Entanglement and Symmetry in Multiple-Qubit States: a geometrical approach Gregg Jaeger Quantum Imaging Laboratory and College of General Studies Boston University, Boston MA 015 U. S. A. Abstract. The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite quantum systems. In particular, invariants under these operations have provided insight into quantum entanglement in multiplequbit states. Relationships between entanglement, mixedness and spin symmetry in multiple-qubit quantum states can be found by exploiting these properties. For example, concurrence and n-tangle are naturally expressed in terms of spin-flip transformations. In the case of specialized entanglement measures and/or special families of states, complementarity relations involving these lengths and state transformations are derivable. Here, the role of geometry in such investigations is explored. Minkowskian geometry is seen to provide elegant representations of entanglement in multiple-qubit systems, particularly the language of twistors. INTRODUCTION. As in all areas of physics, symmetry has been found to play an important role in the investigation of quantum mechanics. This is especially evident now in the study of quantum information and quantum entanglement. The behavior of quantum states under local unitary transformations (LUTs) and stochastic local quantum operations and classical communication (SLOCC) has proven central to the understanding of entanglement in multipartite systems. The associated group-invariant lengths are, geometrically speaking, Euclidean and Minkowskian in character. Spin symmetries also play a role in the relationship between these geometrical quantities. In particular, the relationship between entanglement, mixedness and spin symmetry in multiple-qubit quantum states can be fruitfully explored with these properties and methods in mind. In quantum information theory, stochastic local operations and classical communication (SLOCC) on density matrices [1] are described by elements of the group, SL(,C), which is homomorphic to the Lorentz group. The state of a single qubit has an associated invariant length under transformations of the proper Lorentz group, O o (1,). The Lorentz-group invariant length for every possible finite number of qubits can be naturally written in terms of spin expectation values, or generalized Stokes parameters lying in Minkowski space. The quantum state purity is the corresponding Euclidean length, namely that invariant under the rotation subgroup SU(). Here we focus on the description of these geometrical quantities for multiple-qubit states and the behavior of states of any finite number of qubits under LUTs and SLOCC with an emphasis on their relationship to entanglement. The associated Minkowskian geometry is explored and shown to be a natural one for describing entanglement that is conveniently expressed in twistorial language. From an empirical point of view, the state of a qubit is most naturally characterized by the expectation values of the Pauli spin matrices, the Stokes parameters, as these are the sort of quantities that can in practice be measured by sharp quantum measurements of qubits. Stokes parameters allow one to easily visualize the qubit state geometrically via a Poincaré sphere. The state of a quantum ensemble of individual qubits can be completely described by the set of expectation values S µ Tr(ρσ µ ) (µ 0,1,,), (1)
where σ 0 1 and σ i, i 1,,, are the Pauli matrices. Likewise, one can write the density matrix as ρ 1 S µ σ µ, () µ0 and the vector space for one qubit state-vectors is C. The elements of the group of proper Lorentz transformations O o (1,) represented by six basic matrix forms of matrix, M 1,...,M 6, for example the rotation M 1 and the boost M 4 1 0 0 0 coshχ sinhχ 0 0 0 cosα sinα 0 sinhχ coshχ 0 0 M 1 (α) 0 sinα cosα 0 M 4 (χ) 0 0 1 0. () 0 0 0 1 0 0 0 1 These transformations correspond to transformations on elements of H(), the vector space of all x complex Hermitian matrices, which include the density matrices describing mixed states of single qubits of the traditional quantum formalism. Stokes four-vectors lie in the Minkowskian real vector space, R 4 1,, the four-dimensional real vector space R 4 endowed with the Minkowski metric (+,,, ), i.e. together with a metric tensor g µν possessing, as non-zero elements, the diagonal entries +1, 1, 1, and 1 []. In general, the length of a four-vector x µ in R 4 1, is given by < x,x > g µν x µ x ν. More explicitly, in R 4 1,, the length of a vector x (x 0,x 1,x,x ) is given by x R 4 1, x 0 x 1 x x. (4) Using the standard vector basis for R 4, e 0 (1,0,0,0), e 1 (0,1,0,0), e (0,0,1,0), e (0,0,0,1), there exists a natural vector-space isomorphism, ω : R 4 1, H(), relating the space, R4 1,, of these vectors to the space of state matrices H(); the density matrices of quantum mechanics are just those of trace one within the space of Hermitian matrices H(). Suitably defining a map γ : SL(,C) H() O o (1,) R 4 1,, one arrives at a commuting diagram, of the wellknown relationship between SL(,C) and O 0 (1,), now tailored to the quantum mechanical context (see [] for details). SL(,C) H() α H() γ ω O o (1,) R 4 β 1, R 4 1, The Minkowskian length, l, of the vector of expectation values is then l x 0 x 1 x x. One thus has a well defined Lorentz-invariant length expressible in terms of the Stokes parameters: S (1) S 0 S 1 S S. (5) MULTIPLE-QUBIT STATES AND MINKOWSKIAN GEOMETRY One can extend this picture to multiple-qubit systems to capture state purity and entanglement, beginning with two qubits (see [4] for details). The Lorentz group on two-qubit systems acts on the joint expectation values S µν Tr(ρσ µ σ ν ), where µ,ν 0,1,,, and expressing the matrix of the general state of a two qubit ensemble []: ρ 1 4 µ,ν0 S µν σ µ σ ν, (6)
where σ µ σ ν (µ,ν 0,1,,) are simply tensor products of the identity and Pauli matrices, and the state-vector space for pure states of two qubits is C C. The four-vector, S µ, must then be generalized to a 16-element tensor, S µν. The two-qubit density matrices ρ are positive, unit-trace elements of the 16-dimensional complex vector space of Hermitian 4 4 matrices, H(4). The tensors σ µ σ ν provide a basis for H(4), which is isomorphic to the tensor product space H() H() of the same dimension. A density matrix for the general state of a two-qubit system is an element of H(4) H() H() of the form ρ 1 ( σ 4 0 σ 0 + i1 S i0 σ i σ 0 + j1 S 0 j σ 0 σ j + i, j1 S i j σ i σ j ), (7) in R 4 1, R4 1,, expressed in terms of the elements of standard vector basis for R4, e 0 (1,0,0,0), e 1 (0,1,0,0), e (0,0,1,0), e (0,0,0,1). One has SL(,C) SL(,C) H() H() α α H() H() γ γ ω ω O o (1,) O o (1,) R 4 1, R 4 β β 1, R 4 1, R 4 1, where the invariant length is given by the tensor norm l 1 x R 4 1, R4 1,< x,x > S () (S 00 ) i1 (S i0 ) j1 (S 0 j ) + i1 j1 (S i j ), (8) which is invariant under products of local Lorentz group transformations in O o (1,) O o (1,). This construction is then naturally further generalized. The generalization to the case of n-qubits provides the invariant length for any finite number of qubits. The n-qubit Stokes tensor S i1...i n transforms under the group O o (1,) as S i 1,...,i n L j 1...L j n i j 1,..., j n 0 1 i n S j1... j n, (9) where the L j are such transformations acting in the spaces of qubits 1,..., n. The Minkowskian squared-norm of the i Stokes tensor {S i1...i n } provides the invariant length (here renormalized by the factor n for convenience): S (n) 1 n {(S 0...0 ) + n k,l1 i k,i l 1 n (S 0...ik...0 ) k1 i k 1 (S 0...ik...i l...0 ) + ( 1) n (S i1...i n ) }. (10) i 1,...,i n 1 The purity for a general n-qubit state is the Euclidean length in the space of multiple-qubit Stokes parameters, namely P(ρ) Trρ 1 n Si 1...i n. (11) i 1,...,i n 0
SPIN SYMMETRY, MIXEDNESS AND ENTANGLEMENT: COMPLEMENTARITY RELATIONS Wootters and coworkers (see, for example, [5]) related his concurrence measure of entanglement and its square, the tangle, for two-qubits to the so-called magic basis, which appears naturally in two-qubit systems. This basis is just the Bell basis with a particular choice of norm and phase. (In the next section, we will discuss its deeper connection to Minkowskian geometry.) It arises through the introduction of the spin-flip operation ρ ρ σ ρ σ, (1) where ρ is the complex conjugate of the -qubit density matrix, ρ, and σ is the spin-flipping Pauli matrix. Wootters showed that the concurrence of a pure or mixed two-qubit state, C(ρ AB ), can be expressed in terms of the minimum average pure-state concurrence, C( Ψ AB ), where the minimum is taken over all possible ensemble decompositions of ρ AB and that, in general, C(ρ) max{0,λ 1 λ λ λ 4 }, where the λ i are the square roots of the eigenvalues of the product matrix ρ ρ, the singular values, all of which are non-negative real quantities. It has also been shown that the entanglement of formation of a mixed state ρ of two qubits can be expressed in terms of the concurrence as E f (ρ) h(c(ρ)), (1) where h(x) x log x (1 x) log (1 x) (see [5]). For two-qubit pure states, the Lorentz-group invariant coincides with the tangle measure: S () (P[ ψ ]) τ(p[ ψ ]) C (P[ ψ ]), (14) where P[ ψ ] ψ ψ is the projector corresponding to its state-vector argument, ψ. The tangle, τ, can be generalized to any even number of qubits: taking ψ σ N ψ, (15) where now ψ is a multiple qubit state, one obtains the N-tangle measure [6] generalizing τ so as to apply to N-qubit states, namely τ N ψ ψ, (16) where N is even. The SL(,C) n -invariant quantity S (n) Tr(ρ ρ), (17) is naturally expressed in terms of the generalization of the spin-flip operation to any number of qubits (see [5]), namely ρ ρ σ n ρ σ n. (18) This length is naturally connected to the n-tangle, a general multipartite entanglement measure for even numbers of qubits. For pure states ρ ψ ψ, one has that S (N) ψ ψ τ N (19) (see [6]). The SLOCC (quantum Lorentz group) invariant length also coincides for pure states with the symmetry measure I(ρ, ρ) (see [4]) defined as I(ρ, ρ) 1 D HS(ρ ρ), (0) where D HS (ρ ρ) is the (renormalized) Hilbert-Schmidt distance in the space of density matrices, D HS (ρ ρ ) 1 Tr[(ρ ρ ) ], (1) which measures the indistinguishability of the density operator, ρ, from the density operator, ρ.
The multi-partite entanglement and mixedness can then be related by the square of the Hilbert-Schmidt distance between the state ρ and its corresponding spin-flipped counterpart ρ: D HS(ρ ρ) 1 Tr[(ρ ρ) ] 1 [ ] Trρ + Tr ρ Tr(ρ ρ) Trρ Tr(ρ ρ) P(ρ) S n(ρ). () Thus, we have the following relation between the chosen measure of multi-partite state entanglement and the state purity: Sn (ρ) + D HS (ρ ρ) P(ρ), () where D HS (ρ ρ) can be understood as a measure of distinguishability between the n-qubit state ρ and the corresponding spin-flipped state ρ. The relation of Eq. () gives the following simple, entirely general, relation between multipartite entanglement, (ρ), and mixedness, M(ρ) 1 P(ρ): S (n) S(n) (ρ) + M(ρ) I(ρ, ρ), (4) noting that I(ρ, ρ) measures the indistinguishability of the density matrix, ρ, from the corresponding spin-flipped state, ρ. MULTIPLE-QUBIT INVARIANTS, ENTANGLEMENT AND MINKOWSKIAN GEOMETRY Invariants under LOCC and SLOCC have been considered as algebraic entities by some workers (see, e.g., [7]). For example, Kempe [8] considered LOCC invariants for three-qubit states Of degree two, are the norm of the state, S 000, while of degree four are I i jkmpq α ki j α mi jα mpq α kpq, I ψ α i jk i jk. (5) i jk I 1 α i jk αi jk, (6) i jk i jkmpq α ik j α im jα pmq α pkq, I 4 i jkmpq α i jk α i jmα pqm α pqk, (7) the purities of the density matrices of single particles obtained by partial tracing out the remaining two particles. Note that, labeling the first qubit system A, the second B and the third C, S (1) (ρ A ) 1 I, S (1) (ρ B ) 1 I, S (1) (ρ C ) 1 I 4, (8) where ρ X are the single-qubit reduced density matrices. Rhe natural character of the Minkowskian lengths is evident. Kempe pointed out the LOCC invariant of higher degree for three particles, I 5 i jklmnopq not in general algebraically independent of I,I and I 4. α i jk α ilm α lno α p joα pqm α nqk, (9)
Introducing the Levi-Civita symbol ε, defined by ε 00 0 ε 11 and ε 01 1 ε 10 and related to the σ Pauli matrix by σ iε, the polynomial invariants of pure states (characterized by standard i j... -basis amplitudes α i, j,... ) under SL(,C) can be written (see [7]): K σ allindices1 ε i1,i ε j1, j ε k1,k...ε ir 1,i r ε jr 1, j r ε kr 1,k r α i σ(1) j τ(1) k τ(1)... α i σ() j τ() k τ()...... α i σ(r) j τ(r) d τ(r).... (0) (where the σ and τ are permutations over r elements) which are our Lorentz group invariant lengths S (n), simply written in terms of state amplitudes rather than probabilities (Stokes parameters). All invariants can be written in terms of these basic polynomials. In particular, the three-tangle for three subsystems A, B,C can be written in terms of them, being of the form τ ABC 1 ε i1,i ε j1, j ε k1,k 4 ε i,i 4 ε j, j 4 ε k,k α i σ(1) j τ(1) k τ(1)α i σ() j τ() k τ()α i σ() j τ() d τ()α i σ(4) j τ(4) d τ(4). (1) Note that this expression is symmetric under permutations of qubit indices i, j,k. It should by now not be surprising to find that considerable insight can be achieved by considering such quantities in the Minkowskian context. Levay [9] recently introduced the idea of a twistor formulation of the above considered quantities, allowing the identification of a previously unknown invariant in the space of three qubits. He noticed that twistors Z µ and W µ, invariant under local SL(,C) transformations of a single qubit, can be defined in terms of threequbit pure states. Let this qubit be the first subsystem, A, and let Z 0 1 (α 000 + α 011 ), Z 1 i (α 010 + α 001 ), Z 1 (α 010 α 001 ), Z i (α 000 α 011 ), () and W 0 1 (α 100 + α 111 ), W 1 i (α 110 + α 101 ), W 1 (α 110 α 101 ), W i (α 100 α 111 ). () Following Levay, defining the products over generic twistors U,V, as U V U µv µ,u V U µ V µ, and take U U U, the elements of the reduced state describing the second and third qubits can be written ρ µν BC Zµ Z µ +W ν W ν. (4) We can then compactly write our Lorentz invariant length for the second and third qubits in terms of these twistors as S () (ρ BC ) Z Z + W W + Z W, (5) where S () is given by Eq. (11). Furthermore, taking ρ + ρ AB and ρ ρ AC, one has S () (ρ ±) Tr(ρ ± ρ ± ) (P µν ± P µν ) P µν, (6) where P µν 1 ε µνρσ P ρσ, where we have introduced the Plücker coordinates P µν Z µ W ν Z ν W µ, (7) which are clearly also invariant with respect to SL(,C) transformations on the first particle. Thus, one can express our invariant Stokes lengths naturally in twistorial language. Levay noted that on can write the three tangle as τ ABC P µν P µν, (8) while the tangle between the first qubit and the compound system of qubits two and three can be written τ A(BC) P µν P µν. (9)
Using such an approach, he was able to discover the new invariant σ ABC ψ ψ τ ABC 4Tr(ρ BC P P), (40) which can be written elegantly as the sum of magnitudes of the two principle null directions of the Klein quadric, U µ ± Z νp νµ ± 1 τabc e iφ/ Z µ (41) V µ ± Pµν W ν ± 1 τabc e iφ/ W µ (4) where φ arg[(z W) (Z Z)(W W)], i.e. compactified complexified Minkowski space. That is, σ ABC U + + U + V + + V (4) - a simple compact expression, much as Eq. (7) is in contrast to its unwieldy standard expression, Eq. (1). One can then show that σ ABC 0 for the GHZ states, thus singling them out. CONCLUSION. We have considered the application of the Lorentz group to multiple-qubit states and their representation in the context of Minkowskian geometry. The multiple-qubit state expectation values form Minkowskian tensors and give rise to invariant lengths under the action of the Lorentz group. This length is the Minkowskian analog of the quantum state purity, which is the corresponding Euclidean length. SLOCC invariant quantities have been shown to describe entanglement properties of multiple qubit states, particularly entanglement within two and three qubits systems. By considering multipartite entanglement and mixedness together with the degree of symmetry of quantum states under the n-qubit spin-flip transformation, a general relation was found between these fundamental properties for even numbers of qubits. Multipartite entanglement and state mixedness are seen to be complementary within classes of states possessing the same degree of spin-flip symmetry. For pure states, the value of this multipartite entanglement measure, the degree of spin-flip symmetry, and the n-tangle are seen to coincide. Furthermore, SLOCC invariants naturally describe entanglement within three-qubit systems where they are amenable to a twistorial description. It seems likely that the geometrical characterization of entanglement will continue to be a natural language for describing entanglement, a quantity whose polynomial description rapidly becomes unwieldy, as systems involving greater numbers of qubits are explored in detail. It is hoped that the connections made here between the various representations of Lorentz-group invariants and entanglement will stimulate others to recognize their geometrical character and the value of the language of twistors in describing them. REFERENCES 1. Bennett, C. H., et al., Phys. Rev. A 6, 0107 (001).. Han, D., et al., Phys. Rev. E 56, 6065 (1997).. Jaeger, G. S., et al., Proc. Conf. Foundations of Probability and Physics -, Vaxjo, Sweden 00 (quant-ph/01174) (001). 4. Jaeger, G. S., et al., Phys. Rev. A 67, 007 (00). 5. Hill, S., et al., Phys. Rev. Lett. 78, 50 (1997). 6. Wong, A., et al., Phys. Rev. A 6, 04401 (001). 7. Leifer, M. S., et al., Phys. Rev. A 69, 0504 (004). 8. Kempe, J., Phys. Rev. A 60, 91 (1999). 9. Levay, P., quant-ph/040060 (004).