $+\infty$ outside the support condition, and the displayed inequality uses its real conversion, which sends infinities to zero; every pair used here is required or asserted to be supported. The matrices need not be normalized or nonzero. Zero weights are allowed, but their corresponding matrices must still satisfy all hypotheses; $n=0$ and $k=0$ are included.

QAlgorithms.Papers.Independent.Intakethird100.P51.MONO

This defines the following proposition: for every pair of natural numbers $n,m$, every channel $\Phi$ from complex $(n+1)\times(n+1)$ matrices to complex $(m+1)\times(m+1)$ matrices, and every positive semidefinite pair $X,Y$ of input matrices satisfying $\ker Y\subseteq\ker X$, one has

$$\ker\Phi(Y)\subseteq\ker\Phi(X) \quad\text{and}\quad \mathcal R(\Phi(X),\Phi(Y))\le\mathcal R(X,Y).$$

Here a channel means a map admitting a finite representation $\Phi(Z)=\sum_jK_jZK_j^\dagger$ with $\sum_jK_j^\dagger K_j=I_{n+1}$, and

$$\mathcal R(P,Q)= \begin{cases} \operatorname{Re}\operatorname{tr}\!\bigl(f(P)-P\ell(Q)\bigr),&\ker Q\subseteq\ker P,\\ 0,&\text{otherwise}, \end{cases} \qquad f(x)=x\ln x,\quad\ell(x)=\ln x,$$

with matrix functions given by real continuous functional calculus. This is the real conversion of an extended-real relative entropy whose unsupported value is $+\infty$, converted to zero. Neither $X$ nor $Y$ must have trace one or be nonzero. The cases $n=0$ and $m=0$ give one-dimensional registers, and zero matrices are included.

QAlgorithms.Papers.Independent.Intakethird100.P51.MPT

This proposition says that for every pair of natural numbers $n,m$, including zero, and all positive semidefinite complex matrices $X,Y$ indexed by $\{0,\ldots,n\}\times\{0,\ldots,m\}$ satisfying $\ker Y\subseteq\ker X$, their partial traces over the first factor satisfy

$$\ker(\operatorname{Tr}_1Y)\subseteq\ker(\operatorname{Tr}_1X), \qquad D(\operatorname{Tr}_1X\|\operatorname{Tr}_1Y)\le D(X\|Y).$$

Here $(\operatorname{Tr}_1X)_{bb'}=\sum_aX_{(a,b),(a,b')}$, and, for a supported pair $P,Q$,

$$D(P\|Q)=\operatorname{Re}\operatorname{tr}\!\left(f(P)-P\log Q\right), \qquad f(x)=x\ln x,$$

with both matrix functions defined by continuous functional calculus using the total real logarithm, for which $\ln0=0$. The underlying relative entropy is extended-real-valued and equals $+\infty$ when the kernel inclusion fails; the statement compares its real conversions, which send either infinity to zero, although the stated support conditions make the quantities here finite. Neither $X$ nor $Y$ is required to have trace one or to be nonzero; the two register dimensions are always positive.

QAlgorithms.Papers.Independent.Intakethird100.P51.SSA

This proposition says that for every $n,m,k\in\mathbb N$, including zero, and every positive semidefinite complex matrix $X$ of trace one on registers $A=\{0,\ldots,n\}$, $B=\{0,\ldots,m\}$, and $C=\{0,\ldots,k\}$,

$$0\le S(X_{AB})+S(X_{BC})-S(X)-S(X_B).$$

The marginals are defined entrywise by

$$(X_{AB})_{(a,b),(a',b')}=\sum_cX_{(a,b,c),(a',b',c)},\quad (X_{BC})_{(b,c),(b',c')}=\sum_aX_{(a,b,c),(a,b',c')},\quad (X_B)_{b,b'}=\sum_{a,c}X_{(a,b,c),(a,b',c)}.$$

Here $S(Z)=-\operatorname{Re}\operatorname{tr}f(Z)$, where $f(x)=x\ln x$ is applied through continuous functional calculus and $f(0)=0$. Thus the entropy uses natural logarithms. Singular density matrices and one-dimensional registers are included.

QAlgorithms.Papers.Independent.Intakethird100.P51.SSAlogCondition

For finite types $A,B,C$, each with decidable equality, and any complex matrix $X$ on $A\times(B\times C)$, let $X_{AB}$, $X_{BC}$, and $X_B$ have entries

$$(X_{AB})_{(a,b),(a',b')}=\sum_cX_{(a,(b,c)),(a',(b',c))},\qquad (X_{BC})_{(b,c),(b',c')}=\sum_aX_{(a,(b,c)),(a,(b',c'))},\qquad (X_B)_{b,b'}=\sum_c(X_{BC})_{(b,c),(b',c)}.$$

Writing $L(M)$ for the real natural logarithm applied through continuous functional calculus—using $\log0=0$, $\log x=\log|x|$ for negative real $x$, and the zero matrix for non-Hermitian $M$—this predicate means precisely that, for every $a,a'\in A$, $b,b'\in B$, and $c,c'\in C$,

$$L(X)_{(a,(b,c)),(a',(b',c'))} -\mathbf1_{c=c'}L(X_{AB})_{(a,b),(a',b')} = \mathbf1_{a=a'}L(X_{BC})_{(b,c),(b',c')} -\mathbf1_{a=a'}\mathbf1_{c=c'}L(X_B)_{b,b'}.$$

Here $\mathbf1_P$ is $1$ when $P$ holds and $0$ otherwise. No positivity, normalization, invertibility, or nonemptiness is assumed; if the full index type is empty, this matrix equality holds vacuously.

QAlgorithms.Papers.Independent.Intakethird100.P51.SupportedBy

For any finite type $A$ with decidable equality and any complex matrices $X,Y$ indexed by $A\times A$, this defines “$X$ is supported by $Y$” to mean $\ker Y\subseteq\ker X$: every vector $v:A\to\mathbb C$ satisfying $Yv=0$ also satisfies $Xv=0$. Neither matrix is assumed positive or Hermitian. If $A$ is empty, the condition holds automatically.

QAlgorithms.Papers.Independent.Intakethird100.P51.UEP

For finite nonempty types $A,B$, both with decidable equality, and a supplied channel $\Phi$ from complex matrices on $A$ to complex matrices on $B$, this defines the real supremum of

$$H(\rho)-\sum_{\substack{b=0,\ldots,m-1\\p_b>0}}p_bH(Z_b)$$

over every positive integer $m$, every positive semidefinite $\rho$ with $\operatorname{tr}\rho=1$, and every family of positive semidefinite matrices $E_b$ on $B$ satisfying $\sum_bE_b=I_B$. In this expression,

$$\begin{aligned} Q_b(i,j)&=\sum_{u,v\in B}\overline{\Phi(E^{A}_{ij})_{u,v}}\,(E_b)_{u,v},& p_b&=\operatorname{Re}\operatorname{tr}(Q_b\rho),\\ Z_b&=p_b^{-1}S_\rho Q_bS_\rho,& S_\rho&=(\sqrt{\cdot})(\rho), \end{aligned}$$

where $E^{A}_{ij}$ is the matrix unit and $H(M)=-\operatorname{Re}\operatorname{tr}(f(M))$, $f(x)=x\ln x$, using real continuous functional calculus. Zero reciprocals are zero, and terms with $p_b\le0$ are omitted. The supplied channel hypothesis means that some finite family of rectangular matrices $K_j$ satisfies $\sum_jK_j^\dagger K_j=I_A$ and $\Phi(X)=\sum_jK_jXK_j^\dagger$ for every $X$. The definition takes a supremum and does not itself assert attainment.

QAlgorithms.Papers.Independent.Intakethird100.P51.UpperLogContour

For any finite index type $A$ with decidable equality, any complex matrix $M$ indexed by $A$, any $c\in\mathbb C$, and any $r\in\mathbb R$, this predicate means precisely

$$0<r<\operatorname{Im}c \quad\text{and}\quad \sigma_{\mathbb C}(M)\subseteq\{z\in\mathbb C:|z-c|<r\}.$$

Thus the entire complex spectrum must lie strictly inside an open disk whose closed boundary also lies in the open upper half-plane. No Hermitian or positivity assumption is imposed on $M$, and $A$ is allowed to be empty, in which case its spectrum is empty and the spectral inclusion is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P51.WeakMonotonicity

For finite types $A,B,D$, each with decidable equality, and any complex matrix $X$ indexed by $A\times(B\times D)$, this defines the proposition

$$H(X_D)+H(X_B)\le H(X_{AB})+H(X_{AD}),$$

where

$$\begin{aligned} X_D(d,d')&=\sum_{a,b}X_{(a,b,d),(a,b,d')},& X_B(b,b')&=\sum_{a,d}X_{(a,b,d),(a,b',d)},\\ X_{AB}((a,b),(a',b'))&=\sum_d X_{(a,b,d),(a',b',d)},& X_{AD}((a,d),(a',d'))&=\sum_b X_{(a,b,d),(a',b,d')}. \end{aligned}$$

Here $H(M)=-\operatorname{Re}\operatorname{tr}(f(M))$, with $f(x)=x\ln x$ applied by real continuous functional calculus. No positivity or trace normalization of $X$ is assumed. The types may be empty; if any is empty, all four entropy expressions are zero.

QAlgorithms.Papers.Independent.Intakethird100.P51.basisPVM

For any type $A$, any complex square matrix $U$ indexed by $A$, and any $a\in A$, this defines the matrix

$$(P_a)_{i,j}=U_{i,a}\,\overline{U_{j,a}}\qquad(i,j\in A),$$

the outer product of column $a$ of $U$ with itself. No finiteness, unitarity, orthogonality, or normalization assumption on $U$ is imposed; a zero column gives the zero matrix.

QAlgorithms.Papers.Independent.Intakethird100.P51.coordinatePVM

For types $A,O$ with decidable equality and a chosen $b\in O$, this defines the complex matrix $P_b=I_A\otimes E_{bb}$ indexed by $A\times O$, explicitly

$$(P_b)_{(a,o),(a',o')}= \begin{cases} 1,&a=a',\ o=b,\ o'=b,\\ 0,&\text{otherwise}. \end{cases}$$

Neither type is required to be finite; $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.dualEffects

For any type $J$, a finite type $A$ with decidable equality, any complex matrices $\rho$ and $\sigma_j$ indexed by $A$, and any real weights $w_j$, this defines the family

$$D_j=w_jR_\rho\sigma_jR_\rho,\qquad R_\rho=f(\rho),\quad f(x)=(\sqrt{x})^{-1},$$

where $f(\rho)$ is real continuous functional calculus, with scalar value zero for $x\le0$ and with value zero on non-Hermitian matrices. No probability, density-matrix, or measurement conditions are required of $w,\rho,\sigma$. The type $J$ need not be finite, and $J$ or $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.dualStates

For any type $O$, a finite type $A$ with decidable equality, any complex matrix $\rho$ indexed by $A$, any family $E_b$ of such matrices, and $b\in O$, this defines

$$Z_b=p_b^{-1}S_\rho E_bS_\rho,\qquad p_b=\operatorname{Re}\operatorname{tr}(E_b\rho),\qquad S_\rho=(\sqrt{\cdot})(\rho).$$

The real square root is applied by continuous functional calculus, which returns zero on non-Hermitian matrices. The reciprocal is total, so $p_b=0$ gives $Z_b=0$; negative $p_b$ is also permitted. No density-matrix or measurement assumptions are imposed, $O$ need not be finite, and $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.dualUnnormalized

For any type $O$, a finite type $A$ with decidable equality, any complex matrix $\rho$ indexed by $A$, any family $E_b$ of such matrices, and a chosen $b\in O$, this defines

$$T_b=S_\rho E_bS_\rho,\qquad S_\rho=(\sqrt{\cdot})(\rho),$$

where the square root is applied through real continuous functional calculus. Scalar negative inputs have square root zero, and this functional calculus returns zero on non-Hermitian matrices. No positivity, normalization, or measurement assumptions are imposed. The type $O$ need not be finite, and $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.epsteinTrace

For any finite index type $A$ with decidable equality, arbitrary complex matrices $K,R,B$ indexed by $A$, and any $z\in\mathbb C$, this definition returns the complex number

$$\operatorname{tr}\exp\!\bigl(K+\mathcal L(zR+B)\bigr).$$

Here, if a matrix $M$ has its complex spectrum inside some open disk $|\zeta-c|<r$ with $0<r<\operatorname{Im}c$, then $\mathcal L(M)$ is $(2\pi i)^{-1}\oint_{|\zeta-c|=r}\operatorname{Log}(\zeta)(\zeta I-M)^{-1}\,d\zeta$, using one such disk selected by classical choice; otherwise $\mathcal L(M)=0$. The exponential is the matrix exponential. Consequently the returned value is $\operatorname{tr}e^K$ whenever no qualifying disk exists for $zR+B$. No Hermitian, positivity, or nonemptiness hypothesis is imposed; for empty $A$, the trace is zero.

QAlgorithms.Papers.Independent.Intakethird100.P51.hallBound

For finite types $O,A$, with decidable equality on $A$, any complex matrix $\rho$ indexed by $A$, and any family $E_b$ of such matrices, this defines the real number

$$H(\rho)-\sum_{\substack{b\in O\\p_b>0}}p_bH(Z_b), \qquad p_b=\operatorname{Re}\operatorname{tr}(E_b\rho),\quad Z_b=p_b^{-1}S_\rho E_bS_\rho,\quad S_\rho=(\sqrt{\cdot})(\rho).$$

Here $H(M)=-\operatorname{Re}\operatorname{tr}(f(M))$, $f(x)=x\ln x$, and both matrix functions use real continuous functional calculus, returning zero on non-Hermitian inputs. No positivity, normalization, or measurement assumptions are imposed. Terms with $p_b\le0$ contribute zero; if $O$ is empty, the value is $H(\rho)$. The type $A$ may also be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.hallRelative

For finite types $O,A$, with decidable equality on $A$, any complex matrix $\rho$ indexed by $A$, and any family $E_b$ of such matrices, this defines

$$\sum_{\substack{b\in O\\p_b>0}}p_b\,\mathcal R(Z_b,\rho), \qquad p_b=\operatorname{Re}\operatorname{tr}(E_b\rho),\quad Z_b=p_b^{-1}S_\rho E_bS_\rho,\quad S_\rho=(\sqrt{\cdot})(\rho).$$

For arbitrary matrices $X,Y$, the real number $\mathcal R(X,Y)$ is

$$\mathcal R(X,Y)= \begin{cases} \operatorname{Re}\operatorname{tr}\!\bigl(f(X)-X\ell(Y)\bigr), &\ker Y\subseteq\ker X,\\ 0,&\text{otherwise}, \end{cases} \qquad f(x)=x\ln x,\quad \ell(x)=\ln x.$$

All matrix functions use real continuous functional calculus. The second branch arises because the underlying extended-real relative entropy is $+\infty$ when the kernel inclusion fails, and its conversion to a real number is zero. No density-matrix or measurement assumptions are imposed; terms with $p_b\le0$ are zero, empty $O$ gives zero, and $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.imaginaryExp

For any finite type $A$ with decidable equality, any real number $t$, and any complex matrix $L$ indexed by $A\times A$, this defines

$$\operatorname{imaginaryExp}(t,L)=\exp(itL),$$

where $\exp$ is the matrix exponential. The matrix $L$ need not be Hermitian, so unitarity is not part of the definition. The case $t=0$ gives the identity matrix, and the empty index type is also allowed.

QAlgorithms.Papers.Independent.Intakethird100.P51.inverseRoot

For a finite type $A$ with decidable equality and any complex square matrix $\rho$ indexed by $A$, this defines

$$R_\rho=f(\rho),\qquad f(x)=(\sqrt{x})^{-1},$$

using real continuous functional calculus. The scalar square root and inverse are total: $f(x)=0$ for $x\le0$, including $f(0)=0$. Thus zero eigenvalues are assigned zero, rather than an infinite inverse. No positivity, invertibility, or trace normalization is assumed; the real functional calculus returns zero for a non-Hermitian matrix. The index type may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.krausV

For arbitrary types $A,B,K$ and any family $F_k$ of complex matrices with rows indexed by $B$ and columns indexed by $A$, this defines the complex matrix $V$, with rows indexed by $B\times K$ and columns indexed by $A$, by

$$V_{(b,k),a}=(F_k)_{b,a}.$$

None of the types is required to be finite or nonempty, and no normalization, positivity, or completeness condition on the family is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P51.liftAB

For arbitrary types $A,B,C$, with decidable equality on $C$, and any complex matrix $X$ indexed by $(A\times B)\times(A\times B)$, this defines a matrix on $A\times(B\times C)$ by

$$\operatorname{lift}_{AB}(X)_{(a,(b,c)),(a',(b',c'))} = \begin{cases} X_{(a,b),(a',b')},&c=c',\\ 0,&c\ne c'. \end{cases}$$

No finiteness, positivity, or nonemptiness is required of these types or matrices.

QAlgorithms.Papers.Independent.Intakethird100.P51.liftB

For arbitrary types $A,B,C$, with decidable equality on $A$ and $C$, and any complex matrix $X$ indexed by $B\times B$, this defines a matrix on $A\times(B\times C)$ by

$$\operatorname{lift}_{B}(X)_{(a,(b,c)),(a',(b',c'))} = \begin{cases} X_{b,b'},&a=a'\ \text{and}\ c=c',\\ 0,&\text{otherwise}. \end{cases}$$

No finiteness, positivity, or nonemptiness is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P51.liftBC

For arbitrary types $A,B,C$, with decidable equality on $A$, and any complex matrix $X$ indexed by $(B\times C)\times(B\times C)$, this defines a matrix on $A\times(B\times C)$ by

$$\operatorname{lift}_{BC}(X)_{(a,(b,c)),(a',(b',c'))} = \begin{cases} X_{(b,c),(b',c')},&a=a',\\ 0,&a\ne a'. \end{cases}$$

No finiteness, positivity, or nonemptiness is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P51.logMatrix

For any finite type $A$ with decidable equality and any complex matrix $X$ indexed by $A\times A$, this defines $\logMatrix(X)$ as the continuous functional calculus of the real natural logarithm applied to $X$. For Hermitian $X$, this applies $\log|\lambda|$ to each nonzero eigenvalue $\lambda$ and assigns $0$ to a zero eigenvalue; for non-Hermitian $X$, the functional calculus returns the zero matrix. No positivity, invertibility, or normalization is assumed, and $A$ may be empty.

QAlgorithms.Papers.Independent.Intakethird100.P51.marginalABC

For arbitrary types $A,B,C,D$, with $D$ finite, and any complex matrix $X$ on $A\times(B\times(C\times D))$, this defines a matrix on $A\times(B\times C)$ by

$$\operatorname{marginal}_{ABC}(X)_{(a,(b,c)),(a',(b',c'))} = \sum_{d\in D}X_{(a,(b,(c,d))),(a',(b',(c',d)))}.$$

No density or positivity condition is assumed, and $A,B,C$ need not be finite. If $D$ is empty, the result is the zero matrix.

QAlgorithms.Papers.Independent.Intakethird100.P51.marginalABD

For arbitrary types $A,B,C,D$, with $C$ finite, and any complex matrix $X$ on $A\times(B\times(C\times D))$, this defines a matrix on $A\times(B\times D)$ by

$$\operatorname{marginal}_{ABD}(X)_{(a,(b,d)),(a',(b',d'))} = \sum_{c\in C}X_{(a,(b,(c,d))),(a',(b',(c,d')))}.$$

No density or positivity condition is assumed, and $A,B,D$ need not be finite. If $C$ is empty, the result is the zero matrix.

QAlgorithms.Papers.Independent.Intakethird100.P51.principalLog

For any finite index type $A$ with decidable equality and any complex matrix $M$ indexed by $A$, this definition first asks whether there exist $c\in\mathbb C$ and $r\in\mathbb R$ such that $0<r<\operatorname{Im}c$ and $\sigma_{\mathbb C}(M)\subseteq\{z:|z-c|<r\}$. If such pairs exist, it chooses one by classical choice and returns

$$\frac{1}{2\pi i}\oint_{|z-c|=r}\operatorname{Log}(z)\,(zI-M)^{-1}\,dz,$$

where the circle has the standard positive orientation and $\operatorname{Log}$ is the principal scalar complex logarithm. If no such pair exists, it returns the zero matrix. The inverse in the integrand is the total matrix inverse, agreeing with the usual inverse on invertible matrices and returning zero on singular matrices; on the chosen circle, the spectral inclusion ensures invertibility. The definition allows empty $A$ and imposes no positivity or Hermitian condition on $M$.

QAlgorithms.Papers.Independent.Intakethird100.P51.splitProduct

For arbitrary types $A,B,D,C$, any complex matrix $X$ on $A\times B$, and any complex matrix $Y$ on $D\times C$, this defines a matrix on $A\times((B\times D)\times C)$ whose entries are

$$\operatorname{splitProduct}(X,Y)_{(a,((b,d),c)),(a',((b',d'),c'))} = X_{(a,b),(a',b')}\,Y_{(d,c),(d',c')}.$$

No finiteness, decidable equality, positivity, or nonemptiness assumptions are imposed.

QAlgorithms.Papers.Independent.Intakethird100.P51.ssaGap

For finite types $A,B,C$, each with decidable equality, and any complex matrix $X$ on $A\times(B\times C)$, define matrices

$$(X_{AB})_{(a,b),(a',b')}=\sum_{c\in C}X_{(a,(b,c)),(a',(b',c))},\qquad (X_{BC})_{(b,c),(b',c')}=\sum_{a\in A}X_{(a,(b,c)),(a,(b',c'))},\qquad (X_B)_{b,b'}=\sum_{b_C\in C}(X_{BC})_{(b,b_C),(b',b_C)}.$$

This declaration defines the real number

$$\operatorname{ssaGap}(X)=S(X_{AB})+S(X_{BC})-S(X)-S(X_B), \qquad S(M)=-\operatorname{Re}\operatorname{tr}\!\bigl[f(M)\bigr],\quad f(x)=x\log x,$$

where $f(M)$ uses continuous functional calculus and $\log$ is the real natural logarithm, with $\log0=0$. No density, Hermiticity, or positivity condition is imposed, and no nonnegativity is asserted. The types may be empty, with empty sums equal to zero.