Extending Noether’s theorem by quantifying the asymmetry of quantum states
Noether’s theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law. It is, however, deficient in several respects: (i) it is not applicable to dynamics wherein the system interacts with an environment, and (ii) even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries. To address these deficiencies, we introduce measures of the extent to which a quantum state breaks a symmetry. Such measures yield novel constraints on state transitions: for nonisolated systems, they cannot increase, while for isolated systems they are conserved. We demonstrate that the problem of finding nontrivial asymmetry measures can be solved using the tools of quantum information theory. Applications include deriving model-independent bounds on the quantum noise in amplifiers and assessing quantum schemes for achieving high-precision metrology.
Finding the consequences of symmetries for dynamics is a subject with broad applications in physics, from high energy scattering experiments, through control problems in mesoscopic physics, to issues in quantum cosmology. In many cases, a complete solution of the dynamics is not possible either because it is too complex or because one lacks precise knowledge of all of the relevant parameters. In such cases, one can often still make nontrivial inferences by a consideration of the symmetries. The most prominent example is the inference from dynamical symmetries to constants of the motion in closed system dynamics. For instance, from invariance of the laws of motion under translation in time, translation in space, and rotation, one can infer, respectively, the conservation of energy, linear momentum and angular momentum. This result has its origin in the work of Lagrange in classical mechanics Goldstein (), but when the symmetries of interest are differentiable, the connection is established by Noether’s theorem Noether (). These days, physicists tend to use the term “Noether’s theorem” to refer to the general result, and we follow this convention here. The theorem applies also in the quantum realm, where symmetries of the time evolution imply the existence of a set of observables all of whose moments are conserved WeinbergQFT ().
A symmetric evolution is one that commutes with the action of the symmetry group BRSreview (). For instance, a rotationally-invariant dynamics is such that a rotation of the state prior to the dynamics has the same effect as doing so after the dynamics. An asymmetry measure quantifies how much the symmetry in question is broken by a given state. More precisely, a function from states to real numbers is an asymmetry measure if the existence of symmetric dynamics taking to implies BRST06 (); GourSpekkens (); Vaccaro (); Toloui (). A measure for rotational asymmetry, for instance, is a function over states that is non-increasing under rotationally-invariant dynamics.
For systems interacting with an environment (open-system dynamics), where Noether’s theorem does not apply, every asymmetry measure imposes a non-trivial constraint on what state transitions are possible under the symmetric dynamics, namely that the measure evaluated on the final state be no larger than that of the initial state.
For isolated systems (closed-system dynamics), the existence of a symmetric unitary for some state transition implies the existence of a symmetric unitary for the reverse transition (namely, the adjoint of the unitary), hence each asymmetry measure is a conserved quantity under the symmetric dynamics. We show that for transitions between mixed states, the conserved quantities one obtains in this way are independent of those prescribed by Noether’s theorem. In this way, we find new conservation laws which are not captured by Noether’s theorem.
Our results also allow us to derive constraints on state transitions given discrete symmetries of the dynamics, that is, symmetries associated with finite groups, where there are no generators of the group action and it is less obvious how to apply Noether’s theorem.
Ii The inadequacy of Noether conservation laws for general dynamics
How can we find nontrivial asymmetry measures? In the case of rotational symmetry, one might guess that the (expectation value of) components of angular momentum are good candidates. For one, a state with nonzero angular momentum is necessarily non-invariant under some rotation. For another, in closed system dynamics, any asymmetry measure must be a constant of the motion and angular momentum certainly satisfies this condition. Nonetheless, it turns out that angular momentum is not an asymmetry measure. More generally, it turns out that none of the Noether conserved quantities, nor any functions thereof, provide nontrivial measures of asymmetry. To prove this, it is necessary to provide more precise definitions of the notions of symmetric operations and symmetric states.
One specifies the symmetry of interest by specifying an abstract group of transformations and the appropriate representation thereof. For a general symmetry described by a group the symmetry transformation corresponding to the group element is represented by the map where and is a unitary operator. For instance, under rotation around an axis by angle the density operator of system will be transformed as where is the vector of angular momentum operators. A state does not break the symmetry , or is symmetric relative to group , if for all group elements it holds that . The concept of symmetry transformations and symmetric states can be naturally extended to the case of time evolutions. A time evolution is in general a linear transformation from the space of density operators to itself, . We say the transformation is symmetric relative to group if this transformation commutes with the symmetry transformation for all group elements in group (See Fig. 1). Note that this definition of symmetric time evolution applies equally well to the cases of closed system dynamics, where the system does not interact with an environment, and open system dynamics (See Fig. 2).
For a symmetry described by a Lie group , Noether’s theorem states that every generator of is conserved, or equivalently, the expectation value of any function of is conserved. Therefore, if under a unitary symmetric dynamics, state evolves to state , then all the implications of Noether’s theorem can be summarized as:
for every generator of .
We can now make precise the sense in which Noether conserved quantities yield no nontrivial measures of asymmetry: For a symmetry corresponding to any compact Lie group , any asymmetry measure that is a continuous function of only the Noether conserved quantities is trivial, that is, it takes the same value for all states.
The proof is provided in the supplementary material, but we will here sketch the main idea. States that are asymmetric must necessarily fail to commute with some generator of the group, say , and consequently must have coherence between different eigenspaces of (for example, a state that is noninvariant under phase shifts necessarily has coherence between eigenspaces of the corresponding number operator). A nontrivial asymmetry measure must be able to detect such coherence. If the state were known to be pure, then the presence of such coherences would be revealed by a nonzero variance over . However, an asymmetry measure is a function over all states, and there exist mixed states that have nontrivial variance over even though they have no coherence between the eigenspaces of . Hence the value of the second moment of has no information about the asymmetry properties of the state. By the same logic, no moment of has any information about the asymmetry properties of the state.
It follows that the problem of devising measures of asymmetry is nontrivial. Nonetheless, it can be solved by taking an information-theoretic perspective on symmetric dynamics.
Iii Building asymmetry measures: the power of the information-theoretic perspective
Consider the problem of communicating information about a direction in space (See Fig. 3). It is clear that to be able to succeed in this task, one needs to use states which break the rotational symmetry. Furthermore, intuitively we expect that to transfer more directional information one needs to use states which are more asymmetric. This suggests that one can quantify rotational asymmetry by the amount of information a state encodes about orientation.
To make this connection precise, we note that if by some symmetric dynamics then by definition this dynamics also takes every state in the group orbit of to the corresponding state in the group orbit of , that is, for all . The set of states can be understood as a quantum encoding of the group element , and the dynamical evolution realizing for all can be understood as a kind of data processing. From the existence of such a data processing, it follows that the -based encoding must contain no more information about than the -based encoding.
A measure of the information content of an encoding is a function from encodings to reals that is nonincreasing under data processing. Specifically, is a measure of information if for any two different quantum encodings of a classical random variable , and , the existence of a dynamical evolution that maps to for all implies that . (In the context of information theory, the monotonicity of a measure of information is known as the data processing inequality.) It follows that if we define a real function such that its value on a state is the measure of information of the group orbit of that state, that is, , then is a measure of asymmetry. The proof is simply that if is mapped to by some symmetric dynamics, then for all , the state is mapped to by that dynamics, and consequently , which implies .
Quantum information theorists have defined many measures of information and for each of these we can obtain a measure of asymmetry. We mention a few that can be derived in this fashion (details of the derivation are provided in the supplementary material).
(i) Let be an arbitrary probability density over the group manifold, and define the twirling operation weighted by as . Let be the von Neumann entropy. The function
is an asymmetry measure. We will refer to such a measure as a Holevo asymmetry measure. The intuition behind it is as follows: if a state is close to symmetric, then it is close to invariant under rotations and mixing over all rotations does not change its entropy much, while if it is highly asymmetric, then under rotations it covers a broader manifold of states and hence mixing over all rotations increases the entropy significantly.
(ii) Let the matrix commutator of and be denoted by and the trace norm (or -norm) by . For any generator of the group action, the function
is a measure of asymmetry. This measure formalizes the intuition that the asymmetry of a state can be quantified by the extent to which it fails to commute with the generators of the symmetry.
(iii) For any generator , and for , the function
is a measure of asymmetry. This quantity was introduced by Wigner and Yanase with WignerYanaseDyson () and generalized by Dyson to arbitrary . While it has attracted much interest, its monotonicity under symmetric dynamics—and hence its interpretation as a measure of asymmetry—was not previously recognized.
For all of these examples, if is a symmetric state then the asymmetry measure has value zero. Furthermore, for and , if is a pure state then the measure reduces to the variance over . The measures and can both be understood as quantifying the “coherent spread” over the eigenvalues of . As discussed earlier, this is precisely what the variance over (or any function of ) could not do.
Iv The inadequacy of Noether conservation laws for general closed-system dynamics
Functions of the Noether conserved quantities cannot distinguish symmetric states from asymmetric states. However, the examples we have provided thus far to establish this have relied on considering pairs of states that differ in their degree of purity (or entropy content), and if these are to be the input and output states of some dynamics, then the dynamics must be open. We might hope, therefore, that although conservation of all Noether quantities is neither a necessary nor a sufficient condition for the possibility of a state transition in open-system dynamics, it is necessary and sufficient for closed-system dynamics. Once again, however, we show that such hopes are not fulfilled.
Consider a spin-1/2 system that also has some other independent degree of freedom which is invariant under rotation, denoted by the observable . Let denote eigenstates of spin along the -axis, and let denote orthogonal eigenstates of . Then define
We can easily check that: (i) the state transition is impossible by rotationally-symmetric closed-system dynamics (but possible by dynamics that break rotational symmetry, namely, a rotation around by , so that the transition would not be forbidden were it not for considerations of symmetry) (ii) All constraints implied by Noether’s theorem hold, i.e., the conditions of Eq. (1) for generators of rotations are satisfied, and therefore Noether’s theorem does not forbid this transition.
Condition (ii) is straightforward to verify. The truth of (i) can be made intuitive by noting that is symmetric under rotations about the -axis while is not, such that a transition from to is symmetry-breaking and hence impossible by rotationally-symmetric dynamics. One can derive this same conclusion using asymmetry measures. Consider the Holevo asymmetry measure for the probability density that is uniform over all rotations around and vanishing for all other rotations. We find that and , that is, increases in this state transition, thereby demonstrating that it cannot be achieved by rotationally-symmetric dynamics. We have shown that constants of the motion derived from asymmetry measures can capture restrictions on the dynamics that are not captured by Noether’s theorem.
V The adequacy of Noether conservation laws for closed-system dynamics of pure states
One final special case remains to be considered. Might it be that conservation of the Noether conserved quantities are the necessary and sufficient conditions for the possibility of state interconversion under symmetric closed-system dynamics when the states are pure? In this case, the answer is yes.
In MarvianSpekkens2013 (), it was shown that the asymmetry properties of a pure state are completely determined by the complex function over the group manifold, called the characteristic function. Equality of characteristic functions is the necessary and sufficient condition for two pure states to be reversibly interconvertible under symmetric unitary dynamics. Expanding in a power series over the generators, one deduces that for connected compact Lie groups, such as the group of rotations, equality of all moments of the generators is equivalent to equality of the characteristic functions. Consequently, for pure states undergoing reversible dynamics with such a symmetry, Noether’s theorem, i.e. Eq. (1), captures all of the consequences of the symmetry.
In practice, we are always faced with some loss of information under any quantum dynamics, due to the ubiquity of decoherence, and there is always some noise in our preparation of the initial state. Therefore, reversible dynamics of pure states is an idealization that is never achieved in practice, and as soon as one departs from it, Noether’s theorem is inadequate for describing the consequences of symmetry.
Vi Applications of measures of asymmetry
Phase-insensitive quantum amplifiers are examples of open system dynamics that have a symmetry property. Consequently, from the nonincrease of measures of asymmetry, we can derive bounds on their performance. The purpose of a quantum amplifier is to increase the expectation value of some observable, such as the number operator for optical fields Caves (). In most studies, constraints on amplification are obtained for specific physical models of the amplifier. Furthermore, the analysis is typically done separately for linear and nonlinear amplifiers as well as for deterministic and nondeterministic amplifiers Caves2 (). By contrast, the constraints that can be found with our techniques follow from assumptions of symmetry alone. For instance, in optics they follow from the fact that the amplifier is phase-insensitive. They are therefore model-independent and can be applied whether the amplifier is linear or nonlinear, deterministic or nondeterministic.
Here is an example of such a constraint, arising from the Holevo measure of asymmetry. If is mapped to by a symmetric amplifier, then
which asserts that the change in entropy under the transition has a nontrivial lower bound.
For instance, suppose that is a state of a spin-1/2 system, while is a state of a spin- system for . Suppose further that the probability density in is chosen to be the uniform measure over the symmetry group (which in this problem is ), so that while is large (logarithmic in ). The inequality then implies that must be large.
This demonstrates that in the case of rotationally-symmetric open-system dynamics, an increase in the value of the angular momentum along some axis is not prohibited as long as entropy increases. This ensures that the distinguishability of states at the output is not more than the distinguishability of states at the input, so that the information content has not increased (See Fig. 4).
A second application of measures of asymmetry is to quantify quantum coherence Aberg (). Coherence is at the heart of many distinctly quantum phenomena from interference of individual quanta to superconductivity and superfluidity. On the practical side, coherence is the property of quantum states that is critical for quantum phase estimation: a coherent superposition of number eigenstates, such as , is sensitive to phase shifts, while an incoherent mixture, such as , is not. Phase-shifts, however, are symmetry transformations. Therefore, states with coherence are precisely those that are asymmetric relative to the group of phase shifts. It follows that we can define a measure of coherence as any function that is nonincreasing under phase-insensitive time evolutions (See MarvianSpekkensModes () for more details). It follows that the measures of asymmetry proposed here—Eqs. (2), (3) and (4)—can be used as measures of coherence relative to the eigenspaces of the generator (i.e. as the “coherent spread”).
Finally, measures of asymmetry are important because asymmetry is the resource that powers quantum metrology.111This contrasts with the more standard view that the relevant resource is entanglement. Note, however, that measures of entanglement can sometimes provide bounds on measures of asymmetry. Metrology involves estimating a symmetry transformation. In the most general case, the set of symmetry transformations form a non-Abelian group, such as the group of rotations, but in the most common example, it is the Abelian group of phase shifts (in which case asymmetry corresponds to coherence). We focus on this example to illustrate the idea.
To estimate an unknown phase shift of an optical mode, one prepares the mode in the state , subjects it to the unknown phase shift, leaving it in the state , where is the number operator, and finally one measures it. The usefulness of a particular state can be quantified by any measure of the information content of the ensemble , but, as shown above, every such measure is a measure of the asymmetry of relative to phase shifts. The figure of merit for a metrology task is therefore a measure of asymmetry and dictates the optimal . Suppose, for instance, that one seeks an unbiased estimator of and the figure of merit is the variance in , denoted . It has been shown in Luo () that for a state , we have (a quantum generalization of the Cramer-Rao bound) where is the Wigner-Araki-Dyson skew information of order , defined in Eq. (4). So it is the latter measure of asymmetry that is relevant in this case.
Our analysis prompts the question: what are the necessary and sufficient conditions on states and (not both pure) for it to be possible to map to under symmetric dynamics? Such conditions would capture all of the consequences of the symmetry of the dynamics. The question remains open but our results suggest that adopting an information-theoretic perspective may be the most expedient path to a solution.
Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. IM acknowledges support from NSERC, a Mike and Ophelia Lazaridis fellowship, and ARO MURI grant W911NF-11-1-0268.
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Appendix A Supplementary material
a.1 Triviality of asymmetry measures based on Noether conserved quantities
In this section, we prove that for symmetries corresponding to compact Lie groups, functions of Noether conserved quantities yield only trivial measures of asymmetry. In the case of finite groups, there are no generators of the group action, and therefore we cannot generate Noether conserved quantities in the standard way. Nonetheless, we can show that a similar result holds in this case as well.
We phrase our general result (which applies to both compact Lie groups and finite groups) in terms of characteristic functions. Recall that the characteristic function of a state is the expectation value of the unitary representation of the group, , which is a complex function over the group manifold. Then we can prove the following theorem about measures of asymmetry.
Let be an asymmetry measure for a finite or compact Lie group with unitary representation . Assume that is a function of the characteristic function of the state alone, i.e., for some functional . Furthermore, in the case of compact Lie groups, assume that is continuous. Then the monotone is a constant function, i.e., is independent of .
In the case of compact Lie groups, the characteristic function of a state uniquely specifies all the moments for all generators of the symmetry; this can be seen by considering the Taylor expansion of around the identity. Therefore, the theorem implies that if a continuous measure of asymmetry can be expressed entirely in terms of the Noether conserved quantities alone, i.e. in terms of moments of the form , then it should be a constant function independent of .
We now present the proof of theorem 1.
Proof. We first present the proof for the case of finite groups and then we explain how the result can be generalized to the case of compact Lie groups as well.
Suppose is the Hilbert space of a physical system on which the unitary representation of the symmetry group acts as the left regular representation, i.e., has an orthonormal basis denoted by such that
It turns out that on this space we can define another representation of , called the right regular representation denoted by , such that
Then one can easily see that these two representations of on commute, i.e.
Define to be the quantum operation that averages over all symmetry transformations, . Let be the identity element of the group. We have
where to get the second equality we have used the fact and to get the last equality we have used the fact that the two representations commute, Eq. (10). So the characteristic function of the state is equal to the characteristic function of the state . This means that for any measure whose value for a given state depends only on the characteristic function of that state, we have
As we have seen before, however, for any asymmetry measure, the value of the measure is the same for all symmetric states and furthermore this value is the minimum value of that function over all states. Given that is a symmetric state, it follows that
Now consider an arbitrary state on an a Hilbert space where the projective unitary representation of the symmetry group is . Then, one can easily show that there exists symmetric quantum channels which map the state on to the state on . One such channel is described by the map
But the fact that is symmetric together with the fact that is an asymmetry measure implies that for any state it holds that
This together with Eq. (12) implies that for an arbitrary state ,
and so the asymmetry measure is constant over all states. This completes the proof for the case of finite groups.
In the following, we prove that making the extra assumption that the asymmetry measure is also continuous, this result can be extended to the case of compact Lie groups. Note that in this case the regular representation of the group is not finite dimensional. Nonetheless, as was noted in Kitaev-Mayers-Preskill () and later in BRSreview (), there still exists a sequence of finite dimensional spaces, , where is the maximum dimension of irreducible representation (irrep) supported on each space, and for each there is an over-complete basis , such that the unitary representation of the symmetry group acts as222 Let where the summation is over all irreps of whose dimension is less than or equal to , and is the subsystem on which the symmetry acts like its irrep and is the multiplicity subsystem with dimension equal to . Define where is a normalization factor, is an orthonormal basis for subsystem , and is an orthonormal basis for . The properties assumed for in the proof hold if this set of states is generated from a fiducial state of the form of .
Furthermore, as is discussed in Kitaev-Mayers-Preskill () and BRSreview (), for a given pair of distinct group elements, , one can make the inner product arbitrarily close to zero in the limit of large . In this limit, the state has the maximal asymmetry in the sense that for any given state on an aribrary Hilbert space , there exists a symmetric channel such that
The symmetric channel can be defined in a manner similar to how it was defined for the case of finite groups, namely,
Now, similarly to the case of finite groups, we can also define another representation of on , denoted , such that
Then one can easily see that these two representations of on commute,
Therefore, using the same argument that we used for the case of finite groups, we can prove that for any it holds that where is the identity element of the group . Therefore, for any measure whose value for a given state depends only on the characteristic function of that state, it holds that
Furthermore, because is an asymmetry measure and because is a symmetric channel, we have
The above two equations imply that
On the other hand, given that is assumed to be continuous, Eq. (17) implies that
This, together with Eq. (23), proves that for any arbitrary state in an arbitrary finite dimensional space , it holds that . Therefore, we conclude that in the case of compact Lie groups any continuous asymmetry measure which only depends on the characteristic function of the state is a constant function. This completes the proof.
a.2 Some nontrivial families of asymmetry measures
We now apply the recipe described in the article to generate a few interesting measures of asymmetry from measures of information.
Our first example makes use of a family of information measures that are based on the Holevo quantity Nielsen (). For a set of states , and a probability distribution over , the Holevo quantity is defined as
where is the von Neumann entropy of the state (if is a continuous variable, and is a probability density, we simply replace sums by integrals). It is well-known that this quantity is non-increasing under information processing, i.e. it is an information monotone Nielsen (). This yields a family of asymmetry measures, one for every probability density over the group manifold (probability distribution for the case of a finite group), namely,
is the superoperator that performs a -weighted average over the group action (which is sometimes called the twirling operation weighted by ). We call measures of this form Holevo asymmetry measures.
Note that for any symmetric state and any arbitrary probability distribution , . Also, note that for any probability distribution which is nonzero for all , and for any state which breaks the symmetry, . For the special case of a uniform weighting, this measure has been previously proposed in Ref. Vaccaro () and proven to be monotonic under symmetric operations using a different type of argument.
A particularly simple subclass of measures of the information content of a set of states are measures that consider the distinguishability of just a single pair of states within the set. For a pair of states and , a measure of their distinguishability is defined to be a function from pairs of states to the reals, such that for any quantum channel , we have
Specializing to the case of interest here, where the classical variable is a variable (which ranges over the elements of the relevant group), we focus on the distinguishability of two elements in the group orbit of . Without loss of generality, we can choose the pair to be and for some .
We now introduce two more families of asymmetry measures based on this subclass of measures of information.
Consider a Lie group . Take the distinguishability measure to be the trace distance,
where is the trace norm (or -norm). It is well-known that the trace distance satisfies Eq. (26) and hence constitutes a measure of the distinguishability of a pair of states Helstrom (). Therefore, we can define an asymmetry monotone using this distinguishability measure, namely, .
Now recall that for a Lie group , we can consider a pair of states and where is infinitessimally close to the identity element. Specifically, we can always write for some generator and phase , and in the limit where , we have
So we conclude that for any generator of the Lie group the function
is an asymmetry measure.
Any state that is symmetric (i.e. invariant under the group action) necessarily commutes with all the generators, so for such states, for all . Also, any state that is invariant only under some subgroup of has for those that are generators of this subgroup.
A state can only be asymmetric relative to a subgroup of associated with a generator if it has some coherence over the eigenspaces of , that is, if . Therefore, in retrospect one would naturally expect that some operator norm of the commutator should be a measure of asymmetry. This intuition does not, however, tell us which operator norm to use. Our result shows that it is the trace norm that does the job.
also reduces to a simple expression for pure states: it is proportional to the square root of the variance of the observable , that is,
Given that a superposition over the eigenspaces of that is totally incoherent (i.e. a mixture over the eigenspaces) has vanishing asymmetry according to this measure, while a superposition over these eigenspaces that is totally coherent has asymmetry that depends only on the variance over , this asymmetry measure seems to succeed in quantifying the amount of variance over that is coherent, which one might call “coherent spread” over the eigenspaces of .
We turn to our third and final example of a family of asymmetry measures. We take as our measure of distinguishability the relative Renyi entropy of order , defined as
For , it is well-known that satisfies Eq. (26) and is therefore a valid measure of distinguishability Hayashi (). It follows that is an asymmetry measure. As in the previous example, by considering for group elements which are infinitessimally close to the identity element, we can derive a measure of asymmetry for any arbitrary generator of the group. Using this argument, we can show that for any and any generator ,
is a measure of asymmetry.
This family of measures has been previously studied under the name of the Wigner-Yanase-Dyson skew information WignerYanaseDyson (), but their status as measures of asymmetry had not been recognized. If is a symmetric state, then it commutes with , and . We also find this measure to be zero when is not symmetric, but has some subgroup of as a symmetry and is a generator of that subgroup.
For pure states, the Wigner-Yanase-Dyson skew information reduces to the variance of the observable , that is,
Again, we see that a mixture over the eigenspaces of has vanishing , while a coherent superposition over these eigenspaces has equal to the variance over . Consequently, this asymmetry measure, like , in some sense quantifies the coherent spread over the eigenspaces of .