Nonlocal Properties and Local Invariants

for Bipartite Systems

Sergio Albeverio^{a} ^{1}, Shao-Ming Fei^{a,b} ^{2}, Preeti Parashar^{c} ^{3}, Wen-Li Yang^{d,e} ^{4}

a Institut für Angewandte Mathematik, Universität Bonn, D-53115 b Department of Mathematics, Capital Normal University, Beijing 100037 c Max-Planck Institute für Mathematik, D-53111 Bonn d Physikalisches Institut der Universität Bonn, 53115 Bonn e Institute of Modern Physics, Northwest University, Xian 710069

Abstract

The nonlocal properties of generic N-dimensional bipartite quantum systems are investigated. A complete set of invariants under local unitary transformations is presented. It is shown that two generic density matrices are locally equivalent if and only if all these invariants have equal values in these density matrices.

PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w

1SFB 256; SFB 237; BiBoS; CERFIM(Locarno); Acc. Arch.; USI(Mendriso)

e-mail: albeverio@uni-bonn.de 2e-mail: fei@uni-bonn.de 3e-mail: parashar@mpim-bonn.mpg.de 4e-mail: wlyang@th.physik.uni-bonn.de

As a fundamental phenomenon in quantum mechanics, nonlocality has been given a lot of attention in foundational considerations, in the discussion of Bell type inequalities [1] and hidden variable models, see e.g. [2]. Nonlocality turns out to be also very important in quantum information processing such as quantum computation [3], quantum teleportation [4, 5, 6, 7], dense coding [8] and quantum cryptographic schemes [9, 10, 11]. Nonlocal correlations in quantum systems imply a kind of entanglement among the quantum subsystems. The nonlocal properties as well as the entanglement of two parts of a quantum system remain invariant under local transformations of these parts.

The method developed in [12, 13], in principle, allows one to compute all the invariants of local unitary transformations, though it is not easy to perform it operationally. In [14], the invariants for general two-qubit systems are studied and a complete set of 18 polynomial invariants is presented. It is proven that two qubit mixed states are locally equivalent if and only if all these 18 invariants have equal values in these states. Therefore any nonlocal characteristics of entanglement is a function of these invariants. In [15] three qubits states are also discussed in detail from a similar point of view.

In the present paper we discuss the locally invariant properties of arbitrary dimensional bipartite quantum systems. We present a complete set of invariants and show that two generic density matrices are locally equivalent if and only if all these invariants have equal values in these density matrices.

We first consider the case of pure states. Let H be an N-dimensional complex Hilbert space, with e_{i}, i = 1, ..., N, as an orthonormal basis. A general pure state on H ⊗ H is of the form

|Ψ⟩ = ∑_{i,j=1}^{N} a_{ij}e_{i} ⊗ e_{j},  a_{ij} ∈ C  (1)

with the normalization ∑_{i,j=1}^{N} a_{ij}a_{ij}^{∗} = 1 (∗ denoting complex conjugation).

A quantity is called an invariant associated with the state |Ψ⟩ if it is invariant under all local unitary transformations, i.e. all maps of the form U ⊗ U from H ⊗ H to itself, where U is a unitary transformation on the Hilbert space H. Let A denote the matrix given by (A)_{ij} = a_{ij}. The following quantities are known to be invariants associated with the state |Ψ⟩ given by (1), see [16, 17, 18, 19]:

I_{α} = T r(AA^{†})^{α},  α = 1, ..., N;  (2)

(with A^{†} the adjoint of the matrix A).

In terms of the Schmidt decomposition, a given |Ψ⟩ can always be written in the following form, using a suitable orthonormal basis {e_{i}}, i = 1, ..., N:

|Ψ⟩ = ∑_{i=1}^{N} \sqrt{Λ_{i}}e_{i} ⊗ e_{i},

where ∑_{i=1}^{N} Λ_{i} = 1, Λ_{i} ≥ 0. The Λ_{i}, i = 1, ..., N, are the eigenvalues of the matrix AA^{†}. As AA^{†} is self-adjoint, there always exists a unitary matrix V, V V^{†} = V^{†}V = 1, such that V AA^{†}V^{†} = diag{Λ_{1}, ..., Λ_{N}}. The invariants (2) can then be written in the form:

I_{α} = ∑_{i=1}^{N} Λ_{i}^{α}, α = 1, ..., N.

[Lemma 1]. Two pure states (on H ⊗ H) are equivalent under locally unitary transformations if and only if they have the same values of the invariants I_{α}, α = 1, ..., N.

[Proof]. Let

|Ψ′⟩ = ∑_{i,j=1}^{N} a′_{ij}e_{i} ⊗ e_{j}, a′_{ij} ∈ C, ∑_{i,j=1}^{N} a′_{ij}a_{ij}^{′∗} = 1.

Let A′ denote the matrix given by (A)′_{ij} = a′_{ij}, Λ′_{i} the eigenvalues of the matrix A′A^{′†}, and I′_{α} the corresponding invariants: I′_{α} = T r(A′A^{′†})^{α} = ∑_{i=1}^{N} Λ_{i}^{′α}, α = 1, ..., N.

From the equation

(−1)^{N}det(AA^{†} − λ) = λ^{N} + c_{1}λ^{N−1} + ... + c_{N},

where

c_{1} = I_{1}, c_{k} = −[I_{k} + c_{1}I_{k−1} + ... + c_{k−1}I_{1}], k ≥ 2,

we have that if I′_{α} = I_{α}, ∀α = 1, ..., N, then A′A^{′†} and AA^{†} have the same eigenvalues, Λ′_{i} = Λ_{i}, i = 1, ..., N. Therefore |Ψ′⟩ and |Ψ⟩ are equivalent under local unitary transformations: |Ψ′⟩ = u ⊗ w|Ψ⟩ for some u, w ∈ U, U denoting the family of all unitary operations on H. Two pure states on H ⊗ H are equivalent under local unitary transformations if and only if they have the same set of invariants (2).

From the proof of Lemma 1, one can deduce the following conclusion:

[Lemma 2]. Two Hermitian m × m matrices A and B are unitary equivalent (i.e., there exists a unitary matrix u on an m-dimensional complex vector space satisfying A = uBu^{†}) if and only if

T r(A^{α}) = T r(B^{α}), for α = 1, ..., m.

We consider now mixed states on H ⊗ H. Let ρ be a density matrix defined on H ⊗ H with rank(ρ) = n ≤ N^{2}. ρ can be decomposed according to its eigenvalues and eigenvectors:

ρ = ∑_{i=1}^{n} λ_{i}|ν_{i}⟩⟨ν_{i}|,

where λ_{i} and |ν_{i}⟩, i = 1, ..., n, are the nonzero eigenvalues and eigenvectors respectively of the density matrix ρ. |ν_{i}⟩ has the form

|ν_{i}⟩ = ∑_{k,l=1}^{N} a_{kl}^{i}e_{k} ⊗ e_{l}, a_{kl}^{i} ∈ C, ∑_{k,l=1}^{N} a_{kl}^{i}a_{kl}^{i∗} = 1, i = 1, ..., n.

Let A_{i} denote the matrix given by (A_{i})_{kl} = a_{kl}^{i}. We introduce {ρ_{i}}, {θ_{i}},

ρ_{i} = T r_{1}|ν_{i}⟩⟨ν_{i}| = A_{i}A_{i}^{†}, θ_{i} = T r_{2}|ν_{i}⟩⟨ν_{i}| = A_{i}^{†}A_{i}, i, j = 1, ..., n,  (3)

T r_{1} and T r_{2} stand for the traces over the first and second Hilbert spaces respectively, and therefore, ρ_{i} and θ_{i} can be regarded as reduced density matrices. Let Ω(ρ) and Θ(ρ) be two “metric tensor” matrices, with entries given by

Ω(ρ)_{ij} = T r(ρ_{i}ρ_{j}), Θ(ρ)_{ij} = T r(θ_{i}θ_{j}), for i, j = 1, ..., n,  (4)

and

Ω(ρ)_{ij} = Θ(ρ)_{ij} = 0, for N^{2} ≥ i, j > n.

A mixed state ρ is generic if the corresponding “metric tensor” matrices Ω, Θ satisfy det(Ω(ρ)) ≠ 0 and det(Θ(ρ)) ≠ 0. Obviously, a generic state implies n = N^{2} or det(ρ) ≠ 0.

Similarly we also introduce X(ρ) and Y (ρ) as

X(ρ)_{ijk} = T r(ρ_{i}ρ_{j}ρ_{k}), Y (ρ)_{ijk} = T r(θ_{i}θ_{j}θ_{k}), i, j, k = 1, ..., n.  (5)

[Theorem]. Two generic density matrices are equivalent under local unitary transformations if and only if the following invariants have the same values for both density matrices:

\begin{matrix}J^{s}(ρ) = T r_{2}(T r_{1}ρ^{s}), s = 1, ..., N^{2}; \\ Ω(ρ), Θ(ρ), X(ρ), Y (ρ). \end{matrix}  (6)

[Proof]. We first show that the quantities given in (6) are invariant under local unitary transformations. Let u and w be unitary transformations, uu^{†} = u^{†}u = ww^{†} = w^{†}w = 1. Under the local unitary transformation u ⊗ w, we have ρ → ρ′ = u ⊗ w ρ u^{†} ⊗ w^{†}. Correspondingly, we have |ν_{i}⟩ → |ν′_{i}⟩ = u ⊗ w|ν_{i}⟩, or equivalently A_{i} is mapped to A′_{i} = u^{t}A_{i}w, where u^{t} is the transpose of u. Therefore

ρ′_{i} = A′_{i}A_{i}^{′†} = u^{t}A_{i}A_{i}^{†}u^{∗} = u^{t}ρ_{i}u^{∗}, θ′_{i} = A_{i}^{′†}A′_{i} = w^{†}A_{i}^{†}A_{i}w = w^{†}θ_{i}w.  (7)

By using (7), it is straightforward to check the following relations:

J^{s}(ρ) → J^{s}(ρ′) = T r_{2}[∑_{i=1}^{n} λ_{i}^{s}T r_{1}(|ν′_{i}⟩⟨ν′_{i}|)] = T r_{2}[∑_{i=1}^{n} λ_{i}^{s}A′_{i}A_{i}^{′†}] = J^{s}(ρ),

Ω(ρ)_{ij} → Ω(ρ′)_{ij} = T r(ρ′_{i}ρ′_{j}) = T r(u^{t}ρ_{i}ρ_{j}u^{∗}) = Ω(ρ)_{ij}, Θ(ρ)_{ij} → Θ(ρ′)_{ij} = T r(θ′_{i}θ′_{j}) = T r(w^{†}θ_{i}θ_{j}w) = Θ(ρ)_{ij} X(ρ)_{ijk} → X(ρ′)_{ijk} = T r(ρ′_{i}ρ′_{j}ρ′_{k}) = T r(u^{t}ρ_{i}ρ_{j}ρ_{k}u^{∗}) = X(ρ)_{ijk}, Y (ρ)_{ijk} → Y (ρ′)_{ijk} = T r(θ′_{i}θ′_{j}θ′_{k}) = T r(w^{†}θ_{i}θ_{j}θ_{k}w) = Y (ρ)_{ijk},

where i, j, k = 1, ..., n. Hence the quantities in (6) are invariants of local unitary transformations. If two density matrices are equivalent under local unitary transformations, then their corresponding invariants in (6) have the same values.

Now suppose conversely that the states ρ = ∑_{i=1}^{n} λ_{i}|ν_{i}⟩⟨ν_{i}| and ρ′ = ∑_{i=1}^{n} λ′_{i}|ν′_{i}⟩⟨ν′_{i}| give the same values to the invariants in (6). We are going to prove that ρ and ρ′ are equivalent under local unitary transformations.

a) As

J^{s}(ρ) = T r_{2}(∑_{i=1}^{n} λ_{i}^{s}T r_{1}(|ν_{i}⟩⟨ν_{i}|)) = T r_{2}(∑_{i=1}^{n} λ_{i}^{s}A_{i}A_{i}^{†}) = ∑_{i=1}^{n} λ_{i}^{s},

from J^{s}(ρ′) = J^{s}(ρ) we have

∑_{i=1}^{n} λ_{i}^{′s} = ∑_{i=1}^{n} λ_{i}^{s}, ∀ s = 1, ..., N^{2}.

From Lemma 2, we have that ρ′ and ρ have the same nonzero eigenvalues, i.e., λ′_{i} = λ_{i}, i = 1, ..., n.

b) From (4), the generic condition det(Ω(ρ)) ≠ 0, det(Θ(ρ)) ≠ 0, implies that ρ_{i} (resp. θ_{i}), i = 1, ..., n(= N^{2}), span the space of N × N matrices and form an N-dimensional representation of the associative algebra gl(N, C) (in the sense of universal enveloping algebras),

ρ_{i}ρ_{j} = ∑_{k=1}^{n} C_{ij}^{k}ρ_{k}, θ_{i}θ_{j} = ∑_{k=1}^{n} D_{ij}^{k}θ_{k}, C_{ij}^{k} , D_{ij}^{k} ∈ C  (8)

Taking trace of (8) and using the condition T rρ_{i} = T rθ_{i} = 1 one gets

Ω_{ij} = ∑_{k=1}^{n} C_{ij}^{k}, Θ_{ij} = ∑_{k=1}^{n} D_{ij}^{k}.  (9)

From (9) and (5) we obtain

X_{ijk} = ∑_{l=1}^{n} C_{ij}^{l}Ω_{lk}, Y_{ijk} = ∑_{l=1}^{n} D_{ij}^{l}Θ_{lk}.

Therefore

C_{ij}^{l} = ∑_{k=1}^{n} X_{ijk}Ω^{lk}, D_{ij}^{l} = ∑_{k=1}^{n} Y_{ijk}Θ^{lk},  (10)

where the matrices Ω^{ij} and Θ^{ij} are the corresponding inverses of the matrices Ω_{ij} and Θ_{ij} (which exist by the genericity assumption). (10) implies that the structure coefficients C_{ij}^{l} and D_{ij}^{l} of the algebras (8) are given by {Ω_{ij}, X_{ijk}} and {Θ_{ij}, Y_{ijk}} respectively.

The generic density matrix ρ′ also gives rise to N-dimensional representation of the associative algebra gl(N, C),

ρ′_{i}ρ′_{j} = ∑_{k=1}^{n} C_{ij}^{′k}ρ′_{k}, θ′_{i}θ′_{j} = ∑_{k=1}^{n} D_{ij}^{′k}θ′_{k}, C_{ij}^{′k} , D_{ij}^{′k} ∈ C.  (11)

If Ω(ρ′) = Ω(ρ), Θ(ρ′) = Θ(ρ), X(ρ′) = X(ρ) and Y (ρ′) = Y (ρ), we have C_{ij}^{′l} = C_{ij}^{l} and D_{ij}^{′l} = D_{ij}^{l}. Due to the fact that all the Casimir operators of the algebra gl(N, C) can be written as polynomials in the algebraic generators, we conclude that the values of all the Casimir operators given by the two N-dimensional representations {ρ_{i}} and {ρ′_{i}} (resp. {θ_{i}} and {θ′_{i}}) are equal. Hence, these two sets of representations (primed and unprimed) of the algebra gl(N, C) are equivalent, i.e.,

ρ′_{i} = u^{t}ρ_{i}u^{∗}, θ′_{i} = w^{†}θ_{i}w  (12)

for some u, w ∈ U. Therefore |ν′_{i}⟩ = u ⊗ w|ν_{i}⟩, i = 1, ..., n, and ρ′ = u ⊗ w ρ u^{†} ⊗ w^{†}. Hence ρ′ and ρ are equivalent under local unitary transformations. Remark. For a degenerate state ρ, det(Ω(ρ)) = 0 or det(Θ(ρ)) = 0, the above invariants (6) are not complete in the sense that two degenerate density matrices can not be equivalent under local unitary transformations even if they give the same values to the invariants in (6). This is due to the fact that there exist null vectors for the degenerate state. For example in the case det(Ω(ρ)) = 0, there exists at least one Hermitian matrix B which satisfies T r(Bρ_{i}) = 0 for i = 1, ..., n. Hence Ω(ρ′)_{ij} = Ω(ρ)_{ij} and X(ρ′)_{ijk} = X(ρ)_{ijk} are not enough to get the first equivalence relation of (12). In this case some new invariants have to be introduced to get a complete set of invariants. From the algebraic relations (8) and formula (10), other generalized invariants like T r((ρ_{i})^{m_{i}}(ρ_{j})^{m_{j}}...(ρ_{k})^{m_{k}}) and T r((θ_{i})^{m_{i}}(θ_{j})^{m_{j}}...(θ_{k})^{m_{k}}), i, j, ..., k = 1, ..., n; m_{i}, m_{j}, ..., m_{k} ∈ IN, can be represented in terms of {Ω_{ij}, X_{ijk}} and {Θ_{ij}, Y_{ijk}}, for example,

T r (ρ_{i_{1}}ρ_{i_{2}} ⋯ ρ_{i_{m}}) = ∑_{{α_{1},···,α_{m−2}}} C_{i_{1} i_{2}}^{α_{1}}C_{α_{1} i_{3}}^{α_{2}} ⋯ C_{α_{m−3} i_{m−1}}^{α_{m−2}}Ω_{α_{m−2} i_{m}}.

Hence by doing so we do not get new invariants.

To summarize, we have discussed here the local invariants for arbitrary dimensional bipartite quantum systems and have presented a set of invariants of local unitary transformations. The set of invariants is not necessarily independent but it is complete in the sense that two generic density matrices are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices.

Acknowledgments We thank M. Grassl for interesting discussions. P.P. would like to thank Prof. Manin for encouragement and the Max-Planck Institute für Mathematik, Bonn, for warm hospitality. W-L Y. thanks Prof. von Gehlen for his continuous encouragement. He has been supported by the Alexander-von-Humboldt Foundation.

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