Introduction

Observation is at the heart of understanding physical phenomena. More broadly, it serves as the first step in information processing. In the canonical example, a classical observer gathers data about a quantum system by measuring it. This process yields classical information about the state, but at a high price – the observer's information gain is incomplete (it doesn't uniquely identify the state) and accompanied by an irreversible loss of information. It is typically further restricted by locality; most joint measurements on an N-component quantum system are practically impossible when the components are spatially delocalized. This has profound consequences for extracting information from multiple samples of a quantum state. Measuring them individually is strictly less powerful than a joint measurement on the whole ensemble. By contrast, in classical theories, measurement is a passive procedure with no accompanying information loss and complete information can be gathered from independent measurements of the samples. This implies a fundamental asymmetry between data-gathering in quantum and classical theories.

The conventional paradigm of quantum measurement comprises: (i) a controlled unitary interaction between a system and an apparatus ; (ii) decoherence on , which forces its state into a mixture of “pointer basis” states1; and (iii) experimental readout of the classical result from (arguably accompanied by “collapse” of state). But this prescription breaks down if any part of the observing apparatus can sustain coherence. A coherent apparatus has other, more powerful information-gathering strategies available to it. To demonstrate this in the context of traditional information processing, we allow to be a quantum information processor (QIP) – basically a very small (perhaps just 1 qubit) non-scalable quantum computer. We focus on the case where is protected from decoherence until the very end of the protocol. What remains is a coherent measurement, a unitary interaction between and that transfers information from to .

To explore and demonstrate the power of coherent measurement, let us consider the specific problem of quantum state discrimination2,3: Given N quantum systems that were all prepared in one of K distinct states |ψ1〉,…,|ψK〉, decide in which state they were prepared. In principle, finding the optimal measurement is a straightforward convex program. But when N > 1 copies of |ψk〉 are available, this task (famously) requires joint measurement on all N copies, which is prohibitively difficult. Observing each of the N copies independently yields a strictly lower probability of success4,5. This contrasts starkly with the corresponding classical problem of distinguishing K distinct probability distributions, where one-at-a-time observations are completely sufficient. Discrimination thus provides an ideal scenario to test the information-processing utility of more general data gathering paradigms.

Results

Simple discrimination with N copies

Suppose we are given N quantum systems with d-dimensional Hilbert spaces and a promise that they were all identically prepared in one of K nonorthogonal states {|ψ1〉…|ψK〉}. Their joint state is , with k unknown. Identifying k with maximum success probability requires a joint measurement on all N samples. Non-adaptive, one-at-a-time measurement cannot achieve the optimal success probability. For K = 2 candidate states, there is an adaptive local measurement scheme that achieves the optimal success probability6, but no such protocol has been found for K > 2. It has recently been shown that for K = 3, there exist cases for which local measurements cannot achieve the optimal success probability, even with multiple rounds of classical communication7.

All the information about k is contained in a K-dimensional subspace

So while the optimal measurement is a joint measurement, it does not need to explore the majority of . We will implement it by rotating the entire subspace into the state space of our K-dimensional QIP (the coherent measurement apparatus). We do so via sequential independent interactions between and each of the N samples , “rolling up” all information about k into .

is initially prepared in the |0〉 state. We bring it into contact with and execute a SWAP gate between and the {|0〉,…,|d − 1〉} subspace of . This transfers all information from the first sample into , leaving in the |0〉 state.

Now we bring into contact with . Their joint state is , although we do not know k. But we do know that the state lies within , (see Eq. 1), whose dimension is at most K. A basis {|ϕj〉: j = 1…K} for this space can be obtained by Gram-Schmidt orthogonalization. We apply a unitary interaction between and ,

where we have defined U2 only on the subspace of interest (for completeness, it can be extended to the complement, , in any convenient manner). It rotates into , which places all the information about k in and decouples ( is left with no information about k if and only if is left with all the information about k.). is now in one of K possible states , which (as a set) are unitarily equivalent to – e.g., .

The rest of the algorithm is now fairly obvious; we move on and interact with in the same way, etc, etc. At each step, when comes into contact with , their joint state is . These K alternatives reside in a K-dimensional space (see Eq. 1), spanned by a basis , which is then rotated into by applying

Each sample system is left in the |0〉 state, indicating that all its information has been extracted. After every sample has been sucked dry, we simply measure to extract k. This final measurement can be efficiently computed via convex programming, since is only K-dimensional.

The sequence of coherent measurement interactions is independent of what sort of discrimination we want to do – e.g., minimum-error4, unambiguous discrimination8,9,10,11, maximum-confidence12, etc. – because is a sufficient statistic for any inference about k and our protocol simply extracts it whole, leaving the decision rule up to the final measurement on . As in the classical case, data gathering can now be separated from data analysis.

As an example of the data gathering protocol, we consider now the K = 2 case and take the two possible initial states to be single qubit states, parametrized as follows

The joint state of the first n samples is given by

where |n,j〉 is a normalized equal superposition of all tensor product states with n − j systems in state |0〉 and j systems in state |1〉. A natural basis for the two-dimensional subspace is then given by

and the goal of the data gathering algorithm is to transfer the subspace spanned by these states into the processor via sequential interactions with each of the samples .

At the first step, as indicated above, we perform a SWAP gate between and . At the next step, the joint state of and is

where |0〉2, |1〉2 are joint states, given by the n = 2 case of Eqs. 6 and 7. A two-qubit unitary acting as follows

thus leaves in state |0〉 while the processor, in state

now holds all the information previously contained in the two samples read so far. Now suppose the processor has interacted with and processed the first n − 1 samples. Then we have performed a unitary which acts as follows on the relevant subspace:

Defining and using Eqs. 5–7, the state of the processor is given by

and we wish to update this state through interaction with the nth sample. The basis states |0〉n, |1〉n spanning are related to the corresponding basis states spanning and the basis |0〉, |1〉 of as follows

Thus we wish to implement a 2-qubit unitary on the processor and the nth sample, which acts as follows:

This can be achieved by, e.g., performing a CNOT gate with as the control and as the target, followed by a controlled unitary on to rotate the residual state to |0〉, controlled on the state of . Continuing in this way we can roll up all the information contained in the provided samples into the processor . Each step requires only a 2 qubit unitary, acting on the nth sample and the processor. During the course of this process all the information contained in the joint N qubit state is transferred to the processor, where a single qubit measurement is sufficient to perform any allowed data analysis operation.

This protocol can be modified to discriminate non-symmetric product states, e.g., vs. . A more general extension, however, is to discriminating many-body states of the matrix product form.

Matrix product state discrimination

The information about k can be “rolled up” using sequential interactions because it is contained in a subspace with Schmidt rank at most K across any division of the N systems. By this we mean that, given any pure or mixed state ρ on , if we trace out , the reduced state for has rank at most K. Low Schmidt rank is critical. Consider distinguishing between two states that are each maximally entangled between the first N/2 samples and the last N/2 samples. They lie in a 2-dimensional subspace, but it is not accessible through our protocol with a low-dimensional QIP. The first N/2 samples are maximally entangled with the rest, so their reduced state has rank dN/2. At least N log2 d/2 qubits would be needed to store the information extracted from the first N/2 samples.

But whenever the Schmidt rank condition is satisfied, a variation of our algorithm will work. For product states (above), each state has Schmidt rank 1 and the span of K such states has Schmidt rank at most K.

This property of low Schmidt rank is generalized by matrix product states (MPS)13,14. An N-particle MPS with bond dimension D is guaranteed to have Schmidt rank at most D across any division of the 1D lattice. Thus, the span of K such MPS, each with bond dimension ≤ D, has Schmidt rank at most DK. We denote such a subspace,

a matrix product subspace with bond dimension DK. Such a set of MPS can be distinguished optimally with coherent measurements and (log2 D + log2 K) qubits.

Our algorithm is a straightforward generalization of the one for product states and proceeds as shown in Figure 1. First, we represent the MP subspace by its purification – a single MPS |Ψ〉 for and a fictitious reference system R,

Information about k, which is contained in , equates to correlation with the imaginary R.

Figure 1
figure 1

Discrimination of MPS.

Our protocol represents an unknown MPS |ψk〉 from a set {|ψ1〉 … |ψK〉} by its purification – a single MPS |Ψ〉 involving a fictitious reference R. The algorithm then successively decorrelates each sample from the rest, storing correlations with the remainder of the lattice in the “apparatus” . Ultimately, all existing information about k (i.e., its correlation with R) is contained in , which can be measured.

Full size image

Now, we initialize in the |0〉 state, then SWAP its state with that of (the first lattice site). This decouples , leaving in a matrix product state,

with Schmidt rank no greater than DK.

Now, to roll up each successive site , we find the Schmidt decomposition of the current state between and the remainder of the lattice, write it (generically) as

and apply a unitary operation to ,

which decouples and leaves all the information previously in in . As in Eq. 2, this unitary is specified only on the subspace of interest and can be completed in any convenient fashion. Doing this successively at each site decorrelates all the and we are left in the state

with all information about k in , where it can be extracted by a simple measurement.

Recent proposals for local tomography15 are also based on sequential interactions. Our protocol, with coherent measurements, offers a tremendous efficiency advantage (at the cost of requiring a small QIP!). It can distinguish near-orthogonal MPS states with a single copy, whereas local tomography requires O(N) copies. Distinguishing non-orthogonal states requires multiple (M) copies. To apply our algorithm, we simply line up the copies (they do not have to exist simultaneously) and treat them as a single NM-particle MPS of bond dimension D.

Mixed state discrimination and tomography

In the context of N-copy states, one may ask:

  1. 1

    Can coherent measurement be used to discriminate mixed states, i.e. ?

  2. 2

    Can coherent measurement be used for full state tomography (rather than discrimination)?

The answer to both is “Yes, but it seems to require an O(log N)-qubit apparatus.” This is a very favorable scaling, but less remarkable (and less immediately useful) than the O(1) scaling for pure state discrimination.

This is possible because the order of the samples is completely irrelevant. As we scan through the samples, we can discard ordering information, keeping only a sufficient statistic for inference about ρ. The quantum Schur transform does this16. It is based on Schur-Weyl duality17, which states that because permutations of the N samples commute with collective rotations on them, the Hilbert space can be refactored as

The factors are irreducible representation spaces (irreps) of SU(d), the factors are irreps of SN and λ labels the various irreps. The Schur transform can be implemented by a unitary circuit that acts sequentially on the samples, mapping N qudit registers into three quantum registers containing (respectively) λ, and :

The “ordering” register accounts for nearly all the Hilbert space dimension of and since it is irrelevant to inference it can be discarded as rapidly as it is produced. What remains to be stored in is:

  1. 1

    a “label” register λ (a sufficient statistic for the spectrum of ρ),

  2. 2

    a SU(d) register (a sufficient statistic for the eigenbasis of ρ).

The λ register requires a Hilbert space with dimension ≥ the number of Young diagrams with N boxes in at most d rows, which is approximately

The register must hold the largest N-copy irrep of SU(d), whose size can be calculated using hook-length formulae18 and upper bounded by

Together, these registers require O(d2 log N) qubits of memory (although for pure state tomography, O(d log N) qubits of memory are sufficient).

O(log N) memory appears to be necessary for optimal accuracy. Consider the simplest possible case – discrimination of two classical 1-bit probability distributions

The sufficient statistic is frequencies of “0” and “1”, {n, N −n}. For any given problem, there is a threshold value nc such that the answer depends only on whether n < nc, so only one bit of information is required. However, extracting that bit via sequential queries requires storing n exactly at every step (using O(log N) bits of memory). Any loss of precision could cause a ±1 error at the final step and thus a wrong decision. In this example, classical storage is sufficient. But in the general case, where the candidate ρk do not commute, no method is known to compress the intermediate data into classical memory without loss (previous work suggests it is probably impossible19).

Discussion

Quantum information science is rife with gaps between what is theoretically achievable and what is practically achievable. Our algorithm eliminates performance gaps for pure state discrimination with local measurements – but it requires a new kind of measurement apparatus with at least 1 controllable qubit of quantum storage. Its utility depends on its applications and on the difficulty of implementation.

One immediate application of our protocol is detection of weak forces and transient effects. A simple force detector (e.g., for magnetic fields) might comprise a large array of identical systems (e.g., |↓〉 spins). Each system is only weakly perturbed by the force, so information about the force is distributed across the entire array. Our algorithm efficiently gathers up that information with no loss – whereas local measurements with classical processing waste much information.

A more sensitive N-particle “antenna” would incorporate entanglement between the N particles20. High sensitivity can be achieved by simple MPS states with D = 2, like N00N states21,

Collective forces do not change D, so the final states to be discriminated are also MPS. Our approach can discriminate such states and it can be used to prepare them, by running the “rolling up” process in reverse22.

More ambitious applications include direct probing of many-body states, to test a particular MPS ansatz for a lattice system, or to characterize results of quantum simulations in optical lattices or ion traps. Without fully scalable quantum computers that can couple directly to many-body systems, coherent measurements may be the only way to efficiently probe complex N-particle states. Our protocol does not obviously scale to PEPS, the higher-dimensional analogues of MPS14. Like MPS, PEPS obey an area law – entanglement across a cut scales not with the volume of the lattice (N), but with the area of the cut. For a 1-dimensional MPS on N systems, any cut has area 1, so the Schmidt rank scales as O(1) and our algorithm requires an O(1) qubit QIP. Rolling up a general PEPS on an n-dimensional lattice would require bits of quantum memory. However, some PEPS can be sequentially generated23 and are likely amenable to our protocol.

We note that the engineering requirements for a coherent measurement apparatus are achievable with near-future technology. should be a clean K-dimensional quantum system with:

  1. 1

    Universal local control,

  2. 2

    Long coherence time relative to the gate timescale,

  3. 3

    Controllable interaction with an external d-dimensional “sample” system,

  4. 4

    Sequential coupling to each of N samples,

  5. 5

    Strong measurements (which may be destructive).

Although adaptive local measurements can discriminate 2 states optimally, K = 2 is nevertheless sufficient for a proof of principle experiment demonstrating quantum data gathering. That a sufficient statistic for any desired measurement is stored as quantum data in the processor could be demonstrated by e.g. performing minimum error or unambiguous discrimination through eventual measurement on the processor in different runs of the same experiment. However, K ≥ 3 would be more exciting as optimal discrimination of 3 states via local, adaptive measurements is not possible in all cases7.

These requirements are much weaker than those for scalable quantum computing. Coherent measurement could be an early application for embryonic quantum architectures. Furthermore, scalability is not required, just a single K-dimensional system. Only local control has to be universal, since the interaction with external systems is limited. Error correction is not mandatory, for coherence need only persist long enough to interact with each of the N systems of interest. Since measurements are postponed until the end, they can be destructive.

We do require to be portable – i.e., sequentially coupled to each of the N samples – whereas a quantum computer can be built using only nearest neighbor interactions. Fortunately, most proposed architectures have selective coupling either through frequency space (NMR, ion traps with a phonon bus) or physical motion of the qubits (some ion traps) or flying qubits (photonic architectures). Devices that are not viable for full-scale quantum computing may be even better for coherent measurement. For example, an STM might pick up and transport a single coherent atomic spin along an array of sample atoms, interacting sequentially with each of them.

There are broad implications stemming from the results above. Coherent measurements are a genuinely new way to gather information. We have not just removed collapse from standard quantum measurements! That kind of coherent measurement is used already in quantum error correction, where it's common to replace a measurement of X with a controlled unitary of the form

Such unitaries transfer information about a specific observable X from to . For appropriate and , later measurements of produce exactly the same result as if had been measured directly. The coherent measurements in our discrimination protocols are not of this form. They do not measure (i.e., transfer information to about) a specific basis. For example, in N-copy state discrimination, interacts with the first sample by a SWAP operation, which has no preferred basis. Later interactions are also not of controlled-U form (Eq. 12).

One might ask where the “measurement” occurs, since the interaction between and is purely unitary. The essence of measurement is that an observer or apparatus gains information. Quantum measurements are usually construed as mysterious processes that consume quantum states and excrete specific, definite measurement outcomes. Quantum theories of measurements usually represent them as (i) unitary interaction, (ii) decoherence and superselection and finally (iii) wavefunction collapse or splitting of the universe. Our results suggest that unitary interaction (the only part of this sequence that is really understood) can stand alone as an information-gathering “measurement.” And by avoiding decoherence, we can gather information strictly more effectively.

Decoherence is ubiquitous in human experience. But in its absence, there is no compelling reason why gathering information must be accompanied by collapse or definite outcomes. The whole point of quantum information science is to produce devices that do not decohere and that can process information more efficiently than classical computers. The central message of this paper is that they can also gather information more efficiently. Unfettered by decoherence, they may still be constrained by locality. For such devices, coherent measurements are the natural way to gather information.