Strong subadditivity of quantum entropy

We use the following notation throughout the following: A Hilbert space is denoted by ${\displaystyle {\mathcal {H}}}$, and ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ denotes the bounded linear operators on ${\displaystyle {\mathcal {H}}}$. Tensor products are denoted by superscripts, e.g., ${\displaystyle {\mathcal {H}}^{12}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}}$. The trace is denoted by ${\displaystyle {\rm {Tr}}}$.

Ordinary subadditivity [9] concerns only two spaces ${\displaystyle {\mathcal {H}}^{12}}$ and a density matrix ${\displaystyle \rho ^{12}}$. It states that

${\displaystyle S(\rho ^{12})\leq S(\rho ^{1})+S(\rho ^{2})}$

This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies ${\displaystyle S(\rho ^{12}|\rho ^{1})=S(\rho ^{12})-S(\rho ^{1})}$ and ${\displaystyle S(\rho ^{12}|\rho ^{2})=S(\rho ^{12})-S(\rho ^{2})}$ are both non-negative. In the quantum case, however, both can be negative, e.g. ${\displaystyle S(\rho ^{12})}$ can be zero while ${\displaystyle S(\rho ^{1})=S(\rho ^{2})>0}$. Nevertheless, the subadditivity upper bound on ${\displaystyle S(\rho ^{12})}$ continues to hold. The closest thing one has to ${\displaystyle S(\rho ^{12})-S(\rho ^{1})\geq 0}$ is the Araki–Lieb triangle inequality [9]

${\displaystyle S(\rho ^{12})\geq |S(\rho ^{1})-S(\rho ^{2})|}$

which is derived in [9] from subadditivity by a mathematical technique known as purification.

Suppose that the Hilbert space of the system is a tensor product of three spaces: ${\displaystyle {\mathcal {H}}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}\otimes {\mathcal {H}}^{3}.}$. Physically, these three spaces can be interpreted as the space of three different systems, or else as three parts or three degrees of freedom of one physical system.

Given a density matrix ${\displaystyle \rho ^{123}}$ on ${\displaystyle {\mathcal {H}}}$, we define a density matrix ${\displaystyle \rho ^{12}}$ on ${\displaystyle {\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}}$ as a partial trace: ${\displaystyle \rho ^{12}={\rm {Tr}}_{{\mathcal {H}}^{3}}\rho ^{123}}$. Similarly, we can define density matrices: ${\displaystyle \rho ^{23}}$, ${\displaystyle \rho ^{13}}$, ${\displaystyle \rho ^{1}}$, ${\displaystyle \rho ^{2}}$, ${\displaystyle \rho ^{3}}$.

E. P. Wigner and M. M. Yanase [14] proposed a different definition of entropy, which was generalized by Freeman Dyson.

It was pointed out in [9] that the first statement below is equivalent to SSA and A. Ulhmann in [17] showed the equivalence between the second statement below and SSA.

• ${\displaystyle S(\rho ^{1})+S(\rho ^{3})-S(\rho ^{12})-S(\rho ^{23})\leq 0.}$ Note that the conditional entropies ${\displaystyle S(\rho ^{12}|\rho ^{1})}$ and ${\displaystyle S(\rho ^{23}|\rho ^{3})}$ do not have to be both non-negative.
• The map ${\displaystyle \rho ^{12}\mapsto S(\rho ^{1})-S(\rho ^{12})}$ is convex.

Both of these statements were proved directly in.[3]

As noted by Lindblad[18] and Uhlmann,[19] if, in equation (1), one takes ${\displaystyle K=1}$ and ${\displaystyle r=1-p,A=\rho }$ and ${\displaystyle B=\sigma }$ and differentiates in ${\displaystyle p}$ at ${\displaystyle p=0}$, one obtains the joint convexity of relative entropy: i.e., if ${\displaystyle \rho =\sum _{k}\lambda _{k}\rho _{k}}$, and ${\displaystyle \sigma =\sum _{k}\lambda _{k}\sigma _{k}}$, then

${\displaystyle S{\Bigl (}\sum _{k}\lambda _{k}\rho _{k}||\sum _{k}\lambda _{k}\sigma _{k}{\Bigr )}\leq \sum _{k}\lambda _{k}S(\rho _{k}||\sigma _{k}),}$

(2)

where ${\displaystyle \lambda _{k}\geq 0}$ with ${\displaystyle \sum _{k}\lambda _{k}=1}$.

The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations ${\displaystyle {\mathcal {N}}}$ on density matrices,

${\displaystyle S({\mathcal {N}}(\rho )\|{\mathcal {N}}(\sigma ))\leq S(\rho \|\sigma )}$.

This inequality is called Monotonicity of quantum relative entropy. Owing to the Stinespring factorization theorem, this inequality is a consequence of a particular choice of the CPTP map - a partial trace map described below.

The most important and basic class of CPTP maps is a partial trace operation ${\displaystyle T:{\mathcal {B}}({\mathcal {H}}^{12})\rightarrow {\mathcal {B}}({\mathcal {H}}^{1})}$, given by ${\displaystyle T=1_{{\mathcal {H}}^{1}}\otimes \mathrm {Tr} _{{\mathcal {H}}^{2}}}$. Then

${\displaystyle S(T\rho ||T\sigma )\leq S(\rho ||\sigma ),}$

(3)

which is called Monotonicity of quantum relative entropy under partial trace.

To see how this follows from the joint convexity of relative entropy, observe that ${\displaystyle T}$ can be written in Uhlmann's representation as

${\displaystyle T(\rho ^{12})=N^{-1}\sum _{j=1}^{N}(1_{{\mathcal {H}}^{1}}\otimes U_{j})\rho ^{12}(1_{{\mathcal {H}}^{1}}\otimes U_{j}^{*}),}$

for some finite ${\displaystyle N}$ and some collection of unitary matrices on ${\displaystyle {\mathcal {H}}^{2}}$ (alternatively, integrate over Haar measure). Since the trace (and hence the relative entropy) is unitarily invariant, inequality (3) now follows from (2). This theorem is due to Lindblad [18] and Uhlmann,[17] whose proof is the one given here.

SSA is obtained from (3) with ${\displaystyle {\mathcal {H}}^{1}}$ replaced by ${\displaystyle {\mathcal {H}}^{12}}$ and ${\displaystyle {\mathcal {H}}^{2}}$ replaced ${\displaystyle {\mathcal {H}}^{3}}$. Take ${\displaystyle \rho =\rho ^{123},}$ ${\displaystyle \sigma =\rho ^{1}\otimes \rho ^{23},}$ ${\displaystyle T=1_{{\mathcal {H}}^{12}}\otimes Tr_{{\mathcal {H}}^{3}}}$. Then (3) becomes

${\displaystyle S(\rho ^{12}||\rho ^{1}\otimes \rho ^{2})\leq S(\rho ^{123}||\rho ^{1}\otimes \rho ^{23}).}$

Therefore,

${\displaystyle S(\rho ^{123}||\rho ^{1}\otimes \rho ^{23})-S(\rho ^{12}||\rho ^{1}\otimes \rho ^{2})=S(\rho ^{12})+S(\rho ^{23})-S(\rho ^{123})-S(\rho ^{2})\geq 0,}$

which is SSA. Thus, the monotonicity of quantum relative entropy (which follows from (1) implies SSA.

All of the above important inequalities are equivalent to each other, and can also be proved directly. The following are equivalent:

• Monotonicity of quantum relative entropy (MONO);
• Monotonicity of quantum relative entropy under partial trace (MPT);
• Strong subadditivity (SSA);
• Joint convexity of quantum relative entropy (JC);

The following implications show the equivalence between these inequalities.

• MONO ${\displaystyle \Rightarrow }$ MPT: follows since the MPT is a particular case of MONO;
• MPT ${\displaystyle \Rightarrow }$ MONO: was shown by Lindblad,[20] using a representation of stochastic maps as a partial trace over an auxiliary system;
• MPT ${\displaystyle \Rightarrow }$ SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
• SSA ${\displaystyle \Rightarrow }$ MPT: by choosing ${\displaystyle \rho _{123}}$ to be block diagonal, one can show that SSA implies that the map

${\displaystyle \rho _{12}\mapsto S(\rho _{1})-S(\rho _{12})}$ is convex. In [3] it was observed that this convexity yields MPT;

• MPT ${\displaystyle \Rightarrow }$ JC: as it was mentioned above, by choosing ${\displaystyle \rho _{12}}$ (and similarly, ${\displaystyle \sigma _{12}}$) to be block diagonal matrix with blocks ${\displaystyle \lambda _{k}\rho _{k}}$ (and ${\displaystyle \lambda _{k}\sigma _{k}}$), the partial trace is a sum over blocks so that ${\displaystyle \rho$ :=\rho _{2}=\sum _{k}\lambda _{k}\rho _{k}} , so from MPT one can obtain JC;
• JC ${\displaystyle \Rightarrow }$ SSA: using the 'purification process', Araki and Lieb,[9][21] observed that one could obtain new useful inequalities from the known ones. By purifying ${\displaystyle \rho _{123}}$ to ${\displaystyle \rho _{1234}}$ it can be shown that SSA is equivalent to

${\displaystyle S(\rho _{4})+S(\rho _{2})\leq S(\rho _{12})+S(\rho _{14}).}$

Moreover, if ${\displaystyle \rho _{124}}$ is pure, then ${\displaystyle S(\rho _{2})=S(\rho _{14})}$ and ${\displaystyle S(\rho _{4})=S(\rho _{12})}$, so the equality holds in the above inequality. Since the extreme points of the convex set of density matrices are pure states, SSA follows from JC;

See,[21][22] for a discussion.

E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality,[10] namely,

${\displaystyle S(\rho ^{12})+S(\rho ^{23})-S(\rho ^{123})-S(\rho ^{2})\geq 2\max\{0,S(\rho ^{1})-S(\rho ^{13}),S(\rho ^{3})-S(\rho ^{13})\}}$

If ${\displaystyle S(\rho ^{1})-S(\rho ^{13})\leq 0}$ and ${\displaystyle S(\rho ^{3})-S(\rho ^{13})\leq 0}$, as is always the case for the classical Shannon entropy, this inequality has nothing to say. For the quantum entropy, on the other hand, it is quite possible that the conditional entropies satisfy ${\displaystyle -S(\rho ^{13}|\rho ^{1})=S(\rho ^{1})-S(\rho ^{13})>0}$ or ${\displaystyle -S(\rho ^{13}|\rho ^{3})=S(\rho ^{3})-S(\rho ^{13})>0}$ (but never both!). Then, in this "highly quantum" regime, this inequality provides additional information.

The constant 2 is optimal, in the sense that for any constant larger than 2, one can find a state for which the inequality is violated with that constant.

In his paper [26] I. Kim studied an operator extension of strong subadditivity, proving the following inequality:

For a tri-partite state (density matrix) ${\displaystyle \rho ^{123}}$ on ${\displaystyle {\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}\otimes {\mathcal {H}}^{3}}$,

${\displaystyle Tr_{12}{\Bigl (}\rho ^{123}(-\log(\rho ^{12})-\log(\rho ^{23})+\log(\rho ^{2})+\log(\rho ^{123})){\Bigr )}\geq 0.}$

The proof of this inequality is based on Effros's theorem,[27] for which particular functions and operators are chosen to derive the inequality above. M. B. Ruskai describes this work in details in [28] and discusses how to prove a large class of new matrix inequalities in the tri-partite and bi-partite cases by taking a partial trace over all but one of the spaces.

A significant strengthening of strong subadditivity was proved in 2014,[29] which was subsequently improved in [30] and.[31] In 2017,[32] it was shown that the recovery channel can be taken to be the original Petz recovery map. These improvements of strong subadditivity have physical interpretations in terms of recoverability, meaning that if the conditional mutual information ${\displaystyle I(A;B|E)=S(AE)+S(BE)-S(E)-S(ABE)}$ of a tripartite quantum state ${\displaystyle \rho _{ABE}}$ is nearly equal to zero, then it is possible to perform a recovery channel ${\displaystyle {\mathcal {R}}_{E\to AE}}$ (from system E to AE) such that ${\displaystyle \rho _{ABE}\approx {\mathcal {R}}_{E\to AE}(\rho _{BE})}$. These results thus generalize the exact equality conditions mentioned above.

• Von Neumann entropy
• Conditional quantum entropy
• Quantum mutual information
• Kullback–Leibler divergence