## Abstract

Using spontaneous parametric down-conversion, we produce polarization-entangled states of two photons and characterize them using two-photon tomography to measure the density matrix. A controllable decoherence is imposed on the states by passing the photons through thick, adjustable birefringent elements. When the system is subject to collective decoherence, one particular entangled state is seen to be decoherence-free, as predicted by theory. Such decoherence-free systems may have an important role for the future of quantum computation and information processing.

Quantum computation holds the promise of greatly enhanced speeds for solving certain problems, including factoring (1), simulation of quantum systems (2, 3), and database searching (4, 5). One main obstacle to quantum computation is the problem of decoherence—fragile quantum superpositions are destroyed by unwanted coupling to the environment. In particular, it is the entangling of the quantum system to unobserved degrees of freedom that leads to a loss of coherence. (A related problem is that of dissipation, whereby energy is lost from the system.) Three basic strategies to cope with decoherence in quantum computation have emerged. The first, quantum error correcting codes, relies on trying to detect errors using ancillary quantum bits (qubits) and actively manipulating the interactions to correct these errors (6, 7). The second strategy employs dynamical decoupling, in which rapid switching is used to average out the effects of a relatively slowly decohering environment (8). The final approach attempts to embed the logical qubits into a part of the overall Hilbert space that is inherently immune to noise, a “decoherence-free subspace” (DFS) (9–16) (typically, such a system is also “dissipation-free”). It has been shown that it should be possible to perform quantum computation operations without taking the system out of the DFS (12, 13,15) and that the DFS is robust with respect to perturbations in the interactions (12, 14). Here, we present an experimental demonstration of the existence of a DFS, using entangled photons as our qubits.

The usual condition for a DFS is that the qubits under consideration are subject to collective decoherence—the disturbances affecting them are identical, as each individual qubit couples to the environment “bath” in the same way. When the system-environment interactions possess this sort of permutation symmetry, the decoherence-free states also display symmetry (e.g., are maximally entangled). This is a particular example of the more general requirement that some symmetry in the system-environment interaction decouples the DFS from the environment. Although not always applicable, this model of decoherence is nevertheless relevant for some implementations of quantum information processing, for instance, if the qubits are physically very close to each other and the environment cannot distinguish them.

Although formally one needs density matrices to describe a state after interaction with an environment, it is illustrative to view the action of the environment as adding a random phase shift 〈φ〉 to each term in the state. For example, a decohering environment acting in the 0/1 basis will take |0〉 + |1〉 → e^{i〈φ0〉}|0〉 + e^{i〈φ1〉}|1〉 = e^{i〈φ0〉}[|0〉 + e^{i(〈φ1〉}
^{−〈φ0〉)}|1〉]. The global phase is unobservable, but if the phases 〈φ_{0}〉 and 〈φ_{1}〉 are truly uncorrelated, then the qubit will be left in a mixed state; that is, the off-diagonal elements of the density matrix will vanish because of averaging over the random phases.

Consider two qubits in the “singlet” state |ψ^{−}〉 ≡ (|01〉 − |10〉)/

The same argument predicts that the state |ψ^{+}〉 = (|01〉 + |10〉)/^{±}〉 span a two-dimensional DFS, which may then be used as a logical qubit. However, an environment that acts in the diagonal basis, given by |±〉 ≡ (|0〉 ± |1〉)/^{+}〉 is not decoherence-free, for |ψ^{+}〉 = (|++〉 − |−−〉)/

We can characterize the robustness of an arbitrary initial state ψ_{in} by calculating its overlap with the state after decoherence. This is the fidelity F = 〈ψ_{in}|ρ_{out}|ψ_{in}〉, where ρ_{out} is the final density matrix. If the input is not completely pure, as is typically the case for any experimentally produced state, we use the general definition F = [ Tr(_{out}
^{2}(17). For each of the four “Bell states” [ψ^{±} ≡ (01 ± 10)/^{±} ≡ (00 ± 11)/^{−} is always decoherence free (even in elliptical bases, because it always has the same antisymmetric form). In contrast, the state φ^{+} decoheres in any (linear) basis, where it has the same form, |θθ〉 + |θ^{˔}θ^{˔}〉. The minimum fidelity resulting from this sort of collective decoherence is F = 0.33, for the states φ^{−} and ψ^{+} at θ = 17.6° and 27.4°, respectively. This does not result simply because of a unitary transformation (e.g., a rotation) of the state; a plot of Tr(ρ_{out}
^{2}) yields similar results, confirming that the dips truly arise from decoherence.

In order to have more than one state that is decoherence-free in all bases, one needs to have at least four entangled qubits (11) [though more recently it has been shown that three suffice to make a “noiseless subsystem” (18)]. However, for our proof-of-principle experiment, we restrict ourselves to the simplest system supporting a DFS: two qubits. Our qubits are represented by the polarization states (“0” ≡ H = horizontal; “1” ≡ V = vertical) of two correlated photons. The photon pairs are produced via the process of spontaneous parametric down-conversion in two thin, adjacent, nonlinear optical crystals [beta-barium-borate (BBO)] cut for type I phase matching (19). Inside the crystals, an ultraviolet pump photon (at 351 nm, produced from an 80-mW argon ion laser) may spontaneously split into two correlated daughter photons, emitted into different spatial modes. Because of energy conservation, the sum of the frequencies of these photons must equal that of the (monochromatic) parent photon; thus, the photons' frequencies are entangled.

Because of the details of the conversion process, an incident pump photon polarized at 45° will have equal probability amplitudes to down-convert in the first crystal, producing two H-polarized photons, or in the second crystal, producing two V-polarized photons. The coherence and high spatial overlap between these two processes lead to a very pure [∼99% (19)] maximally entangled state (|HH〉 + |VV〉)/

The quantum mechanical state of the photons is characterized by tomographically measuring the density matrix or, more precisely, the two-photon contribution to the reduced density matrix corresponding to the polarization. This distinction will be important later. In essence, we determine the two-photon analogs of the usual Stokes parameters [see e.g., (20)] characterizing the polarization state of a single photon. We measure polarization correlations between the two photons for 16 analysis settings (e.g., HH, HV, 45°V, etc.), allowing reconstruction of the density matrix (21).

Experimentally, adjustable quarter- and half-wave plates and polarizing beam splitters in the two down-conversion beams allow polarization analysis in any basis (Fig. 2). The photons are detected by using silicon avalanche photodiodes operated in the geiger mode. Each detector is preceded by a small iris to define the spatial mode, a narrowband interference filter [centered at 702 nm, with a full width at half maximum of 5 nm (10 nm) in path 1 (path 2)] to reduce background and define the bandwidth of the photons, and a collection lens. The detector outputs are recorded in coincidence using a time-to-amplitude converter and a single-channel analyzer, leading to an effective coincidence window of ∼5 ns; the resulting rate of accidental coincidences was less than 0.3 s^{−1}, compared to the typical rate of true coincidences, 30 s^{−1}.

The decoherence in our system is introduced as follows: In each arm, the photon passes through a ∼10-mm-thick piece of quartz, with optic axis in the plane of the element. With this thickness, the ordinary and extraordinary polarization components are separated by 140λ_{o}, where λ_{o} is the central wavelength of the photons (the plates were tilt adjusted so that the relative phase modulo 360° was zero). The coherence length of the photons (determined by the frequency filters and irises) is also ∼140λ_{o}. Consequently, after passing through the quartz, the ordinary and extraordinary polarization components acquire a random relative phase. When we detect the photons, we essentially trace over the frequency degree of freedom, which plays the role of the “environment,” and the resulting (reduced) density matrix for the polarization becomes mixed.

Actually, all decoherence is of precisely this sort: coupling to unobserved (and often unobservable) degrees of freedom. Although the total state, including both the quantum system of interest and the “environment” degrees of freedom, is in a pure state, the reduced density matrix of the quantum system alone (obtained by tracing over the environment) can be in a (partially) mixed state. Our quartz elements entangle the relative phase between the two production processes (in down-conversion crystal 1 or 2) with the photon's frequency. In the end, we trace over this degree of freedom; that is, we do not measure it, because our detectors are insensitive to wavelength (over the collection bandwidth). [The curves in Fig. 1 can be calculated by coupling the qubits' states to an imaginary measuring device and tracing over its final state or, alternatively, by using the total (polarization + frequency) state of the down-conversion photons, transformed by the quartz, and tracing over frequency (22). The resulting identical density matrices validate our technique for introducing decoherence.]

By setting the angles of the quartz elements appropriately, we can introduce “collective” decoherence in any desired (linear) basis. There is one subtlety: It is actually necessary to orient the pieces at 90° to each other, so that the decoherence effects are the same. By reversing the roles of the fast and slow axes in the quartz, one compensates for the frequency anticorrelations intrinsic to the down-conversion pairs (22). Otherwise, φ^{+}appears to be the DFS.

Figure 3 shows the measured density matrices for the Bell states. On the left are the states without decoherence, and on the right are the states after being decohered by quartz pieces at 17° (and 107°). The singlet state ψ^{−} is nearly perfectly preserved. This is made more quantitative by the measured fidelities, listed in Table 1. In all cases, there is excellent agreement between experiment and theory.

Our results demonstrate that we can experimentally and quantitatively study various aspects regarding error-free subspaces (of small quantum systems), with measurement accuracies at the percent level. For example, the effect of noncollective decoherence can be studied by changing the relative thickness and/or orientation of our “environments.” The rate of decoherence can be very simply altered by changing the degree to which we trace out the environmental degrees of freedom, i.e., by adjusting the frequency bandwidth of the collection filters. Moreover, we expect that we can readily examine the case of larger DFSs, either by looking at three- and four-photon events or, more easily, by using more degrees of freedom of the photon pairs (23, 24). For example, one might employ the entangled spatial modes to represent additional qubits while relying on frequency techniques to produce decoherence. Finally, we can extend our investigations to include dissipation by introducing controllable polarization-dependent losses.