History states of systems and operators
We examine the generation and entanglement of discrete system-time history states. These states arise for a quantum reference clock of finite dimension and lead to a unitary evolution of system states when satisfying a static discrete Wheeler-DeWitt-type equation. We first analyze their different representations, showing that for any given initial system state there is a “proper time” clock basis for which the evolution corresponds to a constant Hamiltonian. It is also shown that while their entanglement depends normally on the initial state, it becomes maximum and independent of the latter when the evolution operators form a complete orthogonal set, in which case the history state can be generated with a simple two-clock scheme. We then provide an analytic evaluation of the quadratic system-time entanglement entropy, showing it satisfies strict upper and lower bounds determined by the energy spread and the geodesic evolution connecting the initial and final states. We finally show that the unitary operator that generates the history state can itself be considered as an operator history state, whose quadratic entanglement entropy determines its entangling power. Simple measurements on the clock enable to efficiently determine overlaps between system states and also evolution operators at any two times.
The incorporation of time in a fully quantum framework PaW.83 () has recently attracted wide attention Ga.09 (); GL.15 (); Mo.14 (); Ma.15 (); FC.13 (); BR.16 (); Er.17 (); Pa.17 (); Ni.18 (); Ll.18 (). On the one hand, it is interesting as a fundamental problem and a key issue in connection with general relativity. On the other hand, a quantum description of time allows us to exploit the quantum features of superposition and entanglement in the development of new models of parallel-in-time simulation BR.16 ().
The concept of time is related to the quantification of evolution through a reference physical system called clock. Historically, the readings of this clock provided an external classical parameter, called time. Nonetheless, if we aim to introduce time into a fully quantum framework, the clock has to be a quantum system itself.
Here we describe the system and the reference clock through a discrete system-time history state which enforces a discrete unitary evolution on the system states. Moreover, for a fixed initial state it is shown that there is always an adequate selection of clock basis for which the resultant evolution corresponds to a constant Hamiltonian. This defines a “proper” time basis where the history state satisfies a discrete counterpart of a Wheeler-DeWitt type equation DW.67 ().
The entanglement of the history state is a measure of the number of orthogonal states visited by the system at orthogonal times BR.16 (). It depends in general on the initial system state and the evolution operators. It is shown, however, that when the latter form a complete orthogonal set, the entanglement is always maximum, irrespective of the initial state. The corresponding history state admits a simple generation through a two-clock scenario, where the clocks are linked to conjugate system variables.
We then analyze the quadratic entanglement entropy of history states, showing that it can be analytically evaluated for general constant Hamiltonians and that it is upper bounded by the entropy of the energy spread of the initial state and lower bounded by that of the geodesic evolution connecting the initial and final states. And its average over all initial system states is directly proportional to the quadratic operator entanglement entropy Z.00 (); N.03 (); P.07 (); M.13 () of the unitary gate that generates the history state. Through the channel-state duality Ja.72 (); Ch.75 (); Gr.05 (); Du.05 (); Mi.13 (), it is also shown that the pure state which represents the latter is itself an operator history state, whose quadratic entanglement determines its entangling power.
Finally, we show that through measurements on the clock it is possible to use both system and operator history states to efficiently determine the overlap between system states and also the trace of the evolution operator between any two-times. The latter reduces to the trace of a unitary operator (result of the DQC1 circuit KL.98 ()) for the simple case of a qubit clock. The properties of discrete history states and their entanglement are discussed in section II, whereas the entanglement and history states of unitary operators are discussed in III. Conclusions are finally given in IV.
Ii Discrete history states
We consider a system and a reference clock system in a joint pure state , with of finite dimension . Any such state can be written as
where , , are orthogonal states of () and are states of , not necessarily orthogonal or normalized, yet satisfying . Consider now a unitary operator for the whole system of the form
where is identified with and are arbitrary unitary operators on satisfying . If fulfills the eigenvalue equation
the states will undergo a unitary evolution with :
where , with . The states will then have a unit norm if is normalized.
Thus, the state (1) is a discrete finite dimensional version of the history state of the Page-Wootters formalism PaW.83 (); GL.15 (). Moreover, writing , with hermitian (and spectrum ), Eq. (3) is equivalent to
which is a discrete cyclic version of a Wheeler-DeWitt type equation DW.67 ().
A unitary evolution of the states actually occurs if is any eigenstate of : Its eigenvalues are , , and its eigenstates have all the form (1) with satisfying a shifted unitary evolution: . Each eigenvalue has degeneracy equal to the dimension of the system space, with its eigenspace spanned by orthogonal history states generated by orthogonal initial states : BR.16 ().
If is independent of , then
with a fixed hermitian Hamiltonian for system with eigenvalues , integer. The operator (2) becomes then separable: , implying
where is the generator of time translations, satisfying and , with the discrete Fourier transform (DFT) of the states :
which is a discrete version of Schrödinger’s equation: As , for and .
ii.1 Representations and entanglement of the history state
By considering an arbitrary orthogonal basis of , we may first rewrite as
where and is a “wave function” satisfying a unitary evolution with : .
We may then obtain the Schmidt decomposition of , which we will here write as
where are the singular values of the matrix and orthonormal states of () derived from the singular value decomposition of , with . They are eigenstates of the reduced states , with their non-zero eigenvalues. While the states are not necessarily orthogonal but are equally probable, the states are all orthogonal but not equally probable, with representing a “permanence” probability.
since with , and hence becomes Eq. (11), with the strictly orthogonal states (8). The Schmidt coefficients represent in this case the distribution of over distinct energy eigenstates (in case of degeneracy, denotes the projection of onto the eigenspace of energy (), with the total probability of measuring this energy in ). It is then apparent from Eqs. (7) and (11) that satisfies Eq. (5), which becomes a zero “total momentum” condition: ().
In the case of arbitrary unitary operators in (2), for any given initial state there is, however, an orthogonal basis of (“proper time” basis) for which the corresponding states of evolve according to a constant Hamiltonian satisfying (12). It is just necessary to use the inverse DFT of the Schmidt states of (11),
with and defined over the Schmidt states by Eq. (12). The Schmidt coefficients can then be interpreted as the distribution of over these energy eigenstates. In terms of the operators and defined by (12), satisfies Eq. (5) also for an effective of the form (7), and can be also generated from with the circuit of Fig. (1).
Assuming now (the Schmidt decomposition selects in any case subspaces of equal dimension on and ) we can also consider the inverse DFT of the system Schmidt states, , which satisfy . They can be regarded as system proper “time” states in the sense that if , then exhibits a perfect correlation with the clock states . We can then rewrite also as
where depends just on , and
is the DFT of the Schmidt coefficients , with orthogonal maximally entangled history states: ( if ).
The representation (16) is then “conjugate” to (11), expressing as a superposition of maximally entangled orthogonal history states. Like (11), it is symmetric in : States evolve unitarily with (Eq. (15)) while states evolve unitarily with :
where , complementing Eq. (15).
where . If happens to be a common eigenstate of all , such that , then becomes separable and (stationary state), whereas if all are orthogonal (i.e. fully distinguishable), becomes maximally entangled, with (1) already the Schmidt decomposition and maximum. Thus, measures the actual system evolution time, in the sense of counting the number of effective equally probable orthogonal states the system visits at orthogonal times. For constant (Eq. (6)), is just a measure of the energy spread () of the initial state, as . A similar interpretation holds for the general case in terms of the effective defined by (12).
On the other hand, the entropy determined by conjugate distribution ,
measures the spread of over maximally entangled evolutions, or equivalently, the spread of system time states for a given clock state (or viceversa), and is a measure of time uncertainty. It vanishes when is maximally entangled ( if ), in which case there is complete synchronization between system and clock times (, and becomes maximum for a stationary state ( if ), in which case system and clock times are completely uncorrelated, as seen from (16). These two entropies satisfy the entropic uncertainty relation BR.16 () (see also DCT.91 (); PDO.01 (); Hi.57 (); Ll.18 ())
which is saturated in the previous limits.
ii.2 The case of a complete set of evolution operators
The dependence of the system-time entanglement (19) on the initial state is softened when the operators do not commute among themselves: If they have no common eigenstate, will be entangled for any . The extreme case is that where the ’s of (4) form a complete set of orthogonal unitaries on , such that
implying . In this case the history state (1) becomes maximally entangled for any initial state :
Proof: We may view Eq. (22) as the scalar product between column vectors of a unitary matrix of elements , with any orthonormal basis of , such that (22) is equivalent to . This matrix then satisfies as well , i.e. , which implies and hence
for any two states , of . In particular, for , Eq. (24) implies a maximally mixed reduced state for any seed state :
Therefore, a complete orthogonal set of ’s ensures that the system will visit orthogonal states irrespective of the initial state . The Schmidt decomposition (11) will then select a subspace of of dimension connected with through . Due to the -fold degeneracy , any orthogonal basis of this subspace can be used in (11), with all states directly orthogonal.
where , , , and , are orthogonal bases of related through a DFT: . They satisfy, for any eigenstate of ,
which implies Eq. (22), i.e. .
The discrete evolution under these operators can then be achieved by application of just two different unitaries to the preceding state (here , integer):
For instance, if is a qubit () we may take , , with , . Hence, is maximally entangled (), with , , and .
In the general case, it is here natural to view system as formed by two clocks with identical Hilbert space dimension , which govern time-independent Hamiltonians and associated with conjugate operators , on . Then we may write the history state (1) for the operators (26) as
which represents a history state of history states. It can then be implemented with the circuit of Fig. 2.
ii.3 The quadratic entanglement entropy: Analytic evaluation and bounds
The analytic evaluation of the entropy (19) in the general case requires the determination of the singular values , i.e., the eigenvalues of or , which is difficult in most cases. It is then convenient to use the quadratic (also called linear) entropy , which does not require explicit knowledge of the eigenvalues and is a linear function of the purity . Like , it vanishes iff is pure and is maximum iff is maximally mixed (with for a maximally mixed single qubit state), satisfying the majorization relation if Bha.97 (); CR.03 (). The associated entanglement entropy is
and can be determined just from the overlaps between the evolved states. For the complete orthogonal set (22), it is easily verified that , so that , which is the value for a maximally mixed .
The overlaps are also experimentally accessible through a measurement at the clock of the non-diagonal operators ():
where , are hermitian Pauli operators for the pair .
with if (in case of degenerate states , , with ), then and Eq. (31) becomes, for equally spaced times , ,
The exact result for a continuous evolution can also be obtained from (LABEL:E2x), by taking the limit :
Eq. (LABEL:E23) provides a good approximation to (LABEL:E2x) if with finite weight .
Eqs. (LABEL:E2x)–(LABEL:E23) are essentially measures of the spread of over distinct energy eigenstates. For small such that , a second order expansion shows they are proportional to the energy fluctuation in :
where and . It is also apparent from (LABEL:E2x) that is upper bounded by the quadratic entropy of the energy distribution :
The maximum (38) is reached for an equally spaced spectrum of the form
with integer , since in this case the bracket in (LABEL:E2x) takes its maximum value .
are not necessarily all orthogonal, so that will become normally smaller BR.16 (). Nonetheless, for large and not too small , the states will typically be almost orthogonal, so that the deviation from the upper bound (38) will not be large, becoming significant only in the presence of quasidegeneracies in the spectrum: The bracket in (LABEL:E23) vanishes just for , becoming close to for , while that in (LABEL:E2x), which is a periodic function of with period , vanishes for , or integer, becoming close to whenever .
On the other hand, Eq. (LABEL:E23) also admits a lower bound for fixed initial and final states and , reached when the evolution (over equally spaced times under a constant ) remains in the subspace spanned by and :
where is determined by the overlap between the initial and final states:
Writing the final state as
where , the lower bound (41) is the result of Eq. (LABEL:E2x) for an evolution under a two level Hamiltonian
with and , such that
The demonstration of (41) is given in the appendix, but the result is physically clear: The entanglement is a measure of the distinguishability between the evolved states, and the minimum value is then obtained for an evolution within the subspace containing the initial and final states, where all intermediate states will be closer than in a general evolution. Such evolution provides precisely the minimum evolution time between and as it proceeds along the geodesic AA.90 ().
As check, for small such that , a fourth order expansion of (LABEL:E2x) and (41) leads to
where denotes the lower bound (41) and . Hence, the difference is of fourth order in and proportional to the fluctuation of , thus vanishing just for the geodesic evolution (, with constant).
The bound (41) is, of course, an increasing function of for any if , i.e., a decreasing function of the overlap , and also a decreasing function of for any if . The minimum value is thus achieved in the continuous limit . Then, we may also write, for any ,