The inequalities of quantum information theory
The inequalities of quantum information theory
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量子信息论的不等式
DOI：
10.1109/tit.2003.809569
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
N. Pippenger
中科院分区：
文献类型：
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作者：
N. Pippenger
令f spl ρ f表示具有n个部分1，.，n.对于I/spl sube/N={1，...，n}，令/spl rho//sub I/=Tr/sub N/spl bsol/I/（/spl rho/）表示包括使得i/spl isin/I的那些部分i的状态的密度矩阵，并且令S（/spl rho//sub I/）表示状态/spl rho//sub I/的冯诺依曼（1927）熵。/spl nu/=2/sup n/ numbers {S（/spl rho//sub I/）}/sub I/spl sube/N/的集合可以被认为是向量空间R/sup /spl nu//中的一个点，称为/spl rho/的熵分配。设A/sub n/表示R/sup /spl nu//中的点的集合，这些点是n部分量子态的熵的分配。证明了A/sub n/（A/sub n/的拓扑闭包）是R/sup /spl nu//中的闭凸锥.这意味着一个点作为熵分配的近似可解性由它所满足的线性不等式决定。Lieb和Ruskai（1973）建立了多体量子态（强次可加性和弱单调性）的若干不等式。我们给出了这些不等式的一个有限的例子集，它是完整的（在这个意义上，任何有效的熵分配的线性不等式都可以通过正线性组合从它们推导出来）和独立的（在这个意义上，它们中没有一个可以通过正线性组合从其他不等式推导出来）。设B/sub n/表示由这些不等式确定的R/sup /spl nu//中的多面体锥。我们证明了对于n/spl les/3，A~/sub n/~=B/sub n/.此相等的状态为开放n/spl ges/4。我们还考虑这种情况的对称版本，其中S（/spl rho//sub I/）仅通过I中索引的数量i=/spl ne/I依赖于I，因此可以表示为S（/spl rho//sub i/）。在这种情况下，我们给每个n一个有限的完整的和独立的不等式集控制的对称分配的熵{S（/spl rho//sub i/）}/sub 0/spl les/i/spl les/n/在R/sup n+1/。
Let /spl rho/ denote the density matrix of a quantum state having n parts 1, ..., n. For I/spl sube/N={1, ..., n}, let /spl rho//sub I/=Tr/sub N/spl bsol/I/(/spl rho/) denote the density matrix of the state comprising those parts i such that i/spl isin/I, and let S(/spl rho//sub I/) denote the von Neumann (1927) entropy of the state /spl rho//sub I/. The collection of /spl nu/=2/sup n/ numbers {S(/spl rho//sub I/)}/sub I/spl sube/N/ may be regarded as a point, called the allocation of entropy for /spl rho/, in the vector space R/sup /spl nu//. Let A/sub n/ denote the set of points in R/sup /spl nu// that are allocations of entropy for n-part quantum states. We show that A~/sub n/~ (the topological closure of A/sub n/) is a closed convex cone in R/sup /spl nu//. This implies that the approximate achievability of a point as an allocation of entropy is determined by the linear inequalities that it satisfies. Lieb and Ruskai (1973) have established a number of inequalities for multipartite quantum states (strong subadditivity and weak monotonicity). We give a finite set of instances of these inequalities that is complete (in the sense that any valid linear inequality for allocations of entropy can be deduced from them by taking positive linear combinations) and independent (in the sense that none of them can be deduced from the others by taking positive linear combinations). Let B/sub n/ denote the polyhedral cone in R/sup /spl nu// determined by these inequalities. We show that A~/sub n/~=B/sub n/ for n/spl les/3. The status of this equality is open for n/spl ges/4. We also consider a symmetric version of this situation, in which S(/spl rho//sub I/) depends on I only through the number i=/spl ne/I of indexes in I and can thus be denoted S(/spl rho//sub i/). In this case, we give for each n a finite complete and independent set of inequalities governing the symmetric allocations of entropy {S(/spl rho//sub i/)}/sub 0/spl les/i/spl les/n/ in R/sup n+1/.