Strong Subadditivity of Quantum Entropy
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Strong subadditivity of entropy (SSA) was long known and appreciated in classical probability theory and information theory. Its extension to quantum mechanical entropy (the von Neumann entropy) was conjectured by Robinson and Ruelle (1966) and Lanford III and Robinson (1968) and proved by Lieb and Ruskai (1973). It is a basic theorem in modern quantum information theory.
SSA concerns the relation between the entropies of various subsystems of a larger system consisting of three subsystems (or of one system with three degrees of freedom). The proof of this relation in the classical case is quite easy but the quantum case is difficult because of the non-commutativity of the density matrices describing the subsystems.
Definitions
We will use the following notation throughout: A Hilbert space is denoted by \(\mathcal{H}\), and \( \mathcal{B}(\mathcal{H})\) denotes the bounded linear operators on \(\mathcal{H}\). Tensor products are denoted by superscripts, e.g., \(\mathcal{H}^{12}=\mathcal{H}^1\otimes \mathcal{H}^2\). The trace is denoted by \({\rm Tr}\).
Density matrix
A density matrix is a Hermitian, positive semi-definite matrix of trace one. It describes a quantum system in a mixed state. Density matrices on a tensor product are denoted by superscripts, e.g., \(\rho^{12}\) is a density matrix on \(\mathcal{H}^{12}\).
Entropy
The von Neumann quantum entropy of a density matrix \(\rho\) is \[S(\rho):=-{\rm Tr}(\rho\log \rho).\]
Relative entropy
Umegaki's quantum relative entropy of two density matrices \(\rho\) and \(\sigma\) is \[S(\rho||\sigma)={\rm Tr}(\rho\log\rho-\rho\log\sigma)\geq 0. \]
Joint concavity
A function \(g\) of two variables is said to be jointly concave if for any \( 0\leq \lambda\leq 1\) the following holds \[ g(\lambda A_1 + (1-\lambda)A_2,\lambda B_1 + (1-\lambda)B_2 ) \geq \lambda g(A_1, B_1) + (1 -\lambda)g(A_2, B_2). \]
Subadditivity of entropy
Ordinary subadditivity, see Araki and Lieb (1970), concerns only two spaces \(\mathcal{H}^{12}\) and a density matrix \(\rho^{12}\). It states that \[ S(\rho^{12}) \leq S(\rho^1) +S(\rho^2). \] This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies \( S(\rho^{12} | \rho^1)= S(\rho^{12} )-S(\rho^1)\) and \( S(\rho^{12} | \rho^2)=S(\rho^{12} ) -S(\rho^2)\) are both non-negative. In the quantum case, however, both can be negative, e.g. \( S(\rho^{12}) \) can be zero while \( S(\rho^1) = S(\rho^{12}) >0\). Nevertheless, the subadditivity upper bound on \( S(\rho^{12}) \) continues to hold. The closest thing one has to \( S(\rho^{12})- S(\rho^1)\geq 0 \) is the Araki–Lieb triangle inequality \[ S(\rho^{12}) \geq |S(\rho^1) -S(\rho^2)|, \] which is derived by Araki and Lieb (1970) from subadditivity by a mathematical technique known as 'purification'.
Strong subadditivity (SSA)
Suppose that the Hilbert space of the system is a tensor product of three spaces\[\mathcal{H}=\mathcal{H}^1\otimes \mathcal{H}^2\otimes \mathcal{H}^3.\] 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 \(\rho^{123}\) on \(\mathcal{H}\), we define a density matrix \(\rho^{12}\) on \(\mathcal{H}^1\otimes \mathcal{H}^2\) as a partial trace \(\rho^{12}={\rm Tr}_{\mathcal{H}^3} \rho^{123}\). Similarly, we can define density matrices \(\rho^{23}\), \(\rho^{13}\), \(\rho^1\), \(\rho^2\), \(\rho^3\).
Statement
For any tri-partite state \(\rho^{123}\) the following holds \[S(\rho^{123})+S(\rho^2)\leq S(\rho^{12})+S(\rho^{23}),\] where \( S(\rho^{12})=-{\rm Tr}_{\mathcal{H}^{12}} \rho^{12} \log \rho^{12}\), for example.
This was improved in the following way by Carlen and Lieb (2012) \[S(\rho^{12})+S(\rho^{23})-S(\rho^{123})-S(\rho^2) \geq 2\max\{S(\rho^1)-S(\rho^{12}),S(\rho^2)-S(\rho^{12}),0 )\}, \] with the optimal constant \(2\).
As mentioned above, SSA was first proved by Lieb and Ruskai (1973), using Lieb's theorem that was proved by Lieb (1973). The extension from a Hilbert space setting to a von Neumann algebra setting, where states are not given by density matrices, was done by Narnhofer and Thirring (1975).
The theorem can also be obtained by proving numerous equivalent statements, some of which are summarized below.
Wigner–Yanase–Dyson conjecture
Wigner and Yanase (1963) proposed a different definition of entropy, which was generalized by F.J. Dyson.
The Wigner–Yanase–Dyson p-skew information
The Wigner–Yanase–Dyson \(p\)-skew information of a density matrix \(\rho\). with respect to an operator \(K\) is \[ I_p(\rho, K)=\frac{1}{2}{\rm Tr}[\rho^p, K^*][\rho^{1-p}, K],\] where \([A,B]=AB-BA\) is a commutator, \( K^* \) is the adjoint of \(K\) and \(0\leq p\leq 1\) is fixed.
Concavity of p-skew information
It was conjectured by Wigner and Yanase (1964) that \(p\)- skew information is concave as a function of a density matrix \(\rho\) for a fixed \(0\leq p\leq 1\).
Since the term \(-\tfrac{1}{2}{\rm Tr}\rho K^2\) is concave (it is linear), the conjecture reduces to the problem of concavity of \(Tr\rho^p K\rho^{1-p}K\). As noted in (Lieb, 1973), this conjecture (for all \( 0 \leq p \leq 1\)) implies SSA, and was proved for \( p= \tfrac{1}{2}\) in (Wigner, Yanase, 1964), and for all \( 0\leq p \leq 1 \) in (Lieb, 1973) in the following more general form: The function of two matrix variables \[\tag{1} A, B \mapsto {\rm Tr} A^{r}K^*B^pK \] is jointly concave in \( A\) and \( B,\) when \(0\leq r\leq 1\) and \(p+r \leq 1\).
This theorem is an essential part of the proof of SSA in (Lieb, Ruskai, 1973).
In their paper Wigner and Yanase (1964) also conjectured the subadditivity of \(p\)-skew information for \(p=\tfrac{1}{2}\), which was disproved by Hansen (2007) by giving a counterexample.
First two statements equivalent to SSA
It was pointed out in (Araki, Lieb, 1970) that the first statement below is equivalent to SSA and Ulhmann (1973) showed the equivalence between the second statement below and SSA.
- \( S(\rho^1)+S(\rho^3)-S(\rho^{12})-S(\rho^{23})\leq 0.\) Note that the conditional entropies \(S(\rho^{12}|\rho^1)\) and \(S(\rho^{23}|\rho^3)\) do not have to be both non-negative.
- The map \( \rho^{12}\mapsto S(\rho^1)-S(\rho^{12}) \) is convex.
Both of these statements were proved directly by Lieb and Ruskai (1973).
Joint convexity of relative entropy
As noted by Lindblad (1974) and Uhlmann (1977), if, in equation (1), one takes \( K=1\) and \( r=1-p, A=\rho\) and \(B=\sigma\) and differentiates in \( p\) at \(p=0\) one obtains the Joint convexity of relative entropy : i.e., if \(\rho=\sum_k\lambda_k\rho_k\), and \(\sigma=\sum_k\lambda_k\sigma_k\), then \[ \tag{2} S\Bigl(\sum_k \lambda_k\rho_k||\sum_k\lambda_k \sigma_k \Bigr)\leq \sum_k\lambda_k S(\rho_k||\sigma_k),\] where \(\lambda_k\geq 0\) with \(\sum_k\lambda_k=1\).
Monotonicity of quantum relative entropy
The relative entropy decreases monotonically under certain operations on density matrices, the most important and basic of which is the following. Consider the map \(T\) from \( \mathcal{B}(\mathcal{H}^{12}) \rightarrow \mathcal{B}(\mathcal{H}^{12})\) given by \(T=1_{\mathcal{H}^1}\otimes Tr_{\mathcal{H}^2}\) . Then
\[ \tag{3} S(T\rho||T\sigma)\leq S(\rho||\sigma),\]
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 \( T\) can be written in Uhlmann's representation as \[ T(\rho^{12} ) = N^{-1} \sum_{j=1}^N (1_{\mathcal{H}^1}\otimes U_j) \rho^{12}(1_{\mathcal{H}^1}\otimes U_j^*), \] for some finite \( N\) and some collection of unitary matrices on \( \mathcal{H}^2 \) (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 (1974) and Ulhmann (1973), whose proof is the one given here.
SSA is obtained from (3) with \( \mathcal{H}^1 \) replaced by \( \mathcal{H}^{12} \) and \( \mathcal{H}^2 \) replaced \( \mathcal{H}^3 \). Take \( \rho = \rho^{123}, \sigma = \rho^1\otimes \rho^{23}, T= 1_{\mathcal{H}^{12}}\otimes Tr_{\mathcal{H}^3}\). Then (3) becomes \[ S(\rho^{12}||\rho^1\otimes \rho^2)\leq S(\rho^{123}||\rho^1\otimes\rho^{23}).\]
Therefore, \[S(\rho^{123}||\rho^1\otimes\rho^{23})- S(\rho^{12}||\rho^1\otimes \rho^2)=S(\rho^{12})+S(\rho^{23})-S(\rho^{123})-S(\rho^2)\geq 0, \] which is SSA. Thus, the monotonicity of quantum relative entropy (which follows from (1)) implies SSA.
Owing to the Stinespring factorization theorem, equation (3) is valid not only for partial traces but also when \(T\) is a quantum operation, i.e., a completely positive, trace preserving map. In this general case the inequality is called Monotonicity of quantum relative entropy.
Relationship among inequalities
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 \(\Rightarrow \) MPT: follows since the MPT is a particular case of MONO;
- MPT \(\Rightarrow \) MONO: was showed by Lindblad (1975), using a representation of stochastic maps as a partial trace over an auxiliary system;
- MPT \(\Rightarrow\) SSA: follows by taking a particular choice of tri-partite states in MPT, described in the section above, "Monotonicity of quantum relative entropy";
- SSA \(\Rightarrow\) MPT: by choosing \(\rho_{123}\) to be block diagonal, one can show that SSA implies that the map
\(\rho_{12}\mapsto S(\rho_1)-S(\rho_{12})\) is convex. In (Lieb, Ruskai, 1973) it was observed that this convexity yields MPT;
- MPT \(\Rightarrow\) JC: as it was mentioned above, by choosing \(\rho_{12}\) (and similarly, \(\sigma_{12}\)) to be block diagonal matrix with blocks \(\lambda_k\rho_k\) (and \(\lambda_k\sigma_k\)), the partial trace is a sum over blocks so that \(\rho:=\rho_2=\sum_k\lambda_k\rho_k\), so from MPT one can obtain JC;
- JC \(\Rightarrow\) SSA: using the 'purification process', Araki and Lieb (1970), Lieb (1975) observed that one could obtain new useful inequalities from the known ones. By purifying \(\rho_{123}\) to \(\rho_{1234}\) it can be shown that SSA is equivalent to
\[ S(\rho_4)+S(\rho_2)\leq S(\rho_{12})+S(\rho_{14}). \] Moreover, if \(\rho_{124}\) is pure, then \(S(\rho_2)=S(\rho_{14})\) and \(S(\rho_4)=S(\rho_{12})\), 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 (Lieb, 1975), (Ruskai, 2002) for a discussion.
The case of equality
Equality in monotonicity of quantum relative entropy inequality
In (Petz, 1986) and (Petz, 1986) D. Petz showed that the only case of equality in the monotonicity relation is to have a proper "recovery" channel:
For all states \(\rho\) and \(\sigma\) on a Hilbert space \(\mathcal{H}\) and all quantum operators \(T: \mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{K})\),
\[ S(T\rho||T\sigma)= S(\rho||\sigma), \]
if and only if there exists a quantum operator \(\hat{T}\) such that
\[ \hat{T}T\sigma=\sigma,\] and \(\hat{T}T\rho=\rho.\)
Moreover, \(\hat{T}\) can be given explicitly be the formula
\[ \hat{T}\omega=\sigma^{1/2}T^*\Bigl((T\sigma)^{-1/2}\omega(T\sigma)^{-1/2} \Bigr)\sigma^{1/2}, \]
where \(T^*\) is the adjoint map of \(T\).
Petz (1986) also gave another condition when the equality holds in Monotonicity of quantum relative entropy: the first statement in Theorem below. Differentiating it at \(t=0\) we have the second condition. Moreover, M.B. Ruskai gave another proof of the second statement.
For all states \(\rho\) and \(\sigma\) on \(\mathcal{H}\) and all quantum operators \(T: \mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{K})\),
\[ S(T\rho||T\sigma)= S(\rho||\sigma),\]
if and only if the following equivalent conditions are satisfied:
- \( T^*(T(\rho)^{it}T(\sigma)^{it})=\rho^{it}\sigma^{-it}\) for all real \(t\).
- \( \log\rho-\log\sigma=T^*\Bigl(\log T(\rho)-\log T(\sigma) \Bigr).\)
where \(T^*\) is the adjoint map of \(T\).
Equality in strong subadditivity inequality
Hayden et all (2003) described the states for which the equality holds in SSA.
A state \(\rho^{ABC}\) on a Hilbert space \(\mathcal{H}^A\otimes\mathcal{H}^B\otimes\mathcal{H}^C\) satisfies strong subadditivity with equality if and only if there is a decomposition of second system as \[ \mathcal{H}^B=\bigoplus_j \mathcal{H}^{B^L_j}\otimes \mathcal{H}^{B^R_j} \] into a direct sum of tensor products, such that \[ \rho^{ABC}=\bigoplus_j q_j\rho^{AB^L_j}\otimes\rho^{B^R_jC},\] with states \(\rho^{AB^L_j}\) on \(\mathcal{H}^A\otimes\mathcal{H}^{B^L_j}\) and \(\rho^{B^R_jC}\) on \(\mathcal{H}^{B^R_j}\otimes\mathcal{H}^C\), and a probability distribution \(\{q_j\}\).
Carlen-Lieb Extension
E. H. Lieb and E.A. Carlen have found an explicit error term in the SSA inequality Carlen, Lieb (2012) , namely, \[ S(\rho^{12})+S(\rho^{23})-S(\rho^{123})-S(\rho^2)\geq 2\min\{0, S(\rho^1)-S(\rho^{12}), S(\rho^3)-S(\rho^{23})\}.\] If \(-S(\rho^{12}\|\rho^1)=S(\rho^1)-S(\rho^{12})\leq 0 \) and \(-S(\rho^{23}\|\rho^3)=S(\rho^3)-S(\rho^{23})\leq 0 \), 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 \(-S(\rho^{12}\|\rho^1)>0 \) and \(-S(\rho^{23}\|\rho^3)> 0 \) (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.
Operator extension of strong subadditivity
In his paper Kim (2012) studied an operator extension of strong subadditivity, proving the following inequality:
For a tri-partite state (density matrix) \(\rho^{123}\) on \(\mathcal{H}^1\otimes \mathcal{H}^2\otimes\mathcal{H}^3\), \[ Tr_{12}\Bigl(\rho^{123}(-\log(\rho^{12})-\log(\rho^{23})+\log(\rho^2)+\log(\rho^{123}))\Bigr) \geq 0.\]
The proof of this inequality is based on Effros's theorem, see (Effros, 2009), for which particular functions and operators are chosen to derive the inequality above. Ruskai (2012) describes this work in details 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.
References
- Araki(1970). Entropy Inequalities. Commun. Math. Phys. 18: 160–170. doi:10.1007/bf01646092.
- Carlen(2012). Bounds for Entanglement via an Extension of Strong Subadditivity of Entropy. Lett. Math. Phys. 101 (1): 1-11. doi:10.1007/s11005-012-0565-6.
- Effros, E. (2009). A Matrix Convexity Approach to Some Celebrated Quantum Inequalities Proc. Natl. Acad. Sci. USA 106 (4): 1006–1008. doi:10.1073/pnas.0807965106.
- Hansen, F. (2007). The Wigner-Yanase Entropy is Not Subadditive. J. Stat. Phys. 126: 643–648. doi:10.1007/s10955-006-9265-x.
- Hayden, P.; Jozsa, R.; Petz, D. and winter, A. (2003). Structure of States which Satisfy Strong Subadditivity of Quantum Entropy with Equality. Comm. Math. Phys. 246: 359–374. doi:10.1007/s00220-004-1049-z.
- Kim, I. (2012). Operator Extension of Strong Subadditivity of Entropy : . [http://arxiv.org/abs/1210.5190 arXiv:1210.5190
- Lanford III(1968). Mean Entropy of States in Classical Statistical Mechanics. Jour. Math. Phys. 9: 1120.
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- Lieb, E. H. (1975). Some Convexity and Subadditivity Properties of Entropy. Bull. AMS 81: 1–13. doi:10.1090/s0002-9904-1975-13621-4.
- Lindblad, G. (1974). Expectations and Entropy Inequalities for Finite Quantum Systems. Commun. Math. Phys. 39: 111–119. doi:10.1007/bf01608390.
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- Narnhofer(1985). From Relative Entropy to Entropy. Fizika 17: 258–262.
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- Ruskai, M. B. (2002). Inequalities for Quantum Entropy: A Review With Conditions for Equality. J. Math. Phys. 43(9): 4358–4375. doi:10.1063/1.1497701.
- Ruskai, M. B. (2012). Remarks on Kim’s Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions. : . [http://arxiv.org/abs/1211.0049 arXiv:1211.0049
- Uhlmann, A. (1973). Endlich Dimensionale Dichtmatrizen, II. Wiss. Z. Karl-Marx-University Leipzig 22 Jg. H. 2. 139: .
- Uhlmann, A. (1977). Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory Comm. Math. Phys. 54: 21–32. doi:10.1007/bf01609834.
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Recommended reading
- Carlen, E. (2009). Trace Inequalities and Quantum Entropy: An Introductory Course. Contemp. Math. 529: .
- Ruskai, M. B. (2002). Inequalities for Quantum Entropy: A Review With Conditions for Equality. J. Math. Phys. 43(9): 4358–4375. doi:10.1063/1.1497701.
- Nielsen, M. (2000). Quantum Computation and Quantum Information , Cambr. U. Press.
- Ohya, M. and Petz, D. (1993). Quantum Entropy and Its Use , Springer.
