QUANTUM NON-EQUILIBRIUM STEADY-STATES

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1 S P I N o f f Q u B I T spinoffqubit.info THERMODYNAMICS AND INFORMATION IN QUANTUM NON-EQUILIBRIUM STEADY-STATES Gabriel T. Landi Instituto de Física da Universidade de São Paulo Queen s University in Belfast Trinity College, Dublin December, 2018

2 <latexit sha1_base64="ijy93rae3wxabnemvyoumzf/lqc=">aaacfnicbzdlssnafiynxmu9rv2kmfgen5zebf0ofnx0wcfesillmj1ph84kywyiljcvl+erunw9o3hr1q1p4vsy0nyfbn7+cw7nzbckncntof/wwuls8spqya24vrg5tw3v7dzuneokdrrzwlycooczcoqaaq6trairaydmmlge1zspibwlo1s9tmaxpbexkfgitdsxdzzzj/ev9kjjaab32ykxgca4mufzxd6xs07zgqvpg3dqsmiqwsf+9roxtqvemnkivnt1eu1nrgpgoerfl1wqedogpwgbgxebys/g38jxkum6oiylezhg4/t3reaeukmrme5bdf/n1kbhv7vazgzw4ywfsshjnur0sjhmodyxhjhcxsabaj40hldjzo2y9okbpa3joohizikyn43tsuuu3zuzuuvyiqea9tehokyuokcvveu1vecupajn9ijersfrzxq3piatc9z0zg/9kfx5a1cynxk=</latexit> <latexit sha1_base64="ijy93rae3wxabnemvyoumzf/lqc=">aaacfnicbzdlssnafiynxmu9rv2kmfgen5zebf0ofnx0wcfesillmj1ph84kywyiljcvl+erunw9o3hr1q1p4vsy0nyfbn7+cw7nzbckncntof/wwuls8spqya24vrg5tw3v7dzuneokdrrzwlycooczcoqaaq6trairaydmmlge1zspibwlo1s9tmaxpbexkfgitdsxdzzzj/ev9kjjaab32ykxgca4mufzxd6xs07zgqvpg3dqsmiqwsf+9roxtqvemnkivnt1eu1nrgpgoerfl1wqedogpwgbgxebys/g38jxkum6oiylezhg4/t3reaeukmrme5bdf/n1kbhv7vazgzw4ywfsshjnur0sjhmodyxhjhcxsabaj40hldjzo2y9okbpa3joohizikyn43tsuuu3zuzuuvyiqea9tehokyuokcvveu1vecupajn9ijersfrzxq3piatc9z0zg/9kfx5a1cynxk=</latexit> <latexit sha1_base64="ijy93rae3wxabnemvyoumzf/lqc=">aaacfnicbzdlssnafiynxmu9rv2kmfgen5zebf0ofnx0wcfesillmj1ph84kywyiljcvl+erunw9o3hr1q1p4vsy0nyfbn7+cw7nzbckncntof/wwuls8spqya24vrg5tw3v7dzuneokdrrzwlycooczcoqaaq6trairaydmmlge1zspibwlo1s9tmaxpbexkfgitdsxdzzzj/ev9kjjaab32ykxgca4mufzxd6xs07zgqvpg3dqsmiqwsf+9roxtqvemnkivnt1eu1nrgpgoerfl1wqedogpwgbgxebys/g38jxkum6oiylezhg4/t3reaeukmrme5bdf/n1kbhv7vazgzw4ywfsshjnur0sjhmodyxhjhcxsabaj40hldjzo2y9okbpa3joohizikyn43tsuuu3zuzuuvyiqea9tehokyuokcvveu1vecupajn9ijersfrzxq3piatc9z0zg/9kfx5a1cynxk=</latexit> <latexit sha1_base64="ijy93rae3wxabnemvyoumzf/lqc=">aaacfnicbzdlssnafiynxmu9rv2kmfgen5zebf0ofnx0wcfesillmj1ph84kywyiljcvl+erunw9o3hr1q1p4vsy0nyfbn7+cw7nzbckncntof/wwuls8spqya24vrg5tw3v7dzuneokdrrzwlycooczcoqaaq6trairaydmmlge1zspibwlo1s9tmaxpbexkfgitdsxdzzzj/ev9kjjaab32ykxgca4mufzxd6xs07zgqvpg3dqsmiqwsf+9roxtqvemnkivnt1eu1nrgpgoerfl1wqedogpwgbgxebys/g38jxkum6oiylezhg4/t3reaeukmrme5bdf/n1kbhv7vazgzw4ywfsshjnur0sjhmodyxhjhcxsabaj40hldjzo2y9okbpa3joohizikyn43tsuuu3zuzuuvyiqea9tehokyuokcvveu1vecupajn9ijersfrzxq3piatc9z0zg/9kfx5a1cynxk=</latexit> NON-EQUILIBRIUM STEADY STATE (NESS) Its a steady-state: no time-dependence. But it is not in equilibrium: constant flow of heat from hot to cold. NESSs are the closest we have to thermal equilibrium. But for equilibrium we know the basic properties of the state: For the NESS, no such statements can be made. = e Z H One needs to know the full dynamical equations.

3 Clean and natural platform for studying non-equilibrium physics: Fourier s law: diffusive vs. ballistic. Steady-state fluctuation theorems. Transport of excitations (squeezing, magnetization, etc). Connections with quantum information still poorly explored for NESSs: Information transport and quantum state transfer. Entanglement dynamics in quench dynamics. Quantum features of the NESS are now becoming accessible in controlled quantum platforms.

4 CONTROLLED QUANTUM PLATFORMS Optomechanics Cavity BECs Bosonic transport in optical lattices D. Husmann, et. al., Science, 350, 1498 (2015). M. Brunelli, et. al., Phys. Rev. Lett.., 121, (2018) Fermionic transport Proposal for trapped ions A. Bermudez, M. Bruderer and M. B. Plenio, Phys. Rev. Lett., 111, (2013) J. P. Brantut, et. al., Science, 342, 713 (2013).

5 OUTLINE 1. Local vs. Global master equations: thermodynamics of repeated interactions. 2. An analytically soluble model for a Gaussian diffusive NESS. 3. Information transport in the NESS. Work done in collaboration with: William Malouf (master student) and Jader P. Santos (USP) Mauro Paternostro and Gabriele De Chiara (Queens University in Belfast) John Goold (Trinity College Dublin) Gerardo Adesso and Luis Correa (University of Nottingham) arxiv

6 THERMODYNAMICS OF REPEATED INTERACTIONS arxiv

7 LOCAL VS. GLOBAL DILEMA <latexit sha1_base64="iyorvokd3yod6lpx3o6+tglrrd0=">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</latexit> d = dt Di ( ) = i (Ni + 1) <latexit sha1_base64="ixyoxu6ib3r9yo3yesqnorpqrsa=">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</latexit> ai a i 1 {ai ai, } 2 H = H1 + H2 +HI <latexit sha1_base64="mvc7aqrugqt3kvni0su8jm6djlc=">aaacchicbvdlsgnbeoz1gemjqx69dazbemjuepsgepcsywtzggrzzietzmjm7dizk4qlp+avenw7n/hqx3j1s5wke9degm6kqm66qsjhtbvp+3lw1jc2t7ylo8xdvf2dknt41njxqghtkpjhqhnhttmttgmy4bstkipfxgk7gt/n/pyjvzrf8sfmehoipjrswag2vgrduh3dilqpfxrhexwfbtmrehogvelnpaw5gqh73evhjbvugskx1l3fs0yqywuy4xra7kwajpim8zb2lzvyub1k88en6mwqftsils1p0fz9vzfhofverhzsydpsy95m/nelxnjlm7gomiat1fbjfochkucmrrnuuj8psgyfwikjyvz3rezyywjsdkubir8cwsppvsu+v/hvl8u1mzyeapzakzydd1dqgzo0oakeunigf3h1npw35935wiyuofnomfyb8/kd5swxag==</latexit> <latexit sha1_base64="kiy+pgfmqlqw37cl++/se5kgjco=">aaacl3icbvdlsgmxfm3uv62vqks3wsiiqpkzrf0ofny4rgaf0kklk95oq5pjkgseuvon/os/4fb34kbenx9h+hbq64el555zlzc5yckznq777mswlldw17lruy3nre2d/o5evctuuahqyawqh0qdzzfuddmc6okcikiotbb3pfjrd6a0k/gd6sfqfcskwydryqzuyp9d4uakiejlw7bugzajilajjk9+ld+2/ozl41a+4bbdmfai8aakgkyot/jfqvvsvebskcdanzw3mc0buyzrdsnckgpico2rcbqwxksabg7g/xvii6u0cucqw7hby3v2y0ce1n0r2klbtffpeypxxy8uc5dn56i5yhgsgojp5han5dhipaopt5kcanjfekivs2/htesuoczgnlohepmrljkqx/tcond7wihdtupjogn0ii6rh85rcd2gmqogih7rm3pbr86t8+z8oj+t0ywz3dlhf+b8/wdnx6bi</latexit>!1 a 1 a1 +!2 a 2 a2 + <latexit sha1_base64="dej5u4ak4cpbbezcpyo9tajtr7m=">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</latexit> 1 + (a 1 a2 <latexit sha1_base64="ab9cixgkqtkgz2to0kgn5ro7unu=">aaackhicbzdlsgmxfiyz9vbrrerstbailukdkyiuxbtcukxgl9dw4ux62gyzmshjckx0mxwjx8gt7t1jt30s04ugrqcspv7/he7yb7hg2rju2emtrw9sbqw3mzu7e/sh2cojmo4sxbdkihgprgaabzdyndwibmqkiqwe1oon26lff0alesqfzcdgdgg9ybucgbgsn70o0bbgmoti0jz43mora70ekgp+irbs7u3prz2nfjbnft1z0vxwfpaji6r42umre7ekrgmyak2bnhub9hcu4uzgknnknmbanqchtyssqttt4exji3pmlq7trsoeaehm/t0xhfdrqrjyzhbmxy97u/fflwixnpvudxvizzwylgy+ujsiaii6ty12uejmxmacmmxt2ynrgwjmblyzg4q3hmeq1epfzy1695e58s0injq5iacktzxyrcrkjlriltdyqt7io/lwxp1p58szz1ttzmlmmpwpz/incrukda==</latexit> + a1 a 2 ) T2 2 1 {ai a i, } 2 N a i i i ai Ni = <latexit sha1_base64="k6zwi8np+fc/zvzaun7bvp55iye=">aaab/hicbva9swnbfhwxv2l8ilralazbemkdblswcnjelokxbjij7g32kiw7e8funhco+bdstbctw/+lrb/etxkfjg48ggbeyx4tjpxp47pftmftfwnzq7hd2tnd2z8ohx61djwqqn0s81h1qqwpz5l6hhloo4miwisctspx7cxvp1klwswfzcshgcbdysjgslgsf4ea/bt+uejw3tnqkvfyuoeczx75uzeissqonirjrbuem5ggw8owwum01es1ttaz4yhtwiqxodri5s9o0zlvbiiklr1p0fz9fzfhofvehhztydpsy95m/nclxvkyia6djmkknvssrxcucmringscdziixpcjjzgozn9hziqvjsb2vbklemsvrjlwzdvzq959rvk/yespwgmcwjl4cav1aeatfcda4ble4nv5ct6cd+djsvpw8ptj+apn8wfmrjsh</latexit> = T1 i[h, ] + D1 ( ) + D2 ( ) <latexit sha1_base64="dcfaupuvxxtqiihj24rhdwi7ids=">aaacixicbzdlssnafiyn9vbrrerszwarurekiuhcoedglvswf2himexp2qezszizccxkfxwjx8gt7t2jo3hnkzi9llt1h4gf/5zdofmfcwdk2/anvvhaxlldk66xnja3tnfku3stfaesqppgpjadgcjglikmzppdj5farmchhqyvxvx2a0jf4uhojxlwbolhlgsuabp55eqnz/aldknjaobkgdxnbgcamnsnbfsnyfexdnk/xlfr9kr40tgzu0eznfzyt9ulasog0pqtpbqonwgvi1izyievuamchnah6upx2igiuf42+vgoj0zsw2eszys0nqs/jziilbqjwhqkogdqvjyo/60fym6zds+9jevjqigi08vhyrgo8rgx7jejvpormyrkzm7hdeamlg2glgwuzx7bommd1by75tyevuoxmzxfdiaouru56azv0tvqocai6be9oxf0aj1zb9a79tftlvizmx30r9bxd81cou0=</latexit> 1 e i!i 1 Seem to violate the 2nd law. Microscopic derivation yields a global ME. But this also has unphysical properties. A. Levy, R. Kosloff, Europhys.Lett.107, (2014) J. Onam González, L. A. Correa, G. Nocerino, J. P. Palao, D. Alonso, G. Adesso, Open Systems & Inf. Dyn, 24, (2017). M. T. Mitchison and M. B. Plenio, NJP, 20, (2018).

8 BE WISE, DISCRETIZE The LME generates a valid CPTP-semigroup evolution. Is there any physical scenario who is described by a LME? S S environment E Usual approach environment E Repeated interactions (collision model/conveyor belt) RI continuous-time limit always generates local dissipators. This is a physically realization scenario: LMEs have physical realizations! P. Strasberg, G. Schaller, T. Brandes, M. Esposito, Phys. Rev. X, 7, (2017)

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sha1_base64="iobbeq/ppfypoe/fggexznwfp1o=">aaacknicbzdlssnafiyn9vbrlerszwar3vgsexshuhdjsok9qbpkzhlsdp1mwsxekkhp4uv4cm517664ex/eazuftv4w8pgfczhn/idltgnhmvilldw19y3yzmvre2d3z94/akkkkxsanogj7areawccmpppdp1uaokddu1gedett59akpairz1kwy9jx7ciuakn1bndlwsucw7jw+xxivoccat7ck7ny95pz646nwcmvaxuavvuqngzv7wwovkmqlnoloq6tqr9nejnkidxxcsupiqosr+6bgwjqfn57gtjfgkceeejne9ophn/t+qkvmoub6yzjnqgfmtt899aec9s1tg1nzorzhoens+omo51gqe54zbjojqpdbaqmbkd0wgrhgqtbswe4i5gsayti5rr1nyhy2r9poinji7qmtpdlrpcdxspgqijkhpgr+gnvvsv1oc1st7nrswrmdlef2r9/wa3eawx</latexit> <latexit sha1_base64="iobbeq/ppfypoe/fggexznwfp1o=">aaacknicbzdlssnafiyn9vbrlerszwar3vgsexshuhdjsok9qbpkzhlsdp1mwsxekkhp4uv4cm517664ex/eazuftv4w8pgfczhn/idltgnhmvilldw19y3yzmvre2d3z94/akkkkxsanogj7areawccmpppdp1uaokddu1gedett59akpairz1kwy9jx7ciuakn1bndlwsucw7jw+xxivoccat7ck7ny95pz646nwcmvaxuavvuqngzv7wwovkmqlnoloq6tqr9nejnkidxxcsupiqosr+6bgwjqfn57gtjfgkceeejne9ophn/t+qkvmoub6yzjnqgfmtt899aec9s1tg1nzorzhoens+omo51gqe54zbjojqpdbaqmbkd0wgrhgqtbswe4i5gsayti5rr1nyhy2r9poinji7qmtpdlrpcdxspgqijkhpgr+gnvvsv1oc1st7nrswrmdlef2r9/wa3eawx</latexit> <latexit sha1_base64="iobbeq/ppfypoe/fggexznwfp1o=">aaacknicbzdlssnafiyn9vbrlerszwar3vgsexshuhdjsok9qbpkzhlsdp1mwsxekkhp4uv4cm517664ex/eazuftv4w8pgfczhn/idltgnhmvilldw19y3yzmvre2d3z94/akkkkxsanogj7areawccmpppdp1uaokddu1gedett59akpairz1kwy9jx7ciuakn1bndlwsucw7jw+xxivoccat7ck7ny95pz646nwcmvaxuavvuqngzv7wwovkmqlnoloq6tqr9nejnkidxxcsupiqosr+6bgwjqfn57gtjfgkceeejne9ophn/t+qkvmoub6yzjnqgfmtt899aec9s1tg1nzorzhoens+omo51gqe54zbjojqpdbaqmbkd0wgrhgqtbswe4i5gsayti5rr1nyhy2r9poinji7qmtpdlrpcdxspgqijkhpgr+gnvvsv1oc1st7nrswrmdlef2r9/wa3eawx</latexit> FUNDAMENTAL WORK COST In general there will be a work cost associated with turning the interactions on and off: External agent has to pull S and E apart. λ( ) H SEn (t) =H S + H En + (t)v SEn W = hv i hv i 0 The exception is when S+E is a thermal operation. τ [H S + H E,V]=0 Operator version of detailed balance. Can be neatly formulated in terms of eigenoperators. F. Barra, Scientific Reports, 5, (2015) F. Brandão, M. Horodecki, J. Oppenheim, J. Renes, R. Spekkens, Phys. Rev. Lett., 111, (2013).

10 RECONCILIATION WITH THERMODYNAMICS HSE (t) = <latexit sha1_base64="feapwhxc/9fsavjeppw5bah18hi=">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</latexit> N X HSi + HSint + 1,...,SN i=1 N X i=1 HEi + (t) N X VSEi i=1 environment E1 Local detailed balance is satisfied: [HSi + HEi, Vi ] = 0 <latexit sha1_base64="5y7zgrh3kfiioba6goluclpyrwu=">aaacfxicbzdlssnafizp6q3ww9sllgalicgleuexcguruqxol9cgmjlo26ezszizccv040v4cm51707cunbrkzhts9dwhwy+/nmo58wfxjwp7thfvm5hcwl5jb9awfvf2nyyt3fqkkokotus8ug2a6wozygtaay5bcasyhfw2ggg1+n644fkxalwxg9j6gncc1mxeayn5dv7ryqf3vlshi6rortdj6jumw9dice3i07jmqjng5tbetjvffu73yliimioccdktvwn1l6kpwae01ghnsgayzlapdoyggjblzdofjfch8bpog4kzqs1mri/j1islbqkwhqkrptqtjy2/60fymaz7l54kqvjrnoqtbd3e450hmyroq6tlgg+nicjzoz2rppyyqjnkautijsbwtzut0uuu3jvz4rlyyyepozbarybc+dqhgpuoqyehuezxudverlerhfry9qas7kzxfgj6/mh3xgdaw==</latexit> S1 S2 But global detailed balance is broken due to interactions. [HI, Vi ] 6= 0 <latexit sha1_base64="wsdcigoh18dtgf5dkrraitenuse=">aaaccnicbvdlssnafj3uv62vvjdubovgqkoigi5cfnzuxqx7gdaeyfsmhtqzxjmjukl/wf9wq3t34tafcouxog2z0nydfw7n3mu5ncdhtgnh+bikk6tr6xvfzdlw9s7unl3eb6k4lrsanoax7areawccmpppdp1eaokcdu1gdd312w8gfyvfnr4n4evkifjiknfg8u1yt+7fnokwzzzce3cphd+uofvnbrxm3jxuui6gb3/3+jfnixcacqju13us7wveaky5teq9vefc6igmoguoibeol5u9pshhrunjmjzmhmyz9fdfriklxlfgnioih2rrm4r/ekg0kkzdsy9jikk1cdopdloodyynvea+k0a1hxtcqgtmd0yhrbkqtxslu4q7wmeyaz1vxafq3p5xald5puv0ii7qcxlrbaqhomqgjqloet2jf/rqpvlv1rv1mv8twpnnafod6/mhbm2zfw==</latexit> environment E2 In this case there will be a work cost.

11 THE HARMONIC CHAIN WITH SELF-CONSISTENT RESERVOIRS

12 QUANTUM HARMONIC CHAIN T1 <latexit sha1_base64="cgiwu9ti/wimu3a6nvg4bsbbm9u=">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</latexit> d = dt Di ( ) = i (Ni + 1) <latexit sha1_base64="1s1fsz0xlq+nzsnxrakrhlndtgq=">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</latexit> hai ai+1 L-1 2 i[h, ] + D1 ( ) + DL ( ) <latexit sha1_base64="ixyoxu6ib3r9yo3yesqnorpqrsa=">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</latexit> Ji,i+1 = {ai ai, } 2 X TL H=i (a i ai+1 a i+1 ai ), i 1 + i Ni ai ai {ai ai, } 2 <latexit sha1_base64="ei0p1rjalrl0zipquraieqpkkca=">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</latexit> ai ai L <latexit sha1_base64="dej5u4ak4cpbbezcpyo9tajtr7m=">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</latexit> ai+1 ai i This model can be solved analytically for arbitrary L: Z. Rieder, J. L. Lebowitz and E. Lieb, J.. Math. Phys., 8, 1073 (1967) M. Žnidarič, J. Stat. Mech, L05002 (2010) A. Asadian, D. Manzano, M. Tiersch and H. J. Briegel, Phys. Rev. E., 87, (2013) F. Nicacio, A. Ferraro, A. Imparato, M. Paternostro, F. L. Semião, PRE, 91, (2015)

13 SELF-CONSISTENT BATHS We consider instead a modified MEq. of the form d = dt i[h, ] + D1 ( ) + DL ( ) + L X D i ( ) i=1 <latexit sha1_base64="+8atdc1cyimpjvdzhtbtunagsdy=">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</latexit> N i = ha i ai iness D i ( ) = (N i + 1)D[ai ] + N i D[a i ] <latexit sha1_base64="o97y1p0mwvppjxdeervv/uepnru=">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</latexit> <latexit sha1_base64="sprpo5t8qfylckp752bppy9l/9e=">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</latexit> Gives the same 2nd moments as a dephasing model, but has a Gaussian NESS. T1 1 2 L-1 L A. Asadian, D. Manzano, M. Tiersch and H. J. Briegel, Phys. Rev. E., 87, (2013) M. Bolsterli, M. Rich, W. M.Visscher, Phys. Rev. A., 1, 1086 (1970) TL

14 <latexit sha1_base64="cdkx1hqck03snd2hu28pnzmsp48=">aaacrhicbvfna9wwejwdfqtbj2zbyy6is2gxjyvlbpplidbdcwghhwwswdtmlmuoicqbss4sxj+lp6z/pmfkxh82mw4ipb15i43epjxgxgbbh8/fe/b8xcv9v6pxb96+oxi//3btylpttqslkpvtcoyjrtjscivybauzyfswm/the5e/+cw04aw6suukxrikxxnowtoqgf+obkhcmawjv4sykaqmo4wjveg/4sjxqjulhmwvkvo2cvs8hzjsnzy4kkbkwhjqhpgio+msby6xu16mgdyfxvev2yi4+tfv0/mjmmvxaosc8jmz4wq8crzbh/gpiaoyoceuk/hfkctplzmyviaxkxjunm5aw04fa0drbvgf9aektnjqgwqmbnobw/zzmrnos+2wsrhntysakmaszequeuy92c115h9zqxz0cqm4zz1po/3y/cruukpqyxtdtjpxatssd5pdgdemwrf2akjm7key3oo7xrr5jpxvzneyp+a6xjbgqx4et05pbtp20sh6hkaiok/ofj2hs7re1nvzzl7offex/pw/8uon1pegmo/oufj5p747y3y=</latexit> <latexit 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sha1_base64="aqmeetlpkw0/s8fqsmnvrazafv4=">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</latexit> NOISE SOURCES ARE RELEVANT ha i a ii = N 1 + N L (N 1 N L ) (L 1) (L 2i + 1) J = (L 1) (N L N 1 ). We see that for any non-zero Γ the current always scales as 1/L. Diffusive behavior. Γ = Γ = Γ = ( )

15 INFORMATION TRANSPORT IN THE NESS arxiv

16 ENTANGLEMENT DYNAMICS AND QUANTUM STATE TRANSFER When excitations propagate through a chain (e.g. after a quench), they carry information from one part to the other. When excitations propagate through a chain (e.g. after a quench), they carry information from one part to the other. V. Alba and P. Calabrese, PNAS, 114, 7947 (2017) S. Bose, PRL, 91, (2003)

17 INFORMATION TRANSPORT IN THE NESS T1 1 2 L-1 L TL Entanglement dynamics and quantum state transfer are both instances of information transport. It should be possible to formulate an analogous question for NESSs. Heat currents also carry information. But the NESS is fundamentally different: its a steady-state; no dynamics.

18 REPEATED INTERACTIONS SCENARIO The repeated interactions scenario is remarkably similar to quantum state transfer. 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19 <latexit sha1_base64="tpliuusp8egi2uclacfys5oe+pu=">aaacfhicdvblswmxgmzwv62vqkcvwsj6wnal0pzw8olbqwx7ghyp2ttbhuaxjfmhlp0rxjzox/emxr37tzyatiu0rqccw8z3jzmjy0a18bwvj7exubw9k98t7o0fhb4vj09awiykkyawtkpoidrhvjcmoyartqwi4iej7xb8m/pbj0rpkswdmcqk4ggoaeqxmlzq99ri9u8u+8ws73pzqm+twvtkgan58ncqgqynfvg7n5a44uqyzjdwxd+ltzaizshmzfrojzreci/rkhqtfygthatzufn4yzubjksyrxg4v5c3uss1nvdqtnjkrnrdm4l/eiffetkvfjniibywx0tvikuitgwrebenshg0es4aggoqcdzsygncitofqtxc9hpjeywsv/u/av27vuf69+vsvzqvlgdn4bxcar9uqb3cggzoagzg4am8gffn2xlz3p2pxwjoyxzowqqczx8m4z9n</latexit> 0L 0R Joint unitary of each RI stroke: <latexit 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sha1_base64="oxukqtqfbidr8m9fifwwpisojrq=">aaacfhicdvblswmxgmzwv62vqkcvwsj6wnal0pzw8okxin1au5rsmm1d81isrfcw/ggvhvsvebov3v0nhk3bfdqia4fh5vusyyqxo9p43pet29jc2t7j7xb29g8oj4rhjy0te4vje0smvsdemjaqsnnqw0gnvgtxkjf2ol6z+e1hojsv4sfmyhjwnbq0ohgzk7v7ait795f9ysl3vtmg59yskuwm1hz4a5vahka/+n0bsjxwigxmsouu78umsjeyfdmylfqstwkex2hiupykxiko0nnckbywygbgutkjdjyryxsp4lppeggnotijve7nxd+9kk+8naqksaqqxstjomqquhenhgi8ibmldbojzw3bavuegzaxbgff7y8ghif7jbe9fpar+p+0rl3fc/27cqlezurlgznwdq6adyqgdm5bazqbbmpwbf7aq/psvdnvzsdinodko6dgbc7ndxbjn20=</latexit> <latexit sha1_base64="mveb6vvztyeocjpkdvziahg3sqq=">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</latexit> <latexit sha1_base64="ctyee2sps3um2us81lvdvrewdo0=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcopcrqx7ghyomtttxuyxjbmhdp0hny70v9yjwz/ap3fp2o7qfj0qojxzb3jywphrbtzvy8ltbg5t7+r3c3v7b4dhxeotlpajwqsjjzoqeyjngbwkaahhpbmrgnjisdscx8/89inrmkpxbyyxctgachprjiyvwj01kv3bfrhku94c0hnrfpvyrmo+/lvkieojx/zudsroobegm6r11/die6rigyozmrz6isyxwmm0jf1lbejeb+k87rrewguai6nseqbo1ewnfhgtjzy0kxyzkv73zukfxshxxk4fxsrsck/lmve1skmie0mexssjeganhloc4iaqgg2bwikwovzhei+qvcbyggvlvf1pwleu77n+xblur2al5cezoaexwacvuac3oagaaimh8arewkvz7lw5787hyjtnzdunyaxo5w+kij82</latexit> <latexit sha1_base64="ctyee2sps3um2us81lvdvrewdo0=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcopcrqx7ghyomtttxuyxjbmhdp0hny70v9yjwz/ap3fp2o7qfj0qojxzb3jywphrbtzvy8ltbg5t7+r3c3v7b4dhxeotlpajwqsjjzoqeyjngbwkaahhpbmrgnjisdscx8/89inrmkpxbyyxctgachprjiyvwj01kv3bfrhku94c0hnrfpvyrmo+/lvkieojx/zudsroobegm6r11/die6rigyozmrz6isyxwmm0jf1lbejeb+k87rrewguai6nseqbo1ewnfhgtjzy0kxyzkv73zukfxshxxk4fxsrsck/lmve1skmie0mexssjeganhloc4iaqgg2bwikwovzhei+qvcbyggvlvf1pwleu77n+xblur2al5cezoaexwacvuac3oagaaimh8arewkvz7lw5787hyjtnzdunyaxo5w+kij82</latexit> L 1 L R <latexit sha1_base64="8a5qbhly1joopckfj0ilmdbqnsi=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcooyin1ao5rmmmlj8xisjfcg/ombf/or7ssth+cfudrtr2ilhggczrk3otlhzkg2nvfl5dy2t7z38rufvf2dw6pi8ully0rh0ssssdujksamcti01ddsirvbpgskhy6vz377kshnpbg3k5geha0fjshgxkqtnhrj/l2/wpjdbw7outwlsjkjnr/+wiwqodevfvcgeieccimz0rrre7ejuqqmxyxmc71ekxjhmrqsrquccakddj52ci+smocrvpyia+fq8kakunythtpjjsxir3sz8u8v5csvp4jieime1/kyqbqkvmsjiqiv4kqjg0bcwufwqbxbhk0sqvhr+yoir8hey2ynhewq/ietk9f3xp+2xkpxs9ly4aycg0vggwqogxvqae2awqn4ai/g1xl23px352mxmnoynvowaufzb66enzw=</latexit> L In the NESS: R <latexit sha1_base64="8a5qbhly1joopckfj0ilmdbqnsi=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcooyin1ao5rmmmlj8xisjfcg/ombf/or7ssth+cfudrtr2ilhggczrk3otlhzkg2nvfl5dy2t7z38rufvf2dw6pi8ully0rh0ssssdujksamcti01ddsirvbpgskhy6vz377kshnpbg3k5geha0fjshgxkqtnhrj/l2/wpjdbw7outwlsjkjnr/+wiwqodevfvcgeieccimz0rrre7ejuqqmxyxmc71ekxjhmrqsrquccakddj52ci+smocrvpyia+fq8kakunythtpjjsxir3sz8u8v5csvp4jieime1/kyqbqkvmsjiqiv4kqjg0bcwufwqbxbhk0sqvhr+yoir8hey2ynhewq/ietk9f3xp+2xkpxs9ly4aycg0vggwqogxvqae2awqn4ai/g1xl23px352mxmnoynvowaufzb66enzw=</latexit> <latexit sha1_base64="ctyee2sps3um2us81lvdvrewdo0=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcopcrqx7ghyomtttxuyxjbmhdp0hny70v9yjwz/ap3fp2o7qfj0qojxzb3jywphrbtzvy8ltbg5t7+r3c3v7b4dhxeotlpajwqsjjzoqeyjngbwkaahhpbmrgnjisdscx8/89inrmkpxbyyxctgachprjiyvwj01kv3bfrhku94c0hnrfpvyrmo+/lvkieojx/zudsroobegm6r11/die6rigyozmrz6isyxwmm0jf1lbejeb+k87rrewguai6nseqbo1ewnfhgtjzy0kxyzkv73zukfxshxxk4fxsrsck/lmve1skmie0mexssjeganhloc4iaqgg2bwikwovzhei+qvcbyggvlvf1pwleu77n+xblur2al5cezoaexwacvuac3oagaaimh8arewkvz7lw5787hyjtnzdunyaxo5w+kij82</latexit> L R <latexit sha1_base64="8a5qbhly1joopckfj0ilmdbqnsi=">aaace3icdvdlsgmxfm3uv62vqks3wsk4gmak0hzxcooyin1ao5rmmmlj8xisjfcg/ombf/or7ssth+cfudrtr2ilhggczrk3otlhzkg2nvfl5dy2t7z38rufvf2dw6pi8ully0rh0ssssdujksamcti01ddsirvbpgskhy6vz377kshnpbg3k5geha0fjshgxkqtnhrj/l2/wpjdbw7outwlsjkjnr/+wiwqodevfvcgeieccimz0rrre7ejuqqmxyxmc71ekxjhmrqsrquccakddj52ci+smocrvpyia+fq8kakunythtpjjsxir3sz8u8v5csvp4jieime1/kyqbqkvmsjiqiv4kqjg0bcwufwqbxbhk0sqvhr+yoir8hey2ynhewq/ietk9f3xp+2xkpxs9ly4aycg0vggwqogxvqae2awqn4ai/g1xl23px352mxmnoynvowaufzb66enzw=</latexit> trlr U ( L S R )U <latexit sha1_base64="rnq3lpqgl4mahebxmmlojbsvme4=">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</latexit> We now define the NESS map: 0R = <latexit sha1_base64="b8rak8d+7dukobpwjf/6+xjrp9u=">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</latexit> ( L ) = trls U ( L S R )U This is exactly like the noisy channel scenario. I don t think the thermodynamic/transport property of this type of map has ever been explored. = S

20 <latexit sha1_base64="bolgffgf+xxy+t1fdidaplsclwe=">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</latexit> <latexit sha1_base64="bolgffgf+xxy+t1fdidaplsclwe=">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</latexit> <latexit sha1_base64="bolgffgf+xxy+t1fdidaplsclwe=">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</latexit> <latexit sha1_base64="bolgffgf+xxy+t1fdidaplsclwe=">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</latexit> <latexit sha1_base64="rif3bwjqug1tggogf79ew1sqg/4=">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</latexit> <latexit sha1_base64="rif3bwjqug1tggogf79ew1sqg/4=">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</latexit> <latexit sha1_base64="rif3bwjqug1tggogf79ew1sqg/4=">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</latexit> <latexit sha1_base64="rif3bwjqug1tggogf79ew1sqg/4=">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</latexit> SHARED INFORMATION BETWEEN DISTANT PARTS The heat currents transmit information from one part of the chain to the other. A B C The total information can be quantified by the total correlations: T ( ) = LX i=1 S( i ) S( ) =S( 1... L ) For connected parts, shared information can also be quantified by the Mutual Information. A B I(A:B) =S( A )+S( B ) S( AB ) J. Goold, C. Gogolin, S. R. Clark, J. Eisert, A. Scardicchio, and A. Silva, Phys. Rev. B, 92, (2015)

21 HARMONIC CHAIN WITH SELF-CONSISTENT BATHS Total correlations Mutual Information ( ) ~ Γ = Γ = ~ - I( ) ( ) Γ = Γ = Γ = Γ = - Γ =

22 <latexit sha1_base64="twue6ttilymqjyshr35xxjxymri=">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</latexit> <latexit sha1_base64="twue6ttilymqjyshr35xxjxymri=">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</latexit> <latexit sha1_base64="twue6ttilymqjyshr35xxjxymri=">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</latexit> <latexit sha1_base64="twue6ttilymqjyshr35xxjxymri=">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</latexit> CONDITIONAL MUTUAL INFORMATION For disconnected parts, however, the MI is not adequate. Instead, the correct quantifier of this is the Conditional Mutual Information (CMI): A B C I(A:C B) =S( AB )+S( BC ) S( B ) S( ABC ) E. Lieb and Mary B. Ruskai, J. Math. Phys., 14, (1973)

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sha1_base64="qkijrikfwjkkdhlppk8brzzh9ho=">aaaccnicbzdlssnafiyn9vbrldwlm8eitcaleuexcgu3livyc7shtkatdujmjmxm1fj7br6cw927e7e+hfufxembhbb+codjp+dwdr8fmaq043xzuaxlldw1/hphy3nre8cu7jzvgetmgjhkowz7sbfgbwloqhlpr5ig7jps8kexab91s6siobjrsuq8jgacbhqjbayexayxhx7vj5ikviapjpwexxkqzlrwedwmsibtvwd/d/shjjkrgjokvmd1iu2nkdqumzipdgnfiorhaea6bgxirhnj6estegicpgxcaupoohv/b4wrvyrhvpnksa/vfc81/+35fo6yds68mrvrrinas8nbzkaoyzol7fnjsgajayqlnb9dpeqsyw3sk5hq3pkifqf5xhwdqnt9uqqdz/hkwt44agxgglnqa1egdhoagzvwdf7aq/vkvvnv1sdsngdlo3vgj6zph574mty=</latexit> <latexit sha1_base64="qkijrikfwjkkdhlppk8brzzh9ho=">aaaccnicbzdlssnafiyn9vbrldwlm8eitcaleuexcgu3livyc7shtkatdujmjmxm1fj7br6cw927e7e+hfufxembhbb+codjp+dwdr8fmaq043xzuaxlldw1/hphy3nre8cu7jzvgetmgjhkowz7sbfgbwloqhlpr5ig7jps8kexab91s6siobjrsuq8jgacbhqjbayexayxhx7vj5ikviapjpwexxkqzlrwedwmsibtvwd/d/shjjkrgjokvmd1iu2nkdqumzipdgnfiorhaea6bgxirhnj6estegicpgxcaupoohv/b4wrvyrhvpnksa/vfc81/+35fo6yds68mrvrrinas8nbzkaoyzol7fnjsgajayqlnb9dpeqsyw3sk5hq3pkifqf5xhwdqnt9uqqdz/hkwt44agxgglnqa1egdhoagzvwdf7aq/vkvvnv1sdsngdlo3vgj6zph574mty=</latexit> CLASSICAL MARKOV CHAIN We usually write the Markovianity condition as: X Y Z P (z x, y) =P (z y) But this is equivalent to: P (x, z y) =P (z y)p (x y) Markovianity: Past and future are independent, given the present. Non-Markovianity can thus be viewed as the ability of the past to communicate information to the future, through the present. But this does not mean that past and future are (unconditionally) independent: P (x, z) = X y P (z y)p (x y)p (y) Past and future are dependent because they share a common ignorance about the present.

24 <latexit sha1_base64="vnplhrnz6mytbvalfljtf74aai8=">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</latexit> <latexit 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sha1_base64="r4be4o6ny4ftbh+6zutuucu/kni=">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</latexit> <latexit sha1_base64="r4be4o6ny4ftbh+6zutuucu/kni=">aaact3icbvhlsgmxfm3uv1tfvzduokvoqcqmcioofnzoroj9ykeutjppq5pmmgteovtp/amxbrvvvtsx085cwy8k93dotw7uircyqrrtv1uzpewv1bvslr++sbm1xdjzbaggkpjuccac2fkqiowkutdum9ikjuhcy6tpda8tvflepkkbunejkhq46gvqu4y0obqfhsurhmde4ttxqeuewatotocyvikuirjlkkdadepn45cxdesmlahlbhr9ixa8i8zjlhvvwlwryco0u4wixbgnarebk4iiskpwluzcxoajtotgdcnvduxqd2ikncwmmgsjrukeh6hp2gykxinqxnp5x/dimd3ob9isoegu/x0irlypefdmztktmtcs8l/n43odtx/eiakii00enjx2iwz1abnzyy9kgjubgycwpobtea+qcuqbl8gbu5x5cxzb46ti2bxn7rryvuztyyj9cahkwafnoapuqa3uaqavyai+wkf1zn1z35m0ngolya/8iuzub7prsri=</latexit> <latexit sha1_base64="r4be4o6ny4ftbh+6zutuucu/kni=">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</latexit> In a Markovian chain the MI is not zero: I(X :Z) = X x,z p(x, z)ln p(x, z) p(x)p(z) 6=0 But the CMI is zero: I(X :Z Y )= X x,y,z p(x, y, z)ln p(x, y, z)p(y) p(x, y)p(y, z) = X x,y,z p(x, y, z)ln p(x, z y) p(x y)p(z y) =0 In fact, a CMI = 0 can be taken as a definition of Markovianity. For a multi-time chain, the CMI as a function of the size of the present also quantifies the degree of non-markovianity (Markov Chain order). I(X 1,...,X k :X k+`,...,x n X k+1,...,x k+` 1 ) M. Papapetrou, D. Kugiumtzis, Physica A, 392, 1593 (2013)

25 <latexit sha1_base64="ativjzh5d68m1eegtl+gstsnjku=">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</latexit> <latexit sha1_base64="ativjzh5d68m1eegtl+gstsnjku=">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</latexit> <latexit sha1_base64="ativjzh5d68m1eegtl+gstsnjku=">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</latexit> <latexit sha1_base64="ativjzh5d68m1eegtl+gstsnjku=">aaaccxicbvhrahqxfm2mvetq66pp4su1i7bfwdiikfkh7sio+fdbbqvnmmsymd3qtdikd9ous/ohvvjgl/gdzrar6nylyt2cc29ucpkvwnmk9fsu39q4fefu5r3o/qdb2w+7jx4fe1s5iufcautom+6lvkaoukgwp6wtvmi0pmnoh61+cigdv9z8xxkpxwwfgpurwtfqafekfrxnguv6u9m/ym/hhyrtudjchfdagx22zqpljk5nz8ids5fa3mym9chhsaklf9o/ngkdqp5vtqfmurxsw4iyx9ttmf4moit0ujt2e3ralwe3qbicpbkko7t7g02sqappugju/vlcsxzx3keswjydvnlzcnhop/isqmpd7hg9dkibf4gzqg5dwazhyf7dufpc+3mrhcr2qx5da8n/almxnhnzt+nambjcact14lzsgbza+2ginbso5wfw4vs4o4gzd1xg+kromcvzt+amoh41sogg+fk6t7+3smetpcm7pe8s8obsk4/kiiyiin+jjwgr2o5+xk9jiheus+no1foe/bpxy1+kgrq2</latexit> QUANTUM MARKOV CHAIN 1 2 L A NESS density matrix can be viewed as a quantum Markov chain (this chain has no causal order). In the quantum case it can be shown that: I(A:C B) =0() ABC = E BC ( AB 0ih0 C ) P. Hayden, R. Jozsa, D. Petz and A. Winter, Comm. Math. Phys., 246, 359 (2004) O. Fawzi and R. Renner, Comm. Math. Phys., 340, 575 (2015)

26 <latexit sha1_base64="xfqx1xomrjjlgrzoqkkc8nggcg8=">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</latexit> <latexit sha1_base64="xfqx1xomrjjlgrzoqkkc8nggcg8=">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</latexit> <latexit sha1_base64="xfqx1xomrjjlgrzoqkkc8nggcg8=">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</latexit> <latexit sha1_base64="xfqx1xomrjjlgrzoqkkc8nggcg8=">aaacm3icbvdlsgmxfm3ud31vxbqjfqfskdnfuarb7uiffwpwc04td9jmg0xmhiqjlhh+xz/wf9zqvtwvt/6dmdqfth5iojxzl/fe40wckw3b71zuynjqemz2lj+/sli0xfhzvvzhlamtk5chsugbopwftk6z5rqrsqrc4/tgu69l/s0dlyqfwzxurbqpobmwnxhqrmov9l0bukuaj2dp6cjdwpvyflxh4218gf1fakninck9llf7akiapt++s6peuzqmrulrrtgd4hhidekrdxhrkvtddkhiqqnnoch169irbiygnsocpnk3vjqccg8demtoaikqzjk4mcvbrmljp5tmbrop1n8dcqilesizldlfattlxh89t4xm1v5em2fbfgsakj/bfsyxdnewig4zsynmpuoasgz2x6qljihtys6bujzrcmbjdbxi2bxncqd4udemzxato01uqg7aryfoff2goilocb2gv/rmpvsfvt/6/cnnwcoenfqh1tc3ycoobw==</latexit> HARMONIC CHAIN WITH SELF-CONSISTENT RESERVOIRS ( ) - /( + ) I = = Γ I(A:C B) = u (v + L) 2b+2

27 LOCAL THERMALIZATION 1 2 L 1...L <latexit sha1_base64="q5vqs9sra/couuutxqcpvjh/1as=">aaaccnicbvdlssnafj34rpwv6tlnybfcluqexbgouhhhooj9qbpczdjph84jzeyuevoh/ojb3bstt/6ew7/eazufth64cdjnxs7lxbmj2njel7oyura+svnzqm7v7o7tu7wdjpa5wqsnjzoqfynngbwkbahhpjcpgnjmsdcexu/97gnrmkpxb8yzctkacjpsjiyvircwqkgmch8gljfgw9tj5na9hjcdxcz+seqgrctyv4ne4pwtytbdwvd9lznhgzshmjfjncg1yraeoqhpwyoqjzoszq9p4ilvephkzucyofn/xxsiaz3msd3kyaz1ojcv//vivpbs0suwoclldrf4hpzmdbojp73ahcqcdrtbgrci9neih0ghbgx7vvukv1jbmumcnxyv4d+d15txzt0vcasowsnwwqvoghvqam2awsn4bi/g1xly3px352o+uukun4fgd5zphzg5mja=</latexit> Kato-Brandão theorem on approximate Quantum Markov Chains: <latexit sha1_base64="2qvm/qlsosvv1x/agiqe3uz7f2u=">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</latexit> then S <latexit sha1_base64="gjeetce5x+1m+zfcesekm5chgsu=">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</latexit> I(1,..., k P e i hi,i+1 Z 1 : k + 1,..., L k), L 8k [hi,i+1, hj,j+1 ] 6= 0 <latexit sha1_base64="feyj5pp8c6c+nyrpau4hlmfkdou=">aaacfnicbzdlssnafizp6q3ww9slcinfecwleceuc25cvraxaeoytkfttjnjnjkijxtls/gkbnxvtty6deutog2z0nyfbj7+cw7nzb/encntof9wbmv1bx0jv1ny2t7z3bp3dxoqsishdrlxslycrchngty105y2yklxghdadebx03rzgurfingnxzh1qtwxrmci1sby7ep2we9ziz27kxiyocwndxqoi+g9cny76jsdmdayubkuivpnt7873ygkirwackxu23vi7avyaky4nrq6iaixjipcp22daodueensgxn0apwu6kxspkhrzp09kejqqxeymm4q64fare3nf2tbulbz9ypeykscacrifhev4uhhajor6jjjiezja5hizm5hzialjtokwtchuisrlepjouw6zff2slitzphk4qho4axcuiiq3ean6kdgez7hbv6tj+vnerc+5q05k5s5hd+ypn8afcgdsq==</latexit> <latexit sha1_base64="iyw+igowvmtz2tirl7+gyjmp2b4=">aaacanicbvdlsgmxfl3js9zx1awbybeetuxe0ivcwy3lcvyb7vayaaantszdkhhk0j2/4fb37sstp+lwlzf9llt1wixdofdylidmbdfw97+8hcwl5zxv3fp+fwnza7uws1szktwuvakssjdcypjgmatabgvrjjorgqpwd/s3i7/+ylthkr63g4qfknrjhnfkrjoad+ga8vn8wo9xu1d0s/4yaj7gksncfjv24bvvutsvllzuegoa2e9skbftorvsmg+lhiwe9kmxnr2niwqmymyvd9ghuzooutpnbnfy/x2rewnmqizuuxlbm7pespzxc+vmso0ug4zhswpztcfbusqqvwjub+pwzagva0ci1dz9jmipaektay3vssgzfcyt2lkj+yv8d14sx03ryce+hmaryliamtxcbapaqcezvmcr9+s9ee/ex2r1wzve7mefej8/agowjw==</latexit> j=i 1, i + 1 In the ballistic case: I(A : C B = b) L0 <latexit sha1_base64="h8e3oknftydvd1f/xym6ytlxrq4=">aaacihicbvdlssnafj3uv62vqes3u4tynyurwsii1w4uxfswd2himuyn7dczjmxmhbl7cf6ev+bw9+7eps79eidtflo9cofwzr3ce48xmiqvzx0ymbn5hcwl7hjuzxvtfcpc3griibky1hhaathykcsm+qsuqgkkfqqcumdi0xtwe795r4skgx+jrifxoer7tecxulrqmpsor2qaeysvx8uzjw9pojovwnt4dk+hdwadstm8uru6zseqwrpav8rosqgkqhxml6cb4igtx2ggpgzbvqjcgalfmspjnbnjeii8rh3s1trhneg3njw0hnta6cjeiht5ck7unxmx4lkoukc7k/plrjei/3oen9msemu3pn4ykelj6ejexkakyjiw7fjbsgijtrawvn8o8qajhjxonkddswcj+esahyxbktnxr4vkoy0nc3baligcgxydcrganvahgdyaj/amxoxh49v4m96nrrkjndkgv2b8fgmzeaap</latexit> Local states are therefore not necessarily thermal. Diffusive case: <latexit sha1_base64="fvull79ukv4qbndwujc96kuk4n8=">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</latexit> I(A : C B = b) L 4 In the thermodynamic limit, the system becomes locally thermal. <latexit sha1_base64="xfqx1xomrjjlgrzoqkkc8nggcg8=">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</latexit> K. Kato, F. Brandão, u I(A : C B) = (v + L)2b+2

28 CONCLUSIONS Plenty to explore in Quantum NESSs: physics is still not fully understood. I m particularly interested in this notion of information transport. Information transport offers a bridge between non-equilibrium physics and quantum information. Thank you! spinoffqubit.info Acknowledgements: FAPESP Sprint