Bit Threads & Holographic Monogamy

Transcription

1 Bit Threads & Holographic Monogamy Matthew Headrick Brandeis University Entanglement in Quantum Systems GGI Florence, June 2018 Based on arxiv: [hep-th] with M. Freedman & forthcoming work with S. Cui, P. Hayden, T. He, B. Stoica, M. Walter

2 A Contents 1 Bit threads 2 2 MMI 4 3 Proof of MMI 7 4 Structure of holographic states 10 1 Bit threads Throughout this talk: holographic theory dual to classical Einstein gravity static bulk spacetime Ryu-Takayanagi formula: S(A) = 1 4G N area(m(a)) m(a) = minimal surface in bulk homologous to A m(a) sha1_base64="+d+uatgvohed3p8uj7q8ktpzycq=">aaab63icbvbns8naej3ur1q/qh69lbahxkoigh6rxjxwsb/qhrlzbtqlu5uwuxfk6f/w4kerr/4hb/4bn2ko2vpg4pheddpzgpgzbvz32ymtrw9sbpw3kzu7e/sh1cojjo4srwibrdxsvqbrypmkbcmmp71yuswctrvb9c7zu09uarbjrzolqs/wwlkqewwysdrvzofvmttwc6bv4hwkbgvaw+rxybsrrfbpcmda9z03nn6klwge03llkggayzlfy9q3vgjbtz/mt87rmvvgkiyulwlqrv6eslhqeiyc2ymwmehllxp/8/qjca/9lmk4mvssxaiw4chekhscjziixpczjzgozm9fziivjsbgu7ehemsvr5lorcnzg97dza15w8rrhhm4htp4cavnuicwtihabj7hfd4c4bw4787horxkfdph8afo5w8jfy2h</latexit> sha1_base64="+d+uatgvohed3p8uj7q8ktpzycq=">aaab63icbvbns8naej3ur1q/qh69lbahxkoigh6rxjxwsb/qhrlzbtqlu5uwuxfk6f/w4kerr/4hb/4bn2ko2vpg4pheddpzgpgzbvz32ymtrw9sbpw3kzu7e/sh1cojjo4srwibrdxsvqbrypmkbcmmp71yuswctrvb9c7zu09uarbjrzolqs/wwlkqewwysdrvzofvmttwc6bv4hwkbgvaw+rxybsrrfbpcmda9z03nn6klwge03llkggayzlfy9q3vgjbtz/mt87rmvvgkiyulwlqrv6eslhqeiyc2ymwmehllxp/8/qjca/9lmk4mvssxaiw4chekhscjziixpczjzgozm9fziivjsbgu7ehemsvr5lorcnzg97dza15w8rrhhm4htp4cavnuicwtihabj7hfd4c4bw4787horxkfdph8afo5w8jfy2h</latexit> sha1_base64="+d+uatgvohed3p8uj7q8ktpzycq=">aaab63icbvbns8naej3ur1q/qh69lbahxkoigh6rxjxwsb/qhrlzbtqlu5uwuxfk6f/w4kerr/4hb/4bn2ko2vpg4pheddpzgpgzbvz32ymtrw9sbpw3kzu7e/sh1cojjo4srwibrdxsvqbrypmkbcmmp71yuswctrvb9c7zu09uarbjrzolqs/wwlkqewwysdrvzofvmttwc6bv4hwkbgvaw+rxybsrrfbpcmda9z03nn6klwge03llkggayzlfy9q3vgjbtz/mt87rmvvgkiyulwlqrv6eslhqeiyc2ymwmehllxp/8/qjca/9lmk4mvssxaiw4chekhscjziixpczjzgozm9fziivjsbgu7ehemsvr5lorcnzg97dza15w8rrhhm4htp4cavnuicwtihabj7hfd4c4bw4787horxkfdph8afo5w8jfy2h</latexit> sha1_base64="+d+uatgvohed3p8uj7q8ktpzycq=">aaab63icbvbns8naej3ur1q/qh69lbahxkoigh6rxjxwsb/qhrlzbtqlu5uwuxfk6f/w4kerr/4hb/4bn2ko2vpg4pheddpzgpgzbvz32ymtrw9sbpw3kzu7e/sh1cojjo4srwibrdxsvqbrypmkbcmmp71yuswctrvb9c7zu09uarbjrzolqs/wwlkqewwysdrvzofvmttwc6bv4hwkbgvaw+rxybsrrfbpcmda9z03nn6klwge03llkggayzlfy9q3vgjbtz/mt87rmvvgkiyulwlqrv6eslhqeiyc2ymwmehllxp/8/qjca/9lmk4mvssxaiw4chekhscjziixpczjzgozm9fziivjsbgu7ehemsvr5lorcnzg97dza15w8rrhhm4htp4cavnuicwtihabj7hfd4c4bw4787horxkfdph8afo5w8jfy2h</latexit>

3 A Bit threads are an equivalent way to write RT Threads are unoriented 1d objects in bulk Rules: 1. Don t split or join, end only on boundary 2. Density 1/4G N (threads have Planckian thickness) RT v. 2: S(A) = max # threads connecting A to A c Add threads until you can t anymore tightly packed on m(a). Then N AA c = 1 4G N area(m(a)) Guaranteed by Riemannian max flow-min cut theorem (proved using convex relaxation & Lagrangian duality of convex programs) Away from m(a), thread configuration is highly non-unique Each thread represents an entangled pair of qubits in A, A c (these qubits are not local in field theory) Quantities like H(A B), I(A : B), I(A : B C), properties like subadditivity, etc.

4 have direct thread interpretations [Freedman-MH 16] Topics I won t talk about today: Entanglement of purification Bit threads in higher-derivative gravity (forthcoming work with J. Harper, A. Rolph) Covariant bit threads (forthcoming work with V. Hubeny) Lorentzian min flow-max cut [MH-Hubeny 17] All involve generalizations of max flow-min cut 2 MMI Mutual information: I(A : B) := S(A) + S(B) S(AB) Subadditivity: I(A : B) 0 Strong subadditivity: I(A : BC) I(A : B) True for general quantum states Monogamy (superadditivity) of mutual information:

5 I(A : BC) I(A : B) + I(A : C) S(AB) + S(AC) + S(BC) S(A) + S(B) + S(C) + S(ABC) Follows from RT by cutting and pasting minimal surfaces [Hayden-MH-Maloney 11, MH 13] Not true in general quantum states special property of holographic states A B C A B C i) S(AB)+S(BC)+S(AC)! ii) S(A)+S(B)+S(C)+S(ABC)! Tripartite information: I 3 (A : B : C) := S(AB) + S(AC) + S(BC) S(A) S(B) S(C) S(ABC) Figure 1: Illustration of the proof in AdS 3 /CFT 2. (i) The three minimal surfaces corresponding to S(AB), S(BC), and S(AC), with the latter consisting of two connected components in the bulk. (ii) The four minimal surfaces corresponding to S(A), S(B), S(C), and S(ABC). In this example, the surfaces for S(AB) and S(BC) each get decomposed into a purple part and a red part. The purple forms a surface terminating on the boundary of A and the red a surface terminating on the boundary of C. Since these new surfaces need not be minimal, their areas form upper bounds on S(A) and S(C), respectively, by the Ryu-Takayanagi formula. The reader is cautioned, however, that these simple cartoons can be misleading. As we will see in section 6.1, higher-order versions of the monogamy inequality fail even though naive graphical arguments suggest they should hold. Without loss of generality, assume ABCD is pure (can purify holographically) Then I 3 : is symmetric under permutations of A, B, C, D: I 3 = 1 (S(AB) + S(AC) + S(BC) + S(AD) + S(BD) + S(CD)) S(A) S(B) S(C) S(D) 2 We now define the following regions: is sensitive only to fully 4-party entanglement: If ψ ABCD = φ 1 AB φ 2 CD or φ 1 A φ 2 BCD then I 3 = 0 can be positive or negative: ã = ab \ ac \ bc b = ab \ bc \ ac c = ac \ bc \ ab fabc = ab [ bc [ ac. (3.8) As required, these regions are anchored on A, B, C, and ABC X ã = X b = c = X f abc = ABC. (3.9)

6 GHZ: ψ ABCD = 1 2 ( ) I 3 = 1 Perfect-tensor state, maximally entangled across all partitions (e.g. 4-qutrit code): S(A) = S(B) = S(C) = S(D) = S 0 > 0 S(AB) = S(AC) = S(AD) = S(BC) = S(BD) = S(CD) = 2S 0 Apparently in holographic states I 3 = 2S 0 I(A : B) = = 0 perfect-tensor type entanglement dominates over GHZ-type entanglement Interlude: the holographic entropy cone [Bao-Nezami-Ooguri-Stoica-Sully-Walter 15] n-party pure state: 2 n 1 1 independent entropies of subsystems S(A), S(AB),... These form a vector in R 2n 1 1 The vectors that can be realized by holographic states form a convex polyhedral cone in R 2n 1 1 n = 2: 1 independent entropy: S(A) = S(B) = 1 2I(A : B). Any 0 value allowed n = 3: 3 independent entropies: S(A), S(B), S(C) Equivalently coordinates: I(A : B), I(A : C), I(B : C). Any 0 values allowed n = 4: 7 independent entropies: S(A), S(B), S(C), S(D), S(AB), S(AC), S(BC) Equivalently coordinates: I(A : B), I(A : C), I(B : C), I(A : D), I(B : D), I(C : D), I 3 Holographically, any 0 values allowed (subadditive + MMI SSA) (n 5: cone is more complicated)

7 A B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> 3 Proof of MMI Basic theorem about multi-commodity flows: Given regions A 1, A 2,..., A n covering boundary, there exists a thread configuration such that, for all i, N Ai A j = S(A i ) Proof for networks: Kupershtokh 71, Lovász 76, Cherkassky 77 Proof for Riemannian manifolds uses strong duality of convex programs j i n = 2: N AB = S(A) = S(B) = 1 I(A : B) 2

8 B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> A C sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> n = 3: N AB + N AC = S(A) I(A : B) = 2N AB N AB + N BC = S(B) I(A : C) = 2N AC N BC + N AC = S(C) I(B : C) = 2N BC (Since N ij 0, this proves subadditivity)

9 B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> n = 4: S(A) = NAB + NAC + NAD NAB S(AB) = max # threads connecting AB to CD NBD NBC sha1_base64="pttyqepory9f49rgag0lo7bvyjc=">aaab7xicbvbnswmxej34wetx1aoxybe8lv0r9fjrxznusb/qliwbztvyblikwaes/q9epcji1f/jzx9j2u5bwx8mpn6bywzemahurod9o5xvtfwnzcjwcxtnd2+/dhdyncrvldwoekq3q2ky4ji1llectrpnsbwk1gphn1o/9cs04uo+2hhcgpgmji84jdzjzbtedl2b9eplr+lngjejn5my5kj3sl/dvqjpzkslghjt8b3ebhnrllpbjsvualhc6igmwmdrswjmgmx27qsfoqwpi6vdsytn6u+jjmtgjopqdcbeds2inxx/8zqpja6cjmsktuzs+aiofdgqph0d97lm1iqxi4rq7m7fdeg0odyfvhqh+isvl5pmecx3kv79rblay+mowdgcwbn4calvuiu6nidcizzdk7whhv7qo/qyt66gfoyi/gb9/gay9i7e</latexit> NAC + NBC + NAD + NBD D NAD Summing permutations, implies I3 0 A NCD NAC sha1_base64="jycpqr8qqk7+gfomtkxwojidbpc=">aaab7xicbvbnswmxej34wetx1aoxybe8lv0r9fjtxznusb/qliwbztvyblikwaes/q9epcji1f/jzx9j2u5bwx8mpn6bywzemahurod9o5xvtfwnzcjwcxtnd2+/dhdyncrvldwoekq3q2ky4ji1llectrpnsbwk1gphtanfemlaccuf7dhhquwgkkeceuuk5l0vu65neqwyv/fmwmvez0kzctr7pa9ux9e0ztjsqyzp+f5ig4xoy6lgk2i3nswhdeqgrooojdezqta7dojpndlhkdkupmuz9fdermjjxnhoomnih2br sha1_base64="jycpqr8qqk7+gfomtkxwojidbpc=">aaab7xicbvbnswmxej34wetx1aoxybe8lv0r9fjtxznusb/qliwbztvyblikwaes/q9epcji1f/jzx9j2u5bwx8mpn6bywzemahurod9o5xvtfwnzcjwcxtnd2+/dhdyncrvldwoekq3q2ky4ji1llectrpnsbwk1gphtanfemlaccuf7dhhquwgkkeceuuk5l0vu65neqwyv/fmwmvez0kzctr7pa9ux9e0ztjsqyzp+f5ig4xoy6lgk2i3nswhdeqgrooojdezqta7dojpndlhkdkupmuz9fdermjjxnhoomnih2brm4r/ez3urldbxmwswibpffgucmwvnr6o+1wzasxyeui1d7dioisauosckroq/mwxl0nzvoj7ff/+oly9yemowdgcwbn4calvuiu6nidcizzdk7whhv7qo/qyt66gfoyi/gb9/ga0ey7f</latexit> Note that NAB 21 I(A : B) (not =) How can it be >? What do the AB threads represent, if not AB mutual information? Unlike for n = 2, 3, the Nij s are not fixed by the entropies Toy example: (capacity = 1 on each leg) Thread configurations: B B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> C A B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> C A sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> D B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> C A sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> D D NAB = NCD = 1 NAD = NBC = 1 sha1_base64="u4/mfjoashzgmetd1iksij60knq=">aaab6hicdvdlsgnbeoynrxhfuy9ebopgadkvru8s0iphbmwdkixmtnqtmbmpzmafsoqlvhhqxkuf5m2/czjsiiownbrv3xr3+yngsjvol1vywv1b3yhulra2d3b3yvshtrwnkmgdxskwbz8qfdzchuzaydursenfymsf3uz91inkxepoxo8t9ei6ihjagdvgqt/2yhxxdmygjn2xilmyscqqo9yrf3b7mutdjdqtvkmo6ytay6junamcllqpwosyer1gx9cihqi8bhbohjwypu+cwjqknjmpyxmzdzuah77pdkkeqt/evpzl66q6upiyhiwpxojnfwwpidom069jn0tkwownouxycythqyop0yab0nii/5pmme06tls/r1sv8zikcathcaouxeiv7qagdwca8aqv8go9wm/wm/u+by1y+cwh/id18q2e/ozk</latexit> sha1_base64="u4/mfjoashzgmetd1iksij60knq=">aaab6hicdvdlsgnbeoynrxhfuy9ebopgadkvru8s0iphbmwdkixmtnqtmbmpzmafsoqlvhhqxkuf5m2/czjsiiownbrv3xr3+yngsjvol1vywv1b3yhulra2d3b3yvshtrwnkmgdxskwbz8qfdzchuzaydursenfymsf3uz91inkxepoxo8t9ei6ihjagdvgqt/2yhxxdmygjn2xilmyscqqo9yrf3b7mutdjdqtvkmo6ytay6junamcllqpwosyer1gx9cihqi8bhbohjwypu+cwjqknjmpyxmzdzuah77pdkkeqt/evpzl66q6upiyhiwpxojnfwwpidom069jn0tkwownouxycythqyop0yab0nii/5pmme06tls/r1sv8zikcathcaouxeiv7qagdwca8aqv8go9wm/wm/u+by1y+cwh/id18q2e/ozk</latexit> sha1_base64="b8gkrbdanlpjwzvnay8rnekmi3a=">aaab+hicbvdlssnafl2pr1ofjbp0eyycq5kiydefal24kgr2aw0ik+mkhtqzhjmjueo+xi0lrdz6ke78g6dtftp64hip59zl3dl+zkhutv1tfnbwnza3itulnd29/bj5cnirusiwaeoiralni0ky5astqgkkfwucqp+rrj9pzvzuixgsrvxbtwpihmjeauaxulryzpkdl15dz3xdmjdz3fhmil2157bwizotcuroeebxybjhjcrcyyak7dt2rnwucuuxi1lpkegsizxbi9lxlkoqsdedh55zp1ozwkekdhflzdxfgykkpzygvp4mkrrlzw8m/uf1exxu3jtyofge48vdqcisfvmzfkwhfqqrntueyuh1rryei4gw0lmvdajo8pdxsee86thv5/6i0qjlcrthge7gdby4habcqgvagcgbz3ifn+pjedhejy/famhid47gd4zph2glkjy=</latexit> This is a perfect-tensor state with I3 = 2 sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> D sha1_base64="u4/mfjoashzgmetd1iksij60knq=">aaab6hicdvdlsgnbeoynrxhfuy9ebopgadkvru8s0iphbmwdkixmtnqtmbmpzmafsoqlvhhqxkuf5m2/czjsiiownbrv3xr3+yngsjvol1vywv1b3yhulra2d3b3yvshtrwnkmgdxskwbz8qfdzchuzaydursenfymsf3uz91inkxepoxo8t9ei6ihjagdvgqt/2yhxxdmygjn2xilmyscqqo9yrf3b7mutdjdqtvkmo6ytay6junamcllqpwosyer1gx9cihqi8bhbohjwypu+cwjqknjmpyxmzdzuah77pdkkeqt/evpzl66q6upiyhiwpxojnfwwpidom069jn0tkwownouxycythqyop0yab0nii/5pmme06tls/r1sv8zikcathcaouxeiv7qagdwca8aqv8go9wm/wm/u+by1y+cwh/id18q2e/ozk</latexit> sha1_base64="b9z+eh8fgek9mgka52tj7yeqfgs=">aaab+hicbvdlssnafl2pr1ofjbp0eyycq5kiydefal24kgr2aw0ik+mkhtqzhjmjueo+xi0lrdz6ke78g6dtftp64hip59zl3dl+zkhutv1tfnbwnza3itulnd29/bj5cnirusiwaeoiralni0ky5astqgkkfwucqp+rrj9pzvzuixgsrvxbtwpihmjeauaxulryzpkdl17dzhxdrptz3fhmil2157bwizotcuroeebxybjhjcrcyyak7dt2rnwucuuxi1lpkegsizxbi9lxlkoqsdedh55zp1ozwkekdhflzdxfgykkpzygvp4mkrrlzw8m/uf1exxu3jtyofge48vdqcisfvmzfkwhfqqrntueyuh1rryei4gw0lmvdajo8pdxsee86thv5/6i0qjlcrthge7gdby4habcqgvagcgbz3ifn+pjedhejy/famhid47gd4zph2gykjy=</latexit> C A sha1_base64="u4/mfjoashzgmetd1iksij60knq=">aaab6hicdvdlsgnbeoynrxhfuy9ebopgadkvru8s0iphbmwdkixmtnqtmbmpzmafsoqlvhhqxkuf5m2/czjsiiownbrv3xr3+yngsjvol1vywv1b3yhulra2d3b3yvshtrwnkmgdxskwbz8qfdzchuzaydursenfymsf3uz91inkxepoxo8t9ei6ihjagdvgqt/2yhxxdmygjn2xilmyscqqo9yrf3b7mutdjdqtvkmo6ytay6junamcllqpwosyer1gx9cihqi8bhbohjwypu+cwjqknjmpyxmzdzuah77pdkkeqt/evpzl66q6upiyhiwpxojnfwwpidom069jn0tkwownouxycythqyop0yab0nii/5pmme06tls/r1sv8zikcathcaouxeiv7qagdwca8aqv8go9wm/wm/u+by1y+cwh/id18q2e/ozk</latexit> NAC = NBD = 1 sha1_base64="q3ykn1g3vfgbvcashgqzf8jokcw=">aaab+hicbvdlssnafl2pr1ofjbp0eyycq5kiydefal24kgr2aw0ik+mkhtqzhjmjueo+xi0lrdz6ke78g6dtftp64hip59zl3dl+zkhutv1tfnbwnza3itulnd29/bj5cnirusiwaeoiralni0ky5astqgkkfwucqp+rrj9pzvzuixgsrvxbtwpihmjeauaxulryzpkdl141s7pu1zdz3fhmil2157bwizotcuroeebxybjhjcrcyyak7dt2rnwucuuxi1lpkegsizxbi9lxlkoqsdedh55zp1ozwkekdhflzdxfgykkpzygvp4mkrrlzw8m/uf1exxu3jtyofge48vdqcisfvmzfkwhfqqrntueyuh1rryei4gw0lmvdajo8pdxsee86thv5/6i0qjlcrthge7gdby4habcqgvagcgbz3ifn+pjedhejy/famhid47gd4zph2grkjy=</latexit> C sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit>

10 B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> A C sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> sha1_base64="wghunbw9k1gci7etsrko1bhga8i=">aaab6hicdvbntwixeo3if+ix6tflizhxtokiuhajcveiiyajbei3dkhs7w7argnz8au8enayr/4kb/4bu4cjgn3jjc/vzwrmnh9zpjrch1zubx1jcyu/xdjz3ds/kb4edvwusei7jokrvpwxopwj2tfmc3obs4pdn9oep21mfu+essuicannmfvcpbysyarri7wbw2ij2ajqvosurpafwy+jmieixbovbb1dmptacq1h8x0wikgsuqejx0r1hrrrl8vsm8lpvdbifi0xmeix7rsqceivly4oncmzo4xgeeltqsof+n0ixafss9a3nshwe/xby8s/vh6ig5qxmhenmgqyxbqkhooizl/dezouad4zbbpjzk2qtlderjtsciaer0/h/6rbth1ko+1kqxg1iimptsapoacoceedximw6aackhgat+dzurmerrfrddmas1yzx+ahrldp/8enea==</latexit> A B sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> sha1_base64="at37f6zqtc/lxpkexffyrbaqydk=">aaab6hicdvdlssnafl2pr1pfvzdubovgkishtl2wunhzgn1ag8pkomnhth7mtiqs+gvuxcji1k9y5984asoo6ielh3pu5d57vjgzqszrwyhsbg5t7xr3s3v7b4dh5eotnowsqwixrdwsaw9lylliu4optgexodjwoo178+vm799tivku3qpftn0at0pmm4kvljqtcblimbwqc+vukwvak2tedmr1brjzpqi52upy+2gskssgosicszm0rvi5krakeu6xpveiayzjhe/punmqb1s66erqjbrqygt5kdavkrrsv0+kojbyexi6m8bqjn97mfixn0yu33btfsajoifzl/itjlsesq/rhalkff9ogolg+lzezlhgonq2jr3c16fof9jztnsy7u610mzlcrthdm7hemyoqxnuoa1diedhaz7g2bgzho0x43xdwjdymvp4aepte9hfjpu=</latexit> 4 Structure of holographic states The threads suggest a specific entanglement structure for holographic states (in some large-n sense) n = 2: Bipartite pure state with this amount of entanglement: N AB = S(A) = S(B) = 1 I(A : B) 2 n = 3: Recall I(A : B) = 2N AB, I(A : C) = 2N AC, I(B : C) = 2N BC Suggests decomposition into bipartite-entangled subsystems: A = A B A C, B = B A B C, C = C A C B such that ψ ABC = ψ 1 AB B A ψ 2 AC C A ψ 3 BC C B

11 S(A B ) = S(B A ) = 1 2 I(A : B), S(A C) = S(C A ) = 1 2 I(A : C), S(B C) = S(C B ) = 1 I(B : C) 2 (No tripartite entanglement!) Note: Decomposition is not by spatial position within regions (# of lines is fixed, but where they end on boundary is not) n = 4: Bipartite entanglement is not enough; there is also perfect-tensor entanglement Conjectured decomposition: A = A B A C A D A pt, B = B A B C B D B pt, C = C A C B C D C pt, D = D A D B D C D pt ψ ABCD = ψ 1 AB B A ψ 6 CD D C pt AptB ptc ptd pt S(A B ) = S(B A ) = 1 2 I(A : B),..., S(A pt) = S(B pt ) = S(C pt ) = S(D pt ) = 1 2 I 3 Simplest structure consistent with MMI Again, no tripartite entanglement Consistent with n = 3 structure: considering CD as one party, perfect tensor decomposes into bipartite-entangled state: C pt D pt = (CD) A (CD) B, pt AptB ptc ptd pt = ψ 1 Apt(CD) A ψ 2 Bpt(CD) B

12 Nezami-Walter 16: random stabilizer tensor network models of holography indeed admit this decomposition To do: Test/prove/disprove/better define this hypothesis Generalize to more parties