On the permutationally invariant part of a density matrix and nonseparability of -qubit states
We consider the concept of “the permutationally invariant (PI) part of a density matrix,” which has proven very useful for both efficient quantum state estimation and entanglement characterization of -qubit systems. We show here that the concept is, in fact, basis-dependent, but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part of a general (mixed) -qubit state , we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite separability criteria (one of which entails a sufficient criterion for genuine -partite entanglement) that can be experimentally determined by just measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of qubits, and (iv) an explicit example that shows there are, for increasing , genuinely -partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is -nonseparable, then so is the actual state. We further argue to add as requirement on any multi-partite entanglement measure that it satisfy , even though the operation that maps is not local.
Quantum computing experiments are still firmly in the testing phase. One of the practical questions that emerges in the quest to build a quantum computer is what one should measure or verify in an experiment with a number of qubits that is too small to run, say, an unprecompiled Smolin et al. (2013) algorithm in a fault tolerant way, but too large to be amenable to a full simulation or analysis.
Instead of performing quantum-state tomography on the full quantum state of qubits, a useful shortcut is to perform permutationally invariant (PI) tomography Tóth et al. (2010); Moroder et al. (2012) to estimate just the PI part of , defined as
where the denotes the set of all permutations of the qubits.
There are various points to limiting oneself to this procedure: (i) many experiments in the testing phase have as their goal the generation of PI states, with the landmark experiments on 14 entangled ions in an ion trap (GHZ state) Monz et al. (2011) and tomography on 8 ions (W state) Häffner et al. (2005), respectively, being two relevant examples, (ii) there is a simple test to verify whether one’s actual state is close to a PI state Tóth et al. (2010), (iii) the number of parameters needed to describe a PI state is (and so a full numerical analysis of such an experiment is feasible for at least Schwarz and van Enk (2013)), (iv) the number of measurement settings needed is merely , and so the experiment itself is quite feasible, (v) contains useful information about : e.g., if is not fully separable (i.e., entangled), then so is , and (vi) ground states of interesting model Hamiltonians, such as certain Hubbard models, are PI. Moreover, a lot is known about multi-partite entanglement properties of PI states and equivalence classes under SLOCC for such symmetric states Lyons and Walck (2011); Ribeiro and Mosseri (2011); Markham (2011); Novo et al. (2013), and both entanglement witnesses and criteria Tóth and Gühne (2010) as well as tomography Klimov et al. (2013) can be optimized for PI states.
We will extend this list of useful properties of the PI part of a state, but only after pointing out that the procedure of taking the PI part is basis dependent, or, perhaps more accurately, isomorphism dependent. For, the procedure of taking the PI part of a state relies on the simple fact that any two two-dimensional Hilbert spaces are isomorphic. The isomorphism is made explicit when we declare that two particular basis states of qubit , say, the states and are physically the same as the states and of qubit , respectively 111Permutation symmetry must be distinguished from exchange symmetry, which plays a fundamental role in quantum statistics. In the latter case one considers the effect on a wave function of the exchange of unphysical labels of identical particles. Here, in contrast, we distinguish the particles by their locations, and the labels and refer to spatially separated systems and are, therefore, physical.. This is such an obvious convention that it is always left unstated.
That it nonetheless has some nontrivial consequences, is shown by the following example. The state is maximally entangled, but the PI part of this state is an equal mixture of and , and that mixture is not entangled. However, had we made a slight change in the choice of basis states for the first qubit by picking , then the PI part of would have been maximally entangled. In fact, for any pure 2-qubit state, adjusting the basis states so as to be aligned with the Schmidt basis shows that its PI part is always just as much entangled as the original state itself.
We can generalize this example to obtain a much stronger statement. Since the PI part of a state is obtained by mixing states that all have exactly the same (multi-partite) entanglement properties, any quantitative measure of (multi-partite) entanglement ought to satisfy
This bound may be quite weak: the example above shows that the right-hand side (rhs) may be zero, even when is maximally entangled. However, by maximizing the rhs over all possible single-qubit basis changes , we obtain the much stronger bound
While each entanglement measure ought to satisfy these relations, one still has to prove for any individual proposed measure that it indeed does. Note that an interesting and useful intermediate symmetry [intermediate between PI with the basis fixed and PI with arbitrary local bases] for -qubit states was considered in Refs. Eltschka and Siewert (2012a); Siewert and Eltschka (2012): the so-called GHZ symmetry. This in turn leads to a similar bound on entanglement in terms of entanglement of GHZ-symmetric states Eltschka and Siewert (2012b).
As an illustration we will now focus on a particular class of separability criteria and measures (which, as it turns out, can be very efficiently measured). Multi-partite nonseparability is to be distinguished from multi-partite entanglement, as follows (see Huber et al. (2010)). A pure -partite state is called -separable if the parties can be partitioned into groups such that the state can be written as a tensor product . A general mixed state is -separable if it can be written as a mixture of -separable states, and a state is -nonseparable if it is not -separable. One relation between entanglement and -nonseparability is, that 2-nonseparability is equivalent to genuine -partite entanglement.
A quantitative measure of multi-partite -inseparability (with computable lower bounds) was introduced in Hong et al. (2012) (which generalizes to arbitrary the measure for introduced in Ma et al. (2011)). For a pure -partite state , one may define the so-called -ME concurrence in terms of all possible -partitions, as follows:
where is the reduced density matrix of subsystem (and is the complement of ). One then uses the standard convex-roof construction to extend the definition of -ME concurrence to mixed states. One important property is this: is nonzero if and only if is -nonseparable (and so it equals zero if and only if is -separable).
We state now:
Theorem 1 Suppose that is an -partite state. Then the maximum of -ME concurrence Hong et al. (2012) of the permutational invariant part of is a lower bound on the -ME concurrence Hong et al. (2012) of the original state . That is:
The proof is given in the Supplementary Material. An immediate consequence (which can be proven in other ways, too) of this Theorem is:
Corollary If an -partite state is -separable, then so is its PI part. Conversely, if the permutationally invariant part of an -partite state is -nonseparable, then so is the actual state .
(Note that for this result was known (it is stated in Ref. Tóth et al. (2010)).)
Next, we focus on separability criteria that work very well for, e.g., noisy versions of both GHZ states and W states Gao and Hong (2010, 2011); Gao et al. (2013) and that allow for a simple and inexpensive test for the whole hierarchy of -separability with running from down to 2. If an -qubit density matrix is -separable, then its matrix elements (in any given orthonormal basis) have been shown to fulfill the condition Gao et al. (2013)
where we defined
If the above inequality is violated, then is necessarily -nonseparable (i.e., not -separable). In particular, if the inequality for is violated, the state must be genuinely -partite entangled. (We repeat that the notion of -nonseparability is distinct from -partite entanglement, except for the case and .) As was shown in Gao et al. (2013), the number of local measurement settings needed to determine all matrix elements appearing in the inequality (6) is just . The number of parameters needed to fully describe such an experiment on a generic -qubit state would still grow exponentially, as .
By restricting ourselves to PI states (or, in fact, to the PI part of the full density matrix) we can reduce both the number of measurement settings needed and, by much more, the number of parameters needed to describe the experiment. To see this, first write a general PI state of qubits as
where represents the sum of all permutations on qubits. We now rewrite the matrix elements needed for criterion (6) in terms of the coefficients . The off-diagonal elements appearing on the left-hand side of (6) can be written (after some straightforward, even if a little bit tedious, algebra) as
The values of these matrix elements, therefore, do not actually depend on the indices , and they can be determined from just and for . One obtains similarly the diagonal elements appearing on the right-hand side of (6) as
as well as
Finally, for we have
All of this tells us which of the coefficients we have to determine (measure) in order to be able to apply the -separability criterion to a PI state. Noting that is fixed by normalization, we only need the following three types of coefficients
and there are such coefficients. We wish to determine all of these by measurements of local observables of the form . It was shown in Ref. Tóth et al. (2010) that such measurements suffice to determine the PI part of any -qubit state, and thereby all coefficients . One expects that for our more modest purpose fewer measurements are required.
We first note that measuring on all qubits provides us with the first type of coefficients, since
For the remaining coefficients, consider the measurement of the observable with and . We note that
Specializing to the case , we get
The coefficients appearing on the first line of the right-hand side in this equation are already determined from the measurement of . There remain then linear equations for unknowns, for each (for and no useful information is obtained). It suffices, therefore, to measure different observables with to determine the coefficients for each , by linear inversion. Similarly, it suffices to measure other observables with to determine . Hence in total measurement settings suffice to determine everything we need for all -separability criteria (6). (We show explicitly how one may implement this procedure for in the Supplementary Material.)
The number of coefficients of a general PI state that such an experiment with measurement settings actually determines (including many that are not needed to evaluate (6)) is
This, then, equals the number of model parameters one needs to describe this experiment in terms of PI states. Note that, since the number of independent PI measurement outcomes for one measurement setting equals (the number of times one finds “spin up”), not all measurement outcomes corresponding to the measurement settings are independent.
More ambitiously, one may measure all coefficients that determine a general PI state by running experiments of the above type (with measurement settings), where each experiment determines all the coefficients needed for the -separability criteria in a particular basis.
By inverting the relation (6) we can define an effective as follows
This number is efficiently measurable and quantifies the degree of separability of the PI part of one’s actual state. For example, whenever one measures the underlying PI quantum state must be genuinely -partite entangled. For a PI mixture of fully separable -qubit states one should find that .
One can now envisage an experiment in which the degree of separability is varied and measured. For example, along the lines of the experiment of Ref. Choi et al. (2010), one may generate a genuinely -partite entangled PI state, and then observe its decay over time from one separability class to the next, with monotonically increasing with the amount of decoherence. Or, likewise along the lines of Ref. Choi et al. (2010), one may introduce certain imperfections on purpose, in a controlled way, so that one knows that the state one generates falls at best into a certain class of -inseparability. One ought to measure then that .
Moreover, by running the above-mentioned versions of the same experiment, determining the -separability criteria in different bases, one can define just as many different degrees of separability, each one of which ought to increase over time, and each one of which has to obey the inequality if one creates the right type of imperfections. This way one verifies directly both the theory and one’s experimental control, all at a cost growing only polynomially with .
Let us consider here a simple example of how increases with the amount of decoherence, and how it behaves as a function of the number of qubits. Consider a mixture of the -qubit W state and the maximally mixed -qubit state,
for . For this state we obtain the degree of separability as a function of and as
We plot as a function of in Fig. 1, for several values of .
One notable feature visible in the plot is that for large (even for ) the state stays genuinely -partite entangled even when is very close to unity (because ), but then suddenly shoots up to its maximum value for . That is, in the limit of one can find arbitrarily close to the maximally mixed (and fully separable) state a genuinely -partite entangled state. This peculiar behavior agrees with one particular fact about bi-partite entanglement in infinite dimensional systems: arbitrarily close to any separable state lies an entangled state with an arbitrarily large amount of entanglement Eisert et al. (2002); van Enk (2003).
In conclusion, we obtained strong bounds on a multi-partite nonseparability measure, by considering the basis dependence of the permutation operation. We showed that whenever the permutationally invariant (PI) part of a state is -nonseparable, then so is the state itself. Moreover, we have shown it takes just measurement settings to determine all matrix elements needed for checking all -separability criteria for obtained in Gao et al. (2013), as applied to PI -qubit states. The required measurements can be performed and analyzed (using, e.g., model selection Burnham and Anderson (2002)) for a number of qubits that goes well beyond the current state-of-the-art of . From these -separability criteria we obtained in a natural way “degrees of separability,” which are efficiently measurable even for many qubits, and which ought to increase monotonically with the amount of decoherence.
This work was supported by the National Natural Science Foundation of China under Grant No: 11371005, and by the Hebei Natural Science Foundation of China under Grant No: A2012205013.
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