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41-644: [REDACTED] Look up SSA in Wiktionary, the free dictionary. SSA may refer to: Geography [ edit ] Sub-Saharan Africa Organizations [ edit ] Sainsbury's Staff Association, of Sainsbury's , UK Scottish Socialist Alliance , a coalition of left-wing bodies, fore-runner to the Scottish Socialist Party Seismological Society of America , international scientific society founded1906 Shan State Army ,

82-750: A partial trace : ρ 12 = T r H 3 ρ 123 {\displaystyle \rho ^{12}={\rm {Tr}}_{{\mathcal {H}}^{3}}\rho ^{123}} . Similarly, we can define density matrices: ρ 23 {\displaystyle \rho ^{23}} , ρ 13 {\displaystyle \rho ^{13}} , ρ 1 {\displaystyle \rho ^{1}} , ρ 2 {\displaystyle \rho ^{2}} , ρ 3 {\displaystyle \rho ^{3}} . For any tri-partite state ρ 123 {\displaystyle \rho ^{123}}

123-590: A 2004 video game Shared services agreement Slippery slope argument , a rhetorical device (and often a fallacy) Special Service Agreement between the UN and a contractor Special services area or business improvement district SSA, Grand Cross of the Order of the Star of South Africa Supervisory Special Agent Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

164-451: A 2004 video game Shared services agreement Slippery slope argument , a rhetorical device (and often a fallacy) Special Service Agreement between the UN and a contractor Special services area or business improvement district SSA, Grand Cross of the Order of the Star of South Africa Supervisory Special Agent Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

205-534: A Massachusetts ferry service and regulatory body Sudan Studies Association , US professional association Swedish Society of Radio Amateurs , an amateur radio organization Education [ edit ] School of Saint Anthony , Quezon City, Philippines Secular Student Alliance , US Shady Side Academy , Pittsburgh, Pennsylvania, US Government [ edit ] Saudi Space Agency , Saudi government agency Senior Special Assistant , Nigeria government role Social Security Administration of

246-534: A Massachusetts ferry service and regulatory body Sudan Studies Association , US professional association Swedish Society of Radio Amateurs , an amateur radio organization Education [ edit ] School of Saint Anthony , Quezon City, Philippines Secular Student Alliance , US Shady Side Academy , Pittsburgh, Pennsylvania, US Government [ edit ] Saudi Space Agency , Saudi government agency Senior Special Assistant , Nigeria government role Social Security Administration of

287-567: A control character in the C1 control code set Static single-assignment form , a property of intermediate representations used in compilers Stationary Subspace Analysis Strong subadditivity of quantum entropy SubStation Alpha and .ssa file format, a video subtitle editor Super systolic array Science [ edit ] Semantic structure analysis Single-strand annealing in homologous recombination Side-side-angle in geometry for solving triangles Specific surface area ,

328-523: A control character in the C1 control code set Static single-assignment form , a property of intermediate representations used in compilers Stationary Subspace Analysis Strong subadditivity of quantum entropy SubStation Alpha and .ssa file format, a video subtitle editor Super systolic array Science [ edit ] Semantic structure analysis Single-strand annealing in homologous recombination Side-side-angle in geometry for solving triangles Specific surface area ,

369-489: A density matrix ρ {\displaystyle \rho } is Umegaki's quantum relative entropy of two density matrices ρ {\displaystyle \rho } and σ {\displaystyle \sigma } is A function g {\displaystyle g} of two variables is said to be jointly concave if for any 0 ≤ λ ≤ 1 {\displaystyle 0\leq \lambda \leq 1}

410-437: A different definition of entropy, which was generalized by Freeman Dyson . The Wigner–Yanase–Dyson p {\displaystyle p} -skew information of a density matrix ρ {\displaystyle \rho } . with respect to an operator K {\displaystyle K} is where [ A , B ] = A B − B A {\displaystyle [A,B]=AB-BA}

451-813: A former insurgent group in Burma Shipconstructors' and Shipwrights' Association , a former British trade union Singapore Scout Association , youth movement founded 1910 Slovak Society of Actuaries (Slovak: Slovenská spoločnosť aktuárov ), professional association in Slovakia Soaring Society of America , American sporting society founded in 1932 Society for the Study of Addiction , UK learned society with charitable status Society of Scottish Artists , artists society founded in 1891 SSA Global Technologies , American software company acquired by Infor Global Solutions Steamship Authority ,

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492-633: A former insurgent group in Burma Shipconstructors' and Shipwrights' Association , a former British trade union Singapore Scout Association , youth movement founded 1910 Slovak Society of Actuaries (Slovak: Slovenská spoločnosť aktuárov ), professional association in Slovakia Soaring Society of America , American sporting society founded in 1932 Society for the Study of Addiction , UK learned society with charitable status Society of Scottish Artists , artists society founded in 1891 SSA Global Technologies , American software company acquired by Infor Global Solutions Steamship Authority ,

533-401: A function of a density matrix ρ {\displaystyle \rho } for a fixed 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} . Since the term − 1 2 T r ρ K K ∗ {\displaystyle -{\tfrac {1}{2}}{\rm {Tr}}\rho KK^{*}} is concave (it

574-561: A property of solids Medicine [ edit ] Anti-SSA/Ro autoantibodies Senile systemic amyloidosis Sessile serrated adenoma , a type of pre-malignant intestinal polyp Special somatic afferent Sulfosalicylic acid Other uses [ edit ] IATA code of Salvador Bahia Airport ESA Space Situational Awareness Programme Sarva Shiksha Abhiyan , the Government of India's Education for All programme Self-sampling assumption Serious Sam Advance ,

615-508: A property of solids Medicine [ edit ] Anti-SSA/Ro autoantibodies Senile systemic amyloidosis Sessile serrated adenoma , a type of pre-malignant intestinal polyp Special somatic afferent Sulfosalicylic acid Other uses [ edit ] IATA code of Salvador Bahia Airport ESA Space Situational Awareness Programme Sarva Shiksha Abhiyan , the Government of India's Education for All programme Self-sampling assumption Serious Sam Advance ,

656-526: 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: The following implications show the equivalence between these inequalities. ρ 12 ↦ S ( ρ 1 ) − S ( ρ 12 ) {\displaystyle \rho _{12}\mapsto S(\rho _{1})-S(\rho _{12})}

697-401: Is a commutator, K ∗ {\displaystyle K^{*}} is the adjoint of K {\displaystyle K} and 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} is fixed. It was conjectured by E. P. Wigner and M. M. Yanase in that p {\displaystyle p} - skew information is concave as

738-546: 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 {\displaystyle T} can be written in Uhlmann's representation as for some finite N {\displaystyle N} and some collection of unitary matrices on H 2 {\displaystyle {\mathcal {H}}^{2}} (alternatively, integrate over Haar measure ). Since

779-879: 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 T : B ( H 12 ) → B ( H 1 ) {\displaystyle T:{\mathcal {B}}({\mathcal {H}}^{12})\rightarrow {\mathcal {B}}({\mathcal {H}}^{1})} , given by T = 1 H 1 ⊗ T r H 2 {\displaystyle T=1_{{\mathcal {H}}^{1}}\otimes \mathrm {Tr} _{{\mathcal {H}}^{2}}} . Then which

820-487: Is convex. In it was observed that this convexity yields MPT; Moreover, if ρ 124 {\displaystyle \rho _{124}} is pure, then S ( ρ 2 ) = S ( ρ 14 ) {\displaystyle S(\rho _{2})=S(\rho _{14})} and S ( ρ 4 ) = S ( ρ 12 ) {\displaystyle S(\rho _{4})=S(\rho _{12})} , so

861-565: Is denoted by H {\displaystyle {\mathcal {H}}} , and B ( H ) {\displaystyle {\mathcal {B}}({\mathcal {H}})} denotes the bounded linear operators on H {\displaystyle {\mathcal {H}}} . Tensor products are denoted by superscripts, e.g., H 12 = H 1 ⊗ H 2 {\displaystyle {\mathcal {H}}^{12}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}} . The trace

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902-540: Is denoted by T r {\displaystyle {\rm {Tr}}} . A density matrix is a Hermitian , positive semi-definite matrix of trace one. It allows for the description of a quantum system in a mixed state . Density matrices on a tensor product are denoted by superscripts, e.g., ρ 12 {\displaystyle \rho ^{12}} is a density matrix on H 12 {\displaystyle {\mathcal {H}}^{12}} . The von Neumann quantum entropy of

943-622: Is linear), the conjecture reduces to the problem of concavity of T r ρ p K ∗ ρ 1 − p K {\displaystyle Tr\rho ^{p}K^{*}\rho ^{1-p}K} . As noted in, this conjecture (for all 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} ) implies SSA, and was proved for p = 1 2 {\displaystyle p={\tfrac {1}{2}}} in, and for all 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} in in

984-632: Is the Araki–Lieb triangle inequality which is derived in from subadditivity by a mathematical technique known as purification . Suppose that the Hilbert space of the system is a tensor product of three spaces: H = H 1 ⊗ H 2 ⊗ H 3 . {\displaystyle {\mathcal {H}}={\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}\otimes {\mathcal {H}}^{3}.} . Physically, these three spaces can be interpreted as

1025-1150: The joint convexity of relative entropy : i.e., if ρ = ∑ k λ k ρ k {\displaystyle \rho =\sum _{k}\lambda _{k}\rho _{k}} , and σ = ∑ k λ k σ k {\displaystyle \sigma =\sum _{k}\lambda _{k}\sigma _{k}} , then where λ k ≥ 0 {\displaystyle \lambda _{k}\geq 0} with ∑ k λ k = 1 {\displaystyle \sum _{k}\lambda _{k}=1} . The relative entropy decreases monotonically under completely positive trace preserving (CPTP) operations N {\displaystyle {\mathcal {N}}} on density matrices, S ( N ( ρ ) ‖ N ( σ ) ) ≤ S ( ρ ‖ σ ) {\displaystyle S({\mathcal {N}}(\rho )\|{\mathcal {N}}(\sigma ))\leq S(\rho \|\sigma )} . This inequality

1066-674: The US government Social Security Agency (Northern Ireland) Selective Service Act of 1917 , an American piece of legislation signed by President Woodrow Wilson during WWI that established nationwide conscription State Security Agency (South Africa) , the South African intelligence service Computing [ edit ] Stochastic Simulation Algorithm Serial Storage Architecture Singular Spectrum Analysis Software Security Assurance Solid State Array, in flash data storage using solid-state drives Start of Selected Area,

1107-573: The US government Social Security Agency (Northern Ireland) Selective Service Act of 1917 , an American piece of legislation signed by President Woodrow Wilson during WWI that established nationwide conscription State Security Agency (South Africa) , the South African intelligence service Computing [ edit ] Stochastic Simulation Algorithm Serial Storage Architecture Singular Spectrum Analysis Software Security Assurance Solid State Array, in flash data storage using solid-state drives Start of Selected Area,

1148-429: The classical bound of the marginal entropy. The strong subadditivity inequality was improved in the following way by Carlen and Lieb with the optimal constant 2 {\displaystyle 2} . J. Kiefer proved a peripherally related convexity result in 1959, which is a corollary of an operator Schwarz inequality proved by E.H.Lieb and M.B.Ruskai. However, these results are comparatively simple, and

1189-580: The equivalence between the second statement below and SSA. Both of these statements were proved directly in. As noted by Lindblad and Uhlmann, if, in equation ( 1 ), one takes K = 1 {\displaystyle K=1} and r = 1 − p , A = ρ {\displaystyle r=1-p,A=\rho } and B = σ {\displaystyle B=\sigma } and differentiates in p {\displaystyle p} at p = 0 {\displaystyle p=0} , one obtains

1230-983: The following holds Ordinary subadditivity concerns only two spaces H 12 {\displaystyle {\mathcal {H}}^{12}} and a density matrix ρ 12 {\displaystyle \rho ^{12}} . It states that This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies S ( ρ 12 | ρ 1 ) = S ( ρ 12 ) − S ( ρ 1 ) {\displaystyle S(\rho ^{12}|\rho ^{1})=S(\rho ^{12})-S(\rho ^{1})} and S ( ρ 12 | ρ 2 ) = S ( ρ 12 ) − S ( ρ 2 ) {\displaystyle S(\rho ^{12}|\rho ^{2})=S(\rho ^{12})-S(\rho ^{2})} are both non-negative. In

1271-770: The following holds where S ( ρ 12 ) = − T r H 12 ρ 12 log ⁡ ρ 12 {\displaystyle S(\rho ^{12})=-{\rm {Tr}}_{{\mathcal {H}}^{12}}\rho ^{12}\log \rho ^{12}} , for example. Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state ρ A B C {\displaystyle \rho ^{ABC}} , This can also be restated in terms of quantum mutual information , These statements run parallel to classical intuition, except that quantum conditional entropies can be negative, and quantum mutual informations can exceed

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1312-422: The following more general form: The function of two matrix variables is jointly concave in A {\displaystyle A} and B , {\displaystyle B,} when 0 ≤ r ≤ 1 {\displaystyle 0\leq r\leq 1} and p + r ≤ 1 {\displaystyle p+r\leq 1} . This theorem is an essential part of

1353-486: The 💕 [REDACTED] Look up SSA in Wiktionary, the free dictionary. SSA may refer to: Geography [ edit ] Sub-Saharan Africa Organizations [ edit ] Sainsbury's Staff Association, of Sainsbury's , UK Scottish Socialist Alliance , a coalition of left-wing bodies, fore-runner to the Scottish Socialist Party Seismological Society of America , international scientific society founded1906 Shan State Army ,

1394-429: The proof of SSA in. In their paper E. P. Wigner and M. M. Yanase also conjectured the subadditivity of p {\displaystyle p} -skew information for p = 1 2 {\displaystyle p={\tfrac {1}{2}}} , which was disproved by Hansen by giving a counterexample. It was pointed out in that the first statement below is equivalent to SSA and A. Ulhmann in showed

1435-514: The proofs do not use the results of Lieb's 1973 paper on convex and concave trace functionals. It was this paper that provided the mathematical basis of the proof of SSA by Lieb and Ruskai. 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 . The theorem can also be obtained by proving numerous equivalent statements, some of which are summarized below. E. P. Wigner and M. M. Yanase proposed

1476-708: The quantum case, however, both can be negative, e.g. S ( ρ 12 ) {\displaystyle S(\rho ^{12})} can be zero while S ( ρ 1 ) = S ( ρ 2 ) > 0 {\displaystyle S(\rho ^{1})=S(\rho ^{2})>0} . Nevertheless, the subadditivity upper bound on S ( ρ 12 ) {\displaystyle S(\rho ^{12})} continues to hold. The closest thing one has to S ( ρ 12 ) − S ( ρ 1 ) ≥ 0 {\displaystyle S(\rho ^{12})-S(\rho ^{1})\geq 0}

1517-546: The space of three different systems, or else as three parts or three degrees of freedom of one physical system. Given a density matrix ρ 123 {\displaystyle \rho ^{123}} on H {\displaystyle {\mathcal {H}}} , we define a density matrix ρ 12 {\displaystyle \rho ^{12}} on H 1 ⊗ H 2 {\displaystyle {\mathcal {H}}^{1}\otimes {\mathcal {H}}^{2}} as

1558-489: The title SSA . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SSA&oldid=1256751300 " Category : Disambiguation pages Hidden categories: Articles containing Slovak-language text Short description is different from Wikidata All article disambiguation pages All disambiguation pages SSA From Misplaced Pages,

1599-537: The title SSA . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=SSA&oldid=1256751300 " Category : Disambiguation pages Hidden categories: Articles containing Slovak-language text Short description is different from Wikidata All article disambiguation pages All disambiguation pages Strong subadditivity of quantum entropy The classical version of SSA

1640-1126: The trace (and hence the relative entropy) is unitarily invariant, inequality ( 3 ) now follows from ( 2 ). This theorem is due to Lindblad and Uhlmann, whose proof is the one given here. SSA is obtained from ( 3 ) with H 1 {\displaystyle {\mathcal {H}}^{1}} replaced by H 12 {\displaystyle {\mathcal {H}}^{12}} and H 2 {\displaystyle {\mathcal {H}}^{2}} replaced H 3 {\displaystyle {\mathcal {H}}^{3}} . Take ρ = ρ 123 , {\displaystyle \rho =\rho ^{123},} σ = ρ 1 ⊗ ρ 23 , {\displaystyle \sigma =\rho ^{1}\otimes \rho ^{23},} T = 1 H 12 ⊗ T r H 3 {\displaystyle T=1_{{\mathcal {H}}^{12}}\otimes Tr_{{\mathcal {H}}^{3}}} . Then ( 3 ) becomes Therefore, which

1681-400: Was long known and appreciated in classical probability theory and information theory. 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 reduced density matrices describing the quantum subsystems. Some useful references here include: We use the following notation throughout the following: A Hilbert space

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