18 Introduction to Entanglement Entropy - …

1 18 Introduction to Entanglement EntropyThe next few lectures are on Entanglement Entropy in quantum mechanics , in quantumfield theory, and finally in quantum gravity. Here s a brief preview: Entanglemententropy is a measure of how quantum information is stored in a quantum state. Withsome care, it can be defined in quantum field theory, and although it is di cult tocalculate, it can be used to gain insight into fundamental questions like the nature ofthe renormalization group. In holographic systems, Entanglement Entropy is encodedin geometric features of the bulk will start at the beginning with discrete quantum systems and work our way up toquantum :Harlow s lectures on quantum information in quantum gravity, available onthe arxiv, may be useful. See also Nielsen and Chuang s introductory book on quantuminformation for derivations of various statements about matrices, traces, positivity, Definition and BasicsAbipartite systemis a system with Hilbert space equal to the direct product of twofactors,HAB=HA HB.

2 ( )Starting with a general (pure or mixed) state of the full system , the reduced densitymatrix of a subsystem is defined by the partial trace, A=trB ( )and theentanglement entropyis the von Neumann Entropy of the reduced densitymatrix,SA tr Alog A.( )Example: 2 qubit systemIf each subsystemAorBis a single qubit, then the Hilbert space of the full system is165spanned by|00i,|01i,|10i,|11i,( )where the first bit refers toAand the second bit toB, ,we use the shorthand|iji |iiA|jiB |iiA |jiB.( )Suppose the system is in the pure state| i=1p2(|00i+|11i),( )so =| ih |. As a 4x4 matrix, has diagonal and o -diagonal elements. Diago-nal density matrices are just classical probability distributions, but the o -diagonalelements indicate Entanglement and are intrinsically reduced density matrix of systemAis A=trB =12Bh0|(|00i+|11i)(h11|+h00|)|0iB+12Bh1| (|00i+|11i)(h11|+h00|)|1iB=12 |0iAAh0|+|1iAAh1| /12x2.

3 ( )The last line says Ais proportional to the identity matrix of a 2-state system. In thiscase we say Aismaximally mixed, and the initial state| iismaximally Entanglement Entropy of subsystemAis easy to calculate for a diagonal matrix,SA= tr Alog A= 2 14log14= log 2.( )Interpretation of Entanglement entropyIn fact the 2-qubit example illustrates a useful way to put Entanglement Entropy into166words: Entanglement Entropy counts the number of entangled bits we hadkqubits in systemAandkqubits in systemB, then in a maximally entangledstateSA=klog 2. SoSAcounts the number of bits, or equivalently,eSAcounts thenumber of entangled states (sincekqubits have 2kstates).Rephrased slightly:Given a state Awith Entanglement entropySA, the quantityeSAis the minimal numberof auxiliary states that we would need to entangle withAin order to obtain Afrom apure state of the enlarged decompositionA very useful tool is the following theorem, called the Schmidt decomposition: Supposewe have a systemABin a pure state| i.

4 Then there exist orthonormal states|iiAofAand|iiBofBsuch that| i=Xi i|iiA| iiB,( )with ireal numbers in the range [0,1] satisfyingXi 2i=1.( )The number of terms in the sum is (at most) the dimension of the smaller Hilbert :See Wikipedia, or Nielsen and Chuang chapter small andBis big, this is intuitive. It says we can pick a basis for|iiA, and eachof these states will be correlated with a particular state of systemB. The thermofielddouble is an obvious subsystemsAn immediate consequence of the Schmidt decomposition is that a pure state of system167 ABhasSA=SB(pure states).( )To see this, write the reduced density matrices in the Schmidt basis, A=Xi 2i|iiAAhi|, B=Xi 2i|iiBBhi|.( )Both density matrices have eigenvalues 2iso they have the same Entropy . ( ) doesnot hold for mixed states Geometric Entanglement entropyEntanglement Entropy can be defined whenever the Hilbert space splits into two very important example is when we defineAas a subregion of :Nspins on a lattice in 1+1 dimensionsLet s arrangeNspins in a line.

5 DefineAto be a spatial region containingkspins, andB=ACis everything else:The most general state of this system is| i=X{si}cs1 sN|s1i|s2i |sNi( )wheresi= 0 or 1 (meaning up or down ), and thec s are complex with system sizeLet s restrict to 1 |A| |B|, so that we can think of subsystemAas large andBas infinite. In a random state, ,one in which the coe cientscij are drawn froma uniform distribution, we expect any subsystemAto be almost maximally entangledwithB. In the language of the Schmidt decomposition, this means that iis nonzero168and 1/p2kfor a complete basis of states|iiA. In fact this is a theorem, see Harlow slectures for the exact , the Entanglement Entropy scales as the number of spins in +1d this is linear in the size ofA, and more generally,SA Volume(A) (random state).( )In other words, most states in the Hilbert space of the full system have entanglementscaling with , often we are interested in the groundstate.

6 Ground states of a local Hamilto-nian are very non-generic, and the corresponding Entanglement entropies obey specialscaling laws. Usually, if the system is gapped ( ,correlations die o exponentially),the ground state must obey thearea law:SA Area(A) (ground state of local, gapped Hamiltonian).( )(This is a theorem in 1+1d, and usually true in higher dimensions.)Thus groundstates occupy a tiny, special corner of the Hilbert space. This is a cornerwith especially low complexity. Intuitively speaking, a large degree of entnaglementis what makes quantum information exponentially more powerful than classical infor-mation; so states with lower Entanglement Entropy are less complex. More specifically,this actually means that you can encode a groundstate wavefunction with far fewerparameters than the 2 Ncomplex numbers appearing in ( ).DMRGIn 1+1d, the area law becomes simplySA const( )independent of the system size.

7 This special feature is responsible for a hugely impor-tant technique in quantum condensed matter called thedensity matrix renormalizationgroup(DMRG). This technique is used to e ciently compute groundstate wavefunc-169tions of 1+1d systems using a computer. This would not be possible for general states,since (we think) classical computers require exponential time to simulate quantum sys-tems. But ( ) means that, in a precise sense, groundstates of gapped 1d systemsare no more complex than classical at a critical pointThe area law applies to gapped systems. Near a critical point, where dof become mass-less and long-distance correlations are power-law instead of exponentially suppressed,the area law can be violated. In a 1+1d critical system, and therefore also in 1+1dconformal field theory, ( ) is replaced bySA logLA( )whereLAis the size of regionA. This is bigger than the area law, but still much lowerthan the volume-scaling of a random Entropy InequalitiesRelative entropyMuch of the recent progress in QFT based on Entanglement comes from a few inequal-ities obeyed by Entanglement Entropy .

8 Define therelative entropyS( || ) tr log tr log .( )(Note that this is not symmetric in , .) This obeysS( || ) 0( )with equality if and only if = . The proof of this statement is straightforward,see Wikipedia. It just involves some matrix manipulations. The key ingredient is thefact that density matrices in quantum mechanics are very special: they have a positivespectral decomposition, =Xipiviv i( )170wherepiis non-negative andviis a basis vector. This is necessary for quantum me-chanics to have a sensible probabilistic interpretation and is closely related to relative Entropy can be viewed as a measure of how distinguishable and are. In the classical case (diagonal and ), it is error we will make in predictingthe uncertainty of a random process if we think the probability distribution is , butactually it is . Given this interpretation, positivity is obvious clearly we will neverdobetterusing the wrong inequalityPositivity or relative Entropy implies thetriangle inequality,|SA SB| SAB.

9 ( )Mutual informationDefine themutual information,I(A, B) SA+SB SAB.( )This can be written as a relative Entropy , and is therefore non-negative:I(A, B)=S( AB|| A B) 0.( )Roughly,I(A, B) measures the amount of information thatAhas aboutB(or vice-versa, since it is symmetric).In a pure state ofAB, the only correlations betweenAandBcome from Entanglement ,so in this caseI(A, B) measures Entanglement betweenAandB. However, in a mixedstate,I(A, B) also gets classical contributions. For example in a 2-qubit system, it iseasy to check that the classical mixed state AB/|00ih00|+|11ih11|( )has non-zero mutual subadditivitySo far we have discussed partitioning a system into two piecesAandB, but we canpartition further and find new inequalities. Thestrong subadditivityinequality (SSAfor short) applied to a tripartite systemHABC=HA HB HC, isSABC+SB SAB+SBC.( )This is less mysterious if written in terms of the mutual information,I(A, B) I(A, BC).

10 ( )Although this inequality seems obvious clearlyAhas more information aboutBCthan aboutBalone and is just a feature of positive matrices, it is surprisinglydi cult to prove. See Nielsen and Chuang for a totally unenlightening it is useful to express ( ) in di erent notation, whereAandBare twooverlapping subsystems, which are not independent:SA[B+SA\B SA+SB.( )Exercise: Positivity of classical relative entropyProve that the classical relative Entropy is non-negative. That is, prove ( ), as-suming and are : Mutual information practiceConsider a 2-qubit system. First, calculate the mutual information of the two bits inthe classical mixed state =12(|00ih00|+|11ih11|).( )This is a clearly a state with the maximal amount of classical correlation if wemeasure one bit, we know the value of the second , what is the maximal amount of mutual information for aquantum(pure or172mixed) state of 2 qubits?]