1
Entanglement in Quantum Mechanical Systems Johar M. Ashfaque We introduce the notion of bipartite entanglement for pure states in finite-dimensional quantum mechanical systems and classify the states into two types depending on whether they contain non-trivial entanglement entropy that quantifies the amount of quantum entanglement. A set of inequalities of entanglement entropy which play crucial rules will be stated without proofs. I.
BIPARTITE ENTANGLEMENT
Given a lattice model or QFT, suppose the system is in a pure ground state |ψi that is the density matrix for the Hilbert space Htot is given by ρtot = |ψihψ|. We normalize the ground state as hψ|ψi = 1 so that trtot (ρtot ) = 1. We then divide the total system into two subsystems A and B = A complementary to each other. In the spin chain example, we cut off the chain in between the sites and divide the lattice points into two groups. Note that the procedure of cutting is an imaginary process without changing the system at all. In what follows, the total Hilbert space will be assumed to take a direct product form of two Hilbert spaces pf the subsystems Htot = HA ⊗ HB . Let {|iiB , i = 1, 2, ...} be an orthonormal basis in HB and define the reduced density matrix ρA of the system A by taking the partial trace over the system B X ρA ≡ trB (ρtot ) ≡ B hi|ρtot |iiB . i
Note that this definition depends on the choice of the subsystem but does not depend on the choice of the orthonormal basis. For example if ρtot is the tensor product of the density matrices ρA and ρB describing the subsystems that is ρtot = ρA ⊗ ρB then the partial trace simply recovers them trB (ρtot ) = ρA ,
trA (ρtot ) = ρB .
The total density matrix needs not be pure but one may ask once the complete information ρA about a system A is given if it is possible to find a pure density matrix in an enlarged Hilbert space of HA whose partial trace recovers ρA . Indeed one can always construct such an enlarged Hilbert space and the density matrix in the following way. The most general density matrix is of the form X ρA = pi |iiAA hi| i
P where {|iiA } is an orthonormal basis of HA and the coefficients pi ≥ 0 satisfy i pi = 1. We then copy HA into another Hilbert space HA with the basis given by {|iiA } and define the pure density matrix ρ by X√ ρ = |χihχ|, |χi ≡ pi |iiA ⊗ |iiA i
in the enlarged Hilbert space H = HA ⊗ HA . It is easy to check that this construction correctly reproduces ρA under the partial trace over A.The operation is called entanglement purification as it constructs
2 a pure disentangled state from a mixed entangled state by enlarging the Hilbert space. Finally, we can define the entanglement entropy associated with the quantum subsystem A which is given by the von Neumann entropy of the reduced density matrix ρA as SA = − trA [ρA log ρA ]. Note that the entanglement entropy of the total system always vanishes, Stot = 0 for a pure ground state. Entanglement entropy remains finite in a finite-dimensional quantum system but suffers from UV divergences in QFT due to the short range interaction near the boundary of the subsystem A.
II.
SEPARABLE AND ENTANGLED STATES
So what does the entanglement entropy truly measure for a given system. To proceed further consider the pure ground state |ψi in a general form X |ψi = ciµ |iiA ⊗ |µiB i,µ
where |iiA and |µiB are the orthonormal bases for HA = {|iiA , i = 1, ..., dA } and HB = {|µiB , µ = 1, ..., dB } respectively and the coefficient ciµ is a dA × dB matrix with complex entries. There are two different cases depending on the type of the coefficient matrix ciµ . A.
Separable States
B When cij factorizes, cij = cA i cµ , the ground state |ψi is called a separable state (a pure product state) and can be recast into the pure product form
|ψi = |ψA i ⊗ |ψB i where |ψA i ≡ becomes pure
P
A i ci |iiA
and |ψB i ≡
B i cµ |µiB .
P
This is the case where the reduced density matrix also
ρA = |ψA ihψA |. Therefore a separable state has vanishing entanglement entropy SA = 0. Moreover, one show that entanglement entropy vanishes if and only if the pure ground state is separable.
B.
Entangled States
The ground state is called an entangled state (or inseparable state) if it is not separable (with the B coefficient matrix ciµ 6= cA i cµ ). This is the case where the entanglement entropy takes a positive value. Indeed, the form |ψi =
X i,µ
ciµ |iiA ⊗ |µiB
3 can be simplified by changing the bases to Schmidt decomposition form min(dA ,dB )
|ψi =
X
√
pk |ψk iA ⊗ |ψk iB
k=1
P where pk are non-negative real numbers satisfying k pk = 1 and |ψk iA,B are the new orthonormal bases for the subsystems A and B respectively. Note that this decomposition works for any dA ×dB rectangular matrix ciµ . To see how it actually works, we diagonalize the coefficient matrix by the singular-value decomposition c = U ΣV † where U , V are dA × dA amd dB × dB unitary matrices respectively and Σ is a diagonal dA × dB real √ matrix with non-negative real entries pk ≥ 0 with k = 1, ..., min(dA , dB ). The square root for the eigenvalues is purely conventional. The new orthonormal bases |ψk iA,B are the unitary transformation of the original ones. In short, an entangled state is a superposition of several quantum states. An observer who only has access to a subsystem A will be found to be in a mixed state when the pure ground state |ψ|i in the total system is entangled: |ψi : separable ←→ ρA : purestate, |ψi : entangled ←→ ρA : mixedstate. Entanglement entropy therefore measures how much a given state differs from a separable state. It reaches the maximum value when a given state is a superposition of all possible quantum states with an equal weight.
III.
TWO QUBIT SYSTEM
Let us consider a simple example of an entangled state. Consider a system of two particles A and B with spin 12 . The Hilbert spaces HA and HB are spanned by two states: HA,B = {|0iA,B , |1iA,B }. Since the total Hilbert space is the tensor product of the two subsystems Htot = HA ⊗HB it has a four-dimensional orthonormal basis: Htot = {|00i, |01i, |10i, |11i}. Suppose the ground state is given by 1 |ψi = √ |01i − |10i . 2 The reduced density matrix for the particle A is obtained by taking the partial trace over HB of the total density matrix 1 ρA = |0iAA h0| + |1iAA h1| . 2 It is neat to write it in matrix form acting on the two-dimensional vector space HA 1 0 2 ρA = . 0 21 It clearly shows that ρA is not pure and the entanglement entropy does not vanish SA = log 2.
4 IV.
THERMOFIELD DOUBLE STATE
A non-trivial example of an entangled state is the thermofield double state defined by 1 X −βEn /2 |ψi = √ e |niA ⊗ |niB Z n P where we normalize the state with the partition function Z = n e−βEn . The peculiarity of the thermofield double state becomes manifest in taking the partial trace over the subsystem B. Namely the reduced density matrix for the subsystem A becomes a Gibbs state of inverse temperature β ρA =
1 −βHA e Z
where the modular Hamiltonian has been introduced such that HA |niA = En |niA . In this example, the entanglement entropy measures the thermal entropy of the subsystem A SA = − trA [ρA (−βHA − log Z)] = β(hHA i − F ) where F is the thermal free energy βF = − log Z. The thermofield double state is also important to understand the thermal nature of black holes when considering QFT on a background geometry with a horizon.
V.
BELL STATES
In the two qubit system, we saw that the ground state was maximally entangled. There are in total four independent maximally entangled states in the two qubit system 1 |B1 i = √ |00i + |11i , 2 1 |B2 i = √ |00i − |11i , 2 1 |B3 i = √ |01i + |10i , 2 1 |B4 i = √ |01i − |10i . 2 These are known as the Bell states or EPR pairs in quantum information theory. These states manifest their quantum mechanical aspects in a sense that they violate the Bell’s inequalities holding in a local hidden variable theory accounting for the probabilistic features of quantum mechanics with a hidden variable and a probability density. For a system of n qubits, there are entangled states called the GHZ states 1 ⊗n ⊗n |GHZi = √ |0i + |1i . 2 Another type of entangled states are the W state 1 |W i = √ |10...00i + |01...00i + ... + |00...01i . n These two types of states are inequivalent as the GHZ state is fully separable while the W state is not.
5 VI.
PROPERTIES OF ENTANGLEMENT ENTROPY
Entanglement entropy enjoys several useful properties. • If a ground state wave-function is pure then the entanglement entropy of the subsystem A and its complement B = A are the same SA = SB . This follows from the symmetry of the Schmidt decomposition under the exchange of A and B. However, SA is no longer equal to SB when the total system is in a mixed state say at finite temperature. • Given two disjoint subsystems A and B the entanglement entropies satisfy subadditivity SA∪B ≤ SA + SB . Also it satisfies the triangle inequality or the Araki-Lieb inequality |SA − SB | ≤ SA∪B which is symmetric between A and B. • For any three disjoint subsystems A, B, and C the following inequalities hold SA∪B∪C + SB ≤ SA∪B + SB∪C , SA + SC ≤ SA∪B + SB∪C . There are known as the strong subadditivity, the most fundamental inequalities for entanglement entropy. The two inequalities can be shown to be equivalent to each other. The proof is based on convexity of a function built from the density matrix that is Hermitian when a system is unitary. VII.
ENTANGLEMENT ENTROPY AT FINITE TEMPERATURE
The entanglement entropy SA (T ) at finite temperature T = β −1 can be defined just by replacing the total density matrix with thermal density matrix ρthermal =
e−βH tr(e−βH )
where H is the Hamiltonian of the total system. By definition SA (β) equals the thermal entropy when A is the total system Stot = Sthermal . Let |ψi be the ground state with no energy H|ψi = 0 and |φi be the first excited, normalized state with the energy H|φi = Eφ |φi. Then the density matrix has an expansion around the ground state at zero temperature ρthermal =
|ψihψ| + |φihφ|e−βEφ + ... . 1 + e−βRφ + ...
The reduced density matrix allows a similar expansion ρA = ρ0A + e−βEφ (ρφA − ρ0A ) + ... where ρ0A ≡ trB (|ψihψ|) and ρφA ≡ trB (|φihφ|).
6 The Rényi entropy is a one-parameter generalization of the entanglement entropy labelled by an integer n called the replica parameter Sn (A) =
1 log trA (ρnA ). 1−n
In the n → 1 limit with the normalization trA (ρA ) = 1 the Rényi entropy reduces to entanglement entropy SA = lim Sn (A). n→1
This means that the entanglement entropy receives the universal thermal contribution from the excited state SA (T ) = SA (T = 0) + e−βEφ ∆hH0A i + ... where H0A ≡ − log ρ0A is the modular Hamiltonian for the ground state and ∆hH0A i stands for the difference of the modular Hamiltonian ∆hH0A i ≡ trA [H0A (ρφA − ρ0A )].