On monogamy of four-qubit entanglement
Our main result is a monogamy inequality satisfied by the entanglement of a focus qubit (one-tangle) in a four-qubit pure state and entanglement of subsystems. Analytical relations between three-tangles of three-qubit marginal states, two-tangles of two-qubit marginal states and unitary invariants of four-qubit pure state are used to obtain the inequality. The contribution of three-tangle to one-tangle is found to be half of that suggested by a simple extension of entanglement monogamy relation for three qubits. On the other hand, an additional contribution due to a two-qubit invariant which is a function of three-way correlations is found. We also show that four-qubit monogamy inequality conjecture of ref. [PRL 113, 110501 (2014)], in which three-tangles are raised to the power , does not estimate the residual correlations, correctly, for certain subsets of four-qubit states. A lower bound on residual four-qubit correlations is obtained.
Entanglement is a necessary ingredient of any quantum computation and a physical resource for quantum cryptography and quantum communication niel11 (). It has also found applications in other areas such as quantum field theory cala12 (), statistical physics sahl15 (), and quantum biology lamb13 (). Multipartite entanglement that comes into play in quantum systems with more than two subsystems, is a resource for multiuser quantum information tasks. Since the mathematical structure of multipartite states is much more complex than that of bipartite states, the characterization of multipartite entanglement is a far more challenging task horo09 ().
Monogamy is a unique feature of quantum entanglement, which determines how entanglement is distributed amongst the subsystems. Three-qubit entanglement is known to satisfy a quantitative constraint, known as CKW monogamy inequality coff00 (). In recent articles regu14 (); regu15 (); regu16 (), it has been shown that the most natural extension of CKW inequality to four-qubit entanglement is violated by some of the four-qubit states and different ways to extend the monogamy inequality to four-qubits have been conjectured. For a subclass of four-qubit generic states, an extension of strong monogamy inequality to negativity and squared negativity karm16 () is satisfied, however, there exist four-qubit states for which negativity and squared negativity are not strongly monogamous. Three-qubit states show two distinct types of entanglement. As we go to four qubits, additional degrees of freedom make it possible for new entanglement types to emerge. It is signalled by the fact that corresponding to the three-qubit invariant that detects genuine three-way entanglement of a three-qubit pure state, a four-qubit pure state has five three-qubit invariants for each set of three qubits shar16 (). An -qubit invariant is understood to be a function of state coefficients which remains invariant under the action of a local unitary transformation on the state of any one of the qubits. A valid discussion of entanglement monogamy for four qubits, therefore, must include contributions from invariants that detect new entanglement types.
This article is an attempt to identify, analytically, the contributions of two-tangles (pairwise entanglement), three-tangles (genuine three-way entanglement) and four tangles to entanglement of a focus qubit with the three remaining qubits (one-tangle) in a four qubit state. To do this, we express one-tangle in terms of two-qubit invariants. Monogamy inequality constraint on four qubit entanglement is obtained by comparing the one-tangle with upper bounds on two-tangles and three-tangles shar216 () defined on two and three qubit marginal states. Contribution of three-tangles to one-tangle is found to be half of what is expected from a direct generalization of CKW inequality to four qubits. The difference arises due to new entanglement modes that are available to four qubits. It is verified that the ”residual entanglement”, obtained after subtracting the contributions of two-tangles and three-tangles from one-tangle, is greater or equal to genuine four-tangle. Genuine four-tangle shar14 (); shar16 () is a degree eight function of state coefficients of the pure state. Besides that, the ”residual entanglement” also contains contributions from square of degree-two four-tangle shar10 (); luqu03 () and degree-four invariants which quantify the entanglement of a given pair of qubits with its complement in a four-qubit pure state shar10 ().
Ii One-tangle of a focus qubit in a four-qubit State
We start by expressing one-tangle of a focus qubit in a four-qubit state in terms of two-qubit invariant functions of state coefficients. Entanglement of qubit with in a two-qubit pure state
is quantified by two-tangle defined as
where is a two-qubit invariant. Here are the state coefficients. On a four qubit pure state, however, for each choice of a pair of qubits one identifies nine two-qubit invariants. Three-tangle coff00 () of a three-qubit pure state is defined in terms of modulus of a three-qubit invariant. On the most general four qubit state, on the other hand, we have five three-qubit invariants corresponding to a given set of three qubits. Four-qubit invariant that quantifies the sum of three-way and four-way correlations of a three-qubit partition in a pure state is known to be a degree-eight invariant shar16 (), which is a function of three-qubit invariants. It is natural to expect that the monogamy inequality for four qubits takes into account the entanglement modes available exclusively to four-qubit system. To understand, how various two-tangles and three-tangles add up to generate total entanglement of a focus qubit in a pure four-qubit state, we follow the steps listed below:
(1) Write down one-tangle of focus qubit as a sum of two-qubit invariants.
(2) Express two-tangles, three-tangles and four-tangle or the upper bounds on the tangles in terms of two-qubit invariants.
(3) Rewrite one-tangle in terms of tangles defined on two- and three-qubit reduced states and ”residual four-qubit correlations”.
(4) Compare the ”residual four-qubit correlations” with the lower bound on four-qubit correlations written in terms of four-qubit invariants.
To facilitate the identification of two-qubit and three-qubit invariants, we use the formalism of determinants of two by two matrices of state coefficients referred to as negativity fonts. For more on definition and physical meaning of determinants of negativity fonts, please refer to section (VI) of ref. shar16 ().
For the purpose of this article, we write down and use the determinants of negativity fonts of a four-qubit state when qubit is the focus qubit. On a four-qubit pure state, written as
where state coefficients are complex numbers and refers to the basis state of qubit , , we identify the determinants of two-way negativity fonts to be , , and . Besides that we also have (three-way), (three-way), (three-way), and (four-way), as the determinants of negativity fonts.
One-tangle given by , where Tr, quantifies the entanglement of qubit with , and . It is four times the square of negativity of partial transpose of four-qubit pure state with respect to qubit pere96 (). Negativity, in general, does not satisfy the monogamy relation. However, it has been shown by He and Vidal he15 () that negativity can satisfy monogamy relation in the setting provided by disentangling theorem. It is easily verified that
One-tangle depends on way, way and way correlations of focus qubit with the rest of the system.
Iii Definitions of two-tangles and three-tangle
This section contains the definitions of two-tangles and three-tangles for pure and mixed three-qubit states. Consider a three-qubit pure state
Using the notation from ref. shar16 (), we define () (the determinant of a two-way negativity font) and , () (the determinant of a three-way negativity font). Entanglement of qubit with the rest of the system is quantified by one-tangle , where Tr. One can verify that
For qubit pair in , we identify three two-qubit invariants that is
while for the pair two-qubit invariants are
These two-qubit invariants transform under a unitary on the third qubit in a way analogous to the complex functions , and of Appendix A. Then the invariants corresponding to , and are three-qubit invariants for the given choice of qubit pair. In Table I, we enlist the correspondence of two-qubit invariants for qubit pairs and , with complex numbers , and of Appendix A and set the notation for invariants corresponding to , and
|Two-qubit invariants||Three-qubit invariants|
One-tangle in terms of three-qubit invariants listed in column five of Table I reads as
Three tangle coff00 () of pure state is equal to the modulus of the polynomial invariant of degree four that is
The entanglement measure is extended to a mixed state of three qubits via convex roof extension that is
where minimization is taken over all complex decompositions of . Here is the probability of finding the normalized state in the mixed state
Two-tangle of the state is constructed through convex roof extension as
Two-tangle , where is the concurrence hill97 (); woot98 (). One can verify that the invariants , , corresponding two-tangles, and three-tangle saturate the inequalities corresponding to Eq. (66) that is
Since , we obtain
Moduli of two-qubit invariants, which depend only on the determinants of three-way negativity fonts, are used to define new two-tangles on the state via
iii.1 What does measure?
To understand the correlations represented by , we examine a generic three-qubit state in its canonical form. A state is said to be in the canonical form when it is expressed as a superposition of minimal number of local basis product states (LBPS) acin01 (). The state coefficients of this form carry all the information about the non-local properties of the state, and do so minimally. Starting from a generic state in the basis (Eq. (5)), local unitary transformations allow us to write it in a form with the minimal number of LBPS. As a first step towards writing the state in canonical form with respect to qubit, we chose a unitary that results in a state on which one of the two-way two-qubit invariants is zero that is or . For example a unitary with , acting on qubit gives a state U , such that . After eliminating , the state re ads as
It is straight forward to write down the local unitaries and that lead to the canonical form,
Notice that on canonical state, . Next we determine the range of values that takes on a generic state .
Combining the definition of (from Table I) for the state , with the result of Eq. (13), that is
Since (), the value of satisfies
In general, the difference measures the distance of a given three-qubit state from its canonical form with respect to qubit.
On a pure state of three qubits, . The state on which , is obtained by a unitary transformation such that
where three-qubit invariant reads as
Next, consider the three-qubit mixed state , where is an un-normalized state. Let the set of two-qubit invariants for the pair in the state be
New two-qubit invariant (Eq. (17)) on is given by
where from Eq. (23),
with defined as
If and are the local unitaries on the third qubit such that and , then
Obviously, the value of satisfies either the condition
or the constraint
Iv Tangles and Three-qubit invariants of a four-qubit state
In this section, we identify relevant combinations of two-qubit invariants that remain invariant under a local unitary on the third qubit. Three-qubit invariants that we look for are the ones related to tangles of three-qubit reduced states obtained from four-qubit pure state by tracing out the degrees of freedom of the fourth qubit. For any given pair of qubits in a general four-qubit state, there are nine two-qubit invariants. Of the six degree-four three-qubit invariants constructed from the set of nine two-qubit invariants, one is defined only on the pure state. Five remaining invariants are functions of three-tangles and two-tangles. In Table II, we identify sets of two-qubit invariants of a four-qubit state which transform under a unitary, on the third qubit in the same way as the functions , and of Appendix A.
|Two-qubit invariants||Three-qubit invariants|
Three-qubit invariants listed in the last three columns depend on two-qubit invariants of columns two to four in the same way as , and depend on , and , for example three-qubit invariants in the third row of Table II read as
and satisfy the inequality
Here is the un-normalized state defined through .
Upper bound on two-tangle calculated by using the method of ref. shar16 () shows that
Similarly the upper bounds on for the nine families of four-qubit states, calculated in ref. shar216 () satisfy the condition
On , new two-qubit invariant (Eq. (17)) is defined as
where . New three-qubit tangle on a pure state is defined as , where
The invariants , and satisfy the inequality (analogous to Eq. (66)),
Using a similar argument, three-qubit invariants listed in lines 5 and 6 of Table II satisfy the inequalities
where three-tangle defined on pure four-qubit state reads as , and
Using invariants of local unitaries on qubits and , and definitions given in lines 7 and 8 of Table II, we obtain the inequalities
where new three-tangle reads as , and
V Monogamy of four-qubit entanglement
To obtain the relation between tangles of reduced states and one-tangle of the focus qubit, firstly, we identify the three-qubit invariant combinations of two-qubit invariants in Eq. (4). It is found that a four-qubit invariant of degree two, which is defined only on the pure state, is also needed. Genuine four-tangle (Eq. (74) appendix B), defined in refs. shar14 (); shar16 () is a degree-eight function of state coefficients. However, the degree-two four-qubit invariant which is equal to invariant H of refs. shar10 (); luqu03 (), is known to have the form,
Four-tangle defined as , is non zero on a GHZ state and vanishes on W-like states of four qubits. However, since fails to vanish on product of entangled states of two qubits, it is not a measure of genuine four-way entanglement. By direct substitution, one-tangle of Eq. (4) can be rewritten in terms of three-qubit invariants listed in column five of Table II and square of four-qubit invariant that is